%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: tensor.dvi %%Pages: 32 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: XYATIP10 XYBTIP10 XYDASH10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips tensor.dvi -o tensor.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.05.02:1439 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro %! 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TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{ CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end} repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N /@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X /yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet %%BeginFont: XYDASH10 %!PS-AdobeFont-1.1: XYDASH10 001.104 %%CreationDate: 1997 Jul 20 21:19:18 %%RevisionDate: 1997 Aug 28 05:34:12 %%RevisionDate: 1997 Sep 18 10:23:31 % % XYDASH10: line segments for Xy-pic at 10 point % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def /FullName (XYDASH10) readonly def /FamilyName (XYDASH) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch 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1ac6cf81a8eb952a043b1340a7fc883b28638897d15bb4c394d70df7df3d1312 2629b5a8c236d9e91fe466b264d6e5018581bb79e23dd527875cb97ef7962d44 2fb47682afcf9a3869ef323af9fec2d18e8a3613c10d546970216927e6740be8 0e272580db4cdc1b8fece17f94fc78640294a0 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: XYBTIP10 %!PS-AdobeFont-1.1: XYBTIP10 001.104 %%CreationDate: 1997 Jul 20 21:19:18 %%RevisionDate: 1997 Sep 14 19:58:47 % % XYBTIP10: lower arrow tips for Xy-pic at 10 point "technical style". % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def /FullName (XYBTIP10) readonly def /FamilyName (XYBTIP) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -276 def /UnderlineThickness 138 def end readonly def /FontName /XYBTIP10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /d14 put dup 15 /d15 put dup 16 /d16 put dup 33 /d33 put dup 34 /d34 put dup 36 /d36 put dup 38 /d38 put dup 39 /d39 put dup 40 /d40 put dup 41 /d41 put dup 42 /d42 put dup 46 /d46 put dup 47 /d47 put dup 52 /d52 put dup 55 /d55 put dup 58 /d58 put dup 61 /d61 put dup 79 /d79 put dup 97 /d97 put dup 110 /d110 put dup 111 /d111 put dup 116 /d116 put dup 118 /d118 put dup 119 /d119 put dup 120 /d120 put dup 125 /d125 put dup 126 /d126 put readonly def /FontBBox{-542 -542 542 542}readonly def /UniqueXX 5092839 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d78041987a409a2d06b6b3057738213cee08cd789eeaa 097caceb2738a78b2f437638f0d63dd9e45ce613ae94486e726c4ed202501d61 51965c5c865a24933f21e0b1c67ff0d74bea0b8003496a2b1c9e3cde218dfa02 7343f1561243c5419412a440b6d4682c4dd92bf310718d73d28f47559a653346 c8fa6a8e3ec0a68d6661b293a71328a0bd0521249f1263070e67d0c20ca4a48d 221bcd864852e33289496155416b7cc05e73dd2b7f9ba0977ab328be862ac7e0 139c8eef1237e57525cbc853d7cbe3c9a8b54c378e8af02257a8daa736c3d9ae fb18fd198a33681c334984d81e2d783d32adf54549f5bea0bf351b1016032908 81685bde8d44703654d97063c8ebfb896e029b2383f5754d467163ec07f3398a d88196c720fd98b9a2260de8d7d3aa6453f831ce18233cfbf6cb098bc3ca2cd1 495386a279ced386537228ec08f3b3e400cc040ab2e763b0cd93c9a2c5ee0436 f0a2f033ba5d3e4231aacc9b0aa820f7ad72a3cec593a1153ee5527693ad3bc8 eaef55ac2f52fdf27146c04dcc825181a275e632e75a94cb9b3d3f7d17c1c08b 83bbf5c681f864e234d10b0f7c64839aa1671931f39a001e4134030b91d9a473 6c7d5e101e04feb20a04907ab46ab24902c1844b018beefd9014c8b629674e57 f1f0d63ad79dfa8ce4d1fffabeb4315386d494a3ab66cc9f291a714ef0ee4f9f 1687f0ecbcd2acea0e98dd5f94dfd700e546599e58d1f25bc54ef6ec0f12b91e 6690287b7c527a51724cea71da655f2b2974633ba5484cc6c2300ba28dff89e2 0c37542986ec1e4613cc8a16521e5c2720d88fa18111a1083dcee82b011400a2 8b4124ab1a5bdde460e2589f2872b0436396df646b0eb176e75de9af54d7c4ab f628a596d7e1ead5815ec6bb58786913f0125dbe4a6328ea358185ce03fdf5b2 8d3e5cee90066a67f548590d69d197b1503ae8f993a1a7aa4248f0be3be623c0 fe29e1772ade5b00f22b228152d197d6b3e1ddf4dece5c7cbadfec3e7bb38696 91becf5079caf4c910b4a25a1b19e3c64eeb79e5223d56f7aae7411259fda2b2 2e2a4652323e63b97e76d22ca5fa800398fb0b4636201037f11794493ce10d4a 80fc85a0a26686c096d560a3a77cb2cf4fdcd105e98e7e88454412c6a107c5dc 7605eef7932c093879753c20b5397cdeba78d0a798f789304d63956f0f471a10 ad93e8ade240e9185f8028420ecc26e436fcff5421bcd39aeb8ac91800241d34 133cc54e557b68947337eb889fb494948d932519e0e19f3dd891220c04935f13 96f8196a907180e760be47484d144c1039ee7934d0f08397e9b640062ffd670c 1c7fc029acbe5f5c7982e68820e9140313eb390f785901879c614f5461f3e0e6 908f053b64f7f0ec48d567b4125934d21158a2ba50e361b6966e857922618b72 06d412a748706d7eac5fff41ee5c52f4142fce1ce9ba67a3c97ec186b187b80c 41b8ee73ba38e3bf2bc6741de19ef652832faba07c55e7bd7ae5fb49d86b1808 8bb5e80ed536552a211ce36d2961583aff648edd54f1de84ae741b98bbd57e37 577e55ca0f97c4ac1fed15732a0a072a176b0c5f14dbcee98f6d60fe3b3df180 2d93213c7eb7a37ff92ff4f97619596aa0771d0cbc82c5cf55b5be4f44fba75e 631ae71cbc9afbc42312cecdd86b7aec3152ddcf7fa1c6f5e051581597864746 2e975089ad928df03a4073bf06bfe1e262a551f24a12bca29408aacf7ed2e43a 3b7f7ff8e40af2ad2b131622058405cdf249568c2653b0be457fed5352e113fc 1107d6ed67a6ffeef5608109c13e5d5bc2432e65b82d3c3fe6d012e7a84e7730 13ae1201e1ad5ac9c2c11163be1d85f342956954f92651fcde7efbd58975fd01 37afbc60f68065b6b4af9856ef3a425a57a25b3ce8e5375aeccfc1146b78fb7e 6aa315ea438e2826d160cb12ae97fbd9c4704114f4269ae8ed6a114882c3fe34 9aa90531189c9e3b9fd1bb8f1772fdd681245ccfcb8b42165e018ad8af8f2440 954f0390129495317eab98a4d43da3e8b81c4269b19e4069c61c0ecb28a57873 610c578517a8e08f9762dce236f12bdb99718602cba86a56166ddcd42497ebf9 5b52ebedd9f09a6109ae30158fbe7d998aff28df96cb38f706e1b05b685c7df4 8410ed3d5558666c7c85dbae18e6e50330f7fd850d55e2c7a2277fa9fc2372b5 5e8102cd124121243810279c8d7773e43cc5233bdab4d5379ae5bf2f2ef82d78 9621ea5c345dba373f989f82a3f1d6edeb69dc7c9abb0de83996d732589a642c b63d80a8dd4420f3593380d56dabe56ffbf30f6d1bc7b8e3d8da03de5eb33aca 4b1dd4b2ae4b385c05b16d65d336149a68d790bfe240b23f6d38c19945037652 4b54fc9e0a60df4b23b9054d0c05b7081ed97195ef3fa26beecf5ea3fe6cae8b ebea88ed339f7e59494275fa923c8e6d1f6d34d82dd65764941e4578de48351c b2bdd01bcb8f95fbd8adc59d14b5212aa0f4243d1284c6c9325eb8331f11a031 c3c72668b913f9dc2b818ffa189229145614d11cc645beb56e006f2eddb395be f40407977ed4c2018231bdb1b3975e3471c917838f156ce50b6b7881cd5a924e f908a4246961874932ef7694a89836e8b2bdccf9d5e37465d72a051b28eacaa9 0c0499a2caf6f048e16a2d538f93db84d5aa1f5ca2bf5f59e688483eb6f7b76b 86fa39f5bb91ac2cb449490d02cacfae6eeda6cc9f81dd1f23f75dfaa147ed8a 8409b1df8fc6ab93b5e0faa71e6ccd142f1d922463b33e72adb48edb72841e71 6e736726f7502bb9acc792946494e722e21bf0fd127e18a5a841c4eb8e5da923 4b33a1ca992a972d9e9be9ab9ac03490a1ac894a14d0d968505d2e85e031c7ba 214a73d474109de5438f3d6c6fcf7d7c3328c23bb891d9a032b826499fef9f6a 6d2f4b931dac8a9e52f0cdc71237f90c0cc8d3ea2b77300e8eaf805a1c96a531 29116f4d980e6b5d2255be1b414827ec5f1c704957686a2a15eb97163c52aa2e 7ff983316875f3254b75360a5eb964fbbe2512639537c9d747ec797438cc6a1f c29c4ece2df509aca9f4432f2802404525ec3ccdd00c16d5e4b1d988e05a32d0 b3109c438bb4d96461ec7a01f31397d05613041ce4e7a058c76db266c85186ea c120fc8ded67831b7b6eaedc2697dbe21aa0be8c23376566c5c2368571454153 c6376da88f348eba5f988877fe62b7173a34031cf1a438f1d0681e252ba36789 ff08228ba054542b40c4d51023bbcc575acd9b5e48212cbdd008e48b5b455ae4 5e8f7abaac5079262c2f68dab05f1a2bfaf0c701cd9f6212786d56080b547abe de0a0c45b69b6735611e712287a2a2dd7c48bd41b61d5a521465d0056ad19950 f3b69224cceb5e44f4434014634678fad7c159c53fe7f1641358a043c397a25d dfcaaa727be1061288fd97407d8ca2e249e551e25cd28a4013136d80e56a9f03 7b1cf2a97c9d3e555c87c7442f038cda997d27c059816266af66e5af51445134 21ba39ffb81e96b689e4b1e8e6a31a64146f292793e8eef0390cb8cb8ef6ee21 95ffc8dc3707c1b5edd3e55958bafd54976526eaa15c355b4ff695b5dea821e4 78100344d5aa80c8968e88f8084a6c3d4f8498773f922ca08b95e293f39a66c0 7cf7a6471e41f1d213e63b2fc4ed95aaaff8523974de592f8781313988ee1fa1 da857e896e84708b791ce58ef30be9ab9d02ced5d8d6a72833dffeb44db429e3 a9c41e2c05da263b966233f2f52870aff1f5814a7460361ebc8146d9addcde2a 44347bd9eaeffcc9761fdc17a89ea5e8f445f95006e017e9276e079f74c8df0e c40a1c57cd36402eca43f744fa2586a1159bebcb5cdfb5cc7d85a50dcfa3a62d 3ee6c5bfb1867e002e8aa07a2acf3e66b535885f1e226169f48a0128d65481cf dcb8c0c147570b40a573595e69a63d1b7ea70ea2512bf2a3ec433d10ccbd63fb 692e642cb7397d926de8e29fa5320505e0023dc37be3ed06cc5d08b0efd62382 92c8a6fafb49304e2bb94b5a9603c34c7f30fe91bc74b9ba972013ac477ccfac 93541acea4d89fd121931f93bedede6bca4bb76959c2bae0655474de2364b0f2 3f0837b0dfb1ffd40fef2cad7cbd9f4e483227ec15a96a23badbf56d9c46a0b6 71771274545d4e81f830000c86c88934f4eda292bd7f13d93f158f446364831a 887ec603d4bb1308fed4f6effe728ed2460280640addf145df15c70d87970054 f86abf37ce50926a7d7f3a8dafb63ed484128a581d72a90c9ead1890f6551444 8fb113f29934bd2da3fc2a33a83bbcb5ea207c71cad63b05f82d1a55fc81280d 548451d254f5df3c97cd3bf42181bb306f365f419cf90ce7778ff5ffea51ea09 4fab9198f70afaac4dd6a703065f723b7d6e53db961c9699bfe7d79280e2f62f 4a05d7a1c0bbe55ae3be7e851f8bdc5206de3f6f1ea55f26248ca6fbdb061417 eb7689dd9d59d4a6d2e42d8a40b606ff7d4ff5fe0b6f6bf498528e0693b398f6 c74708bcec71b15d2cea5140321727a091677ac067f787001ea089276558c727 51a1e8644e499199d2f896154a8275d8b9b01aa32dec10a0f0196e4474e4b188 ab9a89e29a8edf5c449d7c25ea1091646d2c246599e5dcc4f0167f7ee1081ad8 c267205b3b256b96a3b54281236eb39a700fa43acf8517abc72a845bc110e1ec 153fc374710975248e3401bc8d1e66458fa2198d8885374f1feb57e7f77bf4ab 5edec4d089a223380cdd01d9ce39fdaf30ea1e3512db78a713a40159e3f7f578 fc119c61b32b73c6de5dbdf2899d6e9ccfd5c563887bc57122bc0dbac2091844 6b1ab98b5786c8f18477cb06b0df72dddc8c343deea9161c526b7210c5397fe9 faf8d92b0f886b1cda38491d477c1c8d082a8b542e505c6a8bc5bed72b14f1c3 9f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: XYATIP10 %!PS-AdobeFont-1.1: XYATIP10 001.104 %%CreationDate: 1997 Jul 20 21:19:17 %%RevisionDate: 1997 Sep 14 19:58:47 % % XYATIP10: upper arrow tips for Xy-pic at 10 point "technical style". % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def /FullName (XYATIP10) readonly def /FamilyName (XYATIP) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -276 def /UnderlineThickness 138 def end readonly def /FontName /XYATIP10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /d14 put dup 15 /d15 put dup 16 /d16 put dup 33 /d33 put dup 34 /d34 put dup 36 /d36 put dup 38 /d38 put dup 39 /d39 put dup 40 /d40 put dup 41 /d41 put dup 42 /d42 put dup 47 /d47 put dup 52 /d52 put dup 55 /d55 put dup 58 /d58 put dup 61 /d61 put dup 79 /d79 put dup 97 /d97 put dup 102 /d102 put dup 111 /d111 put dup 116 /d116 put dup 118 /d118 put dup 119 /d119 put dup 120 /d120 put dup 125 /d125 put dup 126 /d126 put readonly def /FontBBox{-542 -542 542 542}readonly def /UniqueXX 5092838 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d78041987a409a2d06b6b3057738213cee08cd789eeaa 097caceb2738a78b2f437638f0d63dd9e45ce613ae94486e726c4ed202501d61 51965c5c865a24933f21e0b1c67ff0d74bea0b8003496a2b1c9e3cde218dfa02 7343f1561243c5419412a440b6d4682c4dd92bf310718d73d28f47559a653346 c8fa6a8e3ec0a68d6661b293a71328a0bd0521249f1263070e67d0c20ca4a48d 221bcd864852e33289496155416b7cc05e73dd2b7f9ba0977ab328be862ac7e0 139d5cb0db6f50b26bd8ab173859c9c94db82970311d7eb0a02bee1be5f0d126 f9079e67107eda14b460e46b03b0422eb45a4f4afa382841c35b0bc0c8639b73 43c819afc69838ce781c2de7d22bf503ef6eec27c83cfd52a77bbc754b4f2050 55341700991f3ef4b5b5a54c21283034b38c8b6a6e65abccc94c0c26836806ec 4df8d8c64f595841395072f8a2d289b7fe5497bc0e061810b16a1e48653bb092 42ffa3ef0bba4e37a5aa4ddc31f3138aaf10998fd66f3817b77012eac677ed2d 7447908c771fbaba4dfcfeca5374a6b87e5657809a82f0ae068c5384c12f4653 2a82645a512212140d815e80ec76b3370c382e9f6d29ec6afc178a622249fda7 775d4f6a554f04748ed4210257fa6e376188a175db3c00f88421820b063a985c 07ad665eff7e2d32a27015c528227c2805aa8df134f4abad9958b841aa4263ec 9ec6d907308b0a5a51049de002cfe60ef35bf33fc863ba14ff361749554abd65 47426fbf3958ddb506ea3e303c932edec2896d3017a57913cde60cbdafeddbd3 cd5ef6dd49299783a92fe9deafb24e7f74a6ca6c0198b2fcda46b51445a2800f d1dc6a092b3192cfb314892e52753b5c7b94edd34c8213b032f8aab5d08753cb 65bd1a1225ba43194efa625ee5dc6eec6d0353d06ef3bef9c0d7df78fe189482 779c9276e83ccd71b50e87ba92cf092d65498e5b43cdc89436019e306c0d628e c7d1470ab322ca2d6adc3e9ff4196c0f7792ca0b20f741ad7bbd4391b0218511 14f0e97a44fd7d03e7003d1fe2c70d6266740dad7b07b3794dc53871887c8eba 2910d6fa654346318753cc752a7d25c1ef970e7dda8d8bbe249e9edf2c8c2ae6 abb93bab2dd8560466a7d08ce3e200d7cc488c6045871a8033ec1dc3a53e7056 3fe265f58c6ac98754d399b47b817b7aed7ebbbb503268e7324d6c0a0931978b 51a7d37187a99f234ae4c46b585383938e7cf59d2dfd20717031966eaa4317e7 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(connections)f(with)h(regular)f(singularities)f(on)i(the)456 1571 y(mo)r(duli)40 b(spaces,)i(instead)e(of)g(the)g(top)r(ological)e (surfaces)h(and)h(mapping)g(class)f(groups)f(in)456 1671 y(Chapter)20 b(5.)34 b(In)21 b(fact,)h(this)f(is)f(ho)n(w)g(the)h(mo)r (dular)f(functor)g(originally)f(app)r(eared)h(in)h(conformal)456 1770 y(\014eld)34 b(theory)-7 b(,)34 b(see,)h(e.g,)g([)p FK(MS1)p FQ(],)g([)p FK(S)q FQ(].)55 b(The)34 b(exp)r(osition)f(in)i (this)f(c)n(hapter)f(is)g(based)h(on)f(the)456 1870 y(unpublished)38 b(man)n(uscript)f([)p FK(BFM)p FQ(];)43 b(similar)37 b(ideas)g(w)n(ere)g(also)g(in)n(tro)r(duced)h(in)g(Deligne's)456 1969 y(letter)27 b(to)h(Drinfeld.)605 2069 y(The)f(complex)f(v)n (ersion)f(of)h(mo)r(dular)g(functor)h(is)f(b)r(est)h(form)n(ulated)f (using)g(the)h(language)456 2169 y(of)e(connections)g(with)h(regular)d (singularities)i(on)g(the)g(Deligne{Mumford)g(compacti\014cation)456 2268 y(of)31 b(the)h(mo)r(duli)g(space)f(of)g(complex)h(curv)n(es.)47 b(F)-7 b(or)31 b(readers')f(con)n(v)n(enience,)h(w)n(e)g(giv)n(e)g(a)g (short)456 2368 y(in)n(tro)r(duction)26 b(to)g(the)g(theory)g(of)g(mo)r (duli)h(spaces)e(and)h(connections)g(with)h(regular)d(singular-)456 2468 y(ities.)605 2567 y(In)34 b(this)f(c)n(hapter,)h(\\complex)f(curv) n(e")f(means)g(\\complex)h(pro)5 b(jectiv)n(e)32 b(curv)n(e")g(\(th)n (us,)j(it)456 2667 y(is)29 b(compact\);)i(unless)e(stated)g(otherwise,) h(the)g(curv)n(es)e(are)h(assumed)g(to)g(b)r(e)h(connected)g(and)456 2766 y(non-singular.)54 b(W)-7 b(e)34 b(remind)g(that)h(b)n(y)e (Riemann's)h(theorem,)i(ev)n(ery)c(\(non-singular\))h(com-)456 2866 y(pact)h(Riemann)h(surface)f(is)h(pro)5 b(jectiv)n(e.)57 b(Ho)n(w)n(ev)n(er,)35 b(unless)f(otherwise)g(sp)r(eci\014ed,)j(w)n(e)d (will)456 2966 y(consider)h(all)g(manifolds)h(with)g(analytic)g(top)r (ology)-7 b(.)60 b(W)-7 b(e)37 b(assume)e(that)h(the)h(reader)d(is)i (fa-)456 3065 y(miliar)26 b(with)h(some)e(basic)h(notions)g(of)h (algebraic)d(geometry)-7 b(,)26 b(suc)n(h)g(as)g(coheren)n(t)f FL(O)r FQ(-mo)r(dules,)456 3165 y(v)n(ector)30 b(bundles,)j(etc.;)i (all)c(the)i(necessary)d(prerequisites)g(can)i(b)r(e)g(found,)h(for)f (example,)g(in)456 3265 y([)p FK(GH)o FQ(].)883 3449 y FK(6.1.)46 b(Mo)s(duli)30 b(spaces)i(and)g(complex)e(T)-8 b(eic)m(hm)s(\177)-50 b(uller)30 b(to)m(w)m(er)605 3599 y FQ(In)k(this)g(section,)h(w)n(e)e(giv)n(e)g(a)g(de\014nition)i(of)e (the)h(T)-7 b(eic)n(hm)r(\177)-44 b(uller)34 b(to)n(w)n(er)e(of)i (group)r(oids)e(in)456 3698 y(terms)d(of)h(mo)r(duli)g(spaces)f(of)h (complex)f(curv)n(es.)43 b(Let)30 b(us)g(\014rst)f(recall)g(the)i (relation)e(b)r(et)n(w)n(een)456 3798 y(the)f(mo)r(duli)g(space)e(and)i (the)g(mapping)f(class)g(group.)605 3898 y(Let)f(\006)g(b)r(e)h(a)e (compact)h(orien)n(ted)f(top)r(ological)f(surface)h(without)i(b)r (oundary)-7 b(.)35 b(A)27 b FO(c)l(omplex)456 3997 y(structur)l(e)g FQ(on)22 b(\006)h(is)g(an)f(isomorphism)g(class)g(of)g(pairs)g(\()p FJ(C)q(;)14 b(')p FQ(\))24 b(where)f FJ(C)29 b FQ(is)22 b(a)h(smo)r(oth)f(compact)456 4104 y(complex)j(curv)n(e)g(and)h FJ(')9 b FQ(:)28 b FJ(C)1390 4057 y FE(\030)1362 4104 y FL(\000)-40 b(!)23 b FQ(\006)k(is)f(a)f(homeomorphism)g(preserving)f (orien)n(tation.)35 b(Equiv-)456 4204 y(alen)n(tly)-7 b(,)24 b(a)f(complex)g(structure)g(on)g(a)g(smo)r(oth)h(surface)e(can)h (b)r(e)h(de\014ned)g(as)f(a)g(p)r(olarization)g(of)456 4303 y(the)29 b(complexi\014ed)g(tangen)n(t)g(space,)f(i.e.,)i(a)f (one-dimensional)f(complex)g(v)n(ector)g(sub-bundle)456 4408 y FJ(T)517 4378 y Fz(C)562 4408 y FQ(\006)23 b FL(\032)f FQ(\()p FJ(T)825 4378 y Fz(R)871 4408 y FQ(\006\))d FL(\012)1047 4420 y Fz(R)1111 4408 y FH(C)48 b FQ(suc)n(h)28 b(that)g(\()p FJ(T)1659 4378 y Fz(R)1704 4408 y FQ(\006\))19 b FL(\012)1880 4420 y Fz(R)1944 4408 y FH(C)44 b FQ(=)22 b FJ(T)2175 4378 y Fz(C)2239 4408 y FL(\010)p 2322 4334 107 4 v 18 w FJ(T)2383 4384 y Fz(C)2428 4408 y FQ(.)605 4508 y(W)-7 b(e)38 b(iden)n(tify)g(t)n(w)n(o)f(complex)g(structures)g(on)g(\006)h 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FQ(is)d(an)h(automorphism)e(of)h (\006)456 525 y(whic)n(h)22 b(is)g(homotopic)f(to)h(iden)n(tit)n(y)-7 b(.)35 b(The)22 b(set)g(of)h(all)e(complex)h(structures)f(on)h(\006)g (up)h(to)f(isotop)n(y)456 624 y(is)d(called)g(the)h FO(T)-6 b(eichm)q(\177)-43 b(ul)t(ler)23 b(sp)l(ac)l(e)e FQ(and)e(will)g(b)r(e) h(denoted)g(b)n(y)f FJ(T)12 b FQ(\(\006\).)33 b(F)-7 b(or)19 b(a)g(connected)g(surface)456 724 y(of)27 b(gen)n(us)g FJ(g)s FQ(,)g(w)n(e)g(will)h(also)f(use)g(the)h(notation)f FJ(T)1985 736 y FI(g)2023 724 y FQ(.)605 824 y(Denote)g(b)n(y)g FL(M)1104 836 y FI(g)1170 824 y FQ(the)g(set)h(of)f(isomorphism)f (classes)g(of)h(complex)f(curv)n(es)g(of)h(gen)n(us)g FJ(g)s FQ(.)36 b(It)456 923 y(is)29 b(w)n(ell)h(kno)n(wn)g(that)g(this) g(set)g(has)f(a)h(natural)f(structure)h(of)g(an)f(analytic)h(v)-5 b(ariet)n(y)e(.)43 b(W)-7 b(e)30 b(will)456 1023 y(call)i FL(M)713 1035 y FI(g)785 1023 y FQ(the)h FO(mo)l(duli)j(sp)l(ac)l(e)d FQ(of)g(curv)n(es)f(of)h(gen)n(us)f FJ(g)k FQ(\(to)d(b)r(e)h(more)e (precise,)i(it)f(is)g(a)g FO(c)l(o)l(arse)456 1123 y FQ(mo)r(duli)d(space)g(in)h(the)f(terminology)f(of)i(Mumford|see)e (Theorem)h(6.1.8\).)44 b(The)31 b(follo)n(wing)456 1222 y(result)c(immediately)h(follo)n(ws)e(from)i(the)g(de\014nitions.)605 1367 y FP(Pr)n(oposition)j FQ(6.1.1)p FP(.)40 b FO(The)34 b(mo)l(duli)f(sp)l(ac)l(e)h FL(M)2143 1379 y FI(g)2214 1367 y FO(is)f(isomorphic)j(to)d FJ(T)2887 1379 y FI(g)2925 1367 y FJ(=)p FQ(\000)3019 1379 y FI(g)3057 1367 y FO(,)h(wher)l(e)g FQ(\000)3406 1379 y FI(g)456 1467 y FO(is)c(the)g(mapping)h(class)f(gr) l(oup)g(of)h(a)f(surfac)l(e)g(of)h(genus)e FJ(g)s FO(.)605 1612 y FQ(The)f(next)f(result)h(is)f(classical,)g(see,)g(e.g.,)g([)p FK(Ab)p FQ(].)605 1757 y FP(Theorem)32 b FQ(6.1.2)e(\(T)-7 b(eic)n(hm)r(\177)-44 b(uller\))p FP(.)41 b FO(The)26 b(set)f FJ(T)2108 1769 y FI(g)2171 1757 y FO(of)h(al)t(l)g(c)l(omplex)g (structur)l(es)k FQ(\()p FO(up)25 b(to)g(iso-)456 1856 y(topy)7 b FQ(\))29 b FO(on)e(a)h(c)l(onne)l(cte)l(d)g(surfac)l(e)g 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y(zero)d(cotangen)n(t)h(v)n(ector:)35 b(it)28 b(can)f(b)r(e)h (de\014ned)g(b)n(y)g FL(h)p FJ(v)2115 3465 y FI(i)2143 3453 y FJ(;)14 b(v)2223 3423 y FE(\003)2220 3475 y FI(i)2261 3453 y FL(i)24 b FQ(=)e(1)p FJ(;)14 b(v)2523 3465 y FI(i)2574 3453 y FL(2)23 b FJ(T)2701 3465 y FI(y)2735 3473 y FG(i)2765 3453 y FJ(C)q(;)14 b(v)2905 3423 y FE(\003)2902 3475 y FI(i)2967 3453 y FL(2)23 b FJ(T)3106 3423 y FE(\003)3094 3474 y FI(y)3128 3482 y FG(i)3158 3453 y FJ(C)6 b FQ(.)605 3553 y(One)27 b(de\014nes)h(isomorphism)e(of)i(p)r(oin)n(ted)g(curv)n (es)e(in)i(an)f(ob)n(vious)g(w)n(a)n(y)-7 b(.)35 b(Let)28 b(us)g(denote)945 3670 y FL(M)1045 3682 y FI(g)r(;n)1167 3670 y FQ(=)o(the)g(set)g(of)f(isomorphism)g(classes)f(of)i(p)r(oin)n (ted)g(curv)n(es)1314 3794 y(of)g(gen)n(us)f FJ(g)j FQ(with)e FJ(n)g FQ(mark)n(ed)e(p)r(oin)n(ts.)456 3741 y(\(6.1.1\))456 3929 y(As)h(b)r(efore,)h(w)n(e)f(will)h(call)f FL(M)1382 3941 y FI(g)r(;n)1509 3929 y FQ(the)h FO(mo)l(duli)i(sp)l(ac)l(e)h(of)f (p)l(ointe)l(d)h(curves)p FQ(.)605 4074 y FP(Remark)h FQ(6.1.5)p FP(.)39 b FQ(This)19 b(mo)r(duli)g(space)f(is)h(di\013eren)n (t)g(from)f(the)h(mo)r(duli)h(space)d(considered)456 4174 y(in)40 b([)p FK(Kn)p FQ(].)76 b(The)41 b(latter)f(space,)j(whic)n (h)d(w)n(e)g(will)h(denote)f FL(M)2457 4144 y FI(n)2457 4194 y(g)2502 4174 y FQ(,)k(is)c(de\014ned)h(as)f(the)h(set)g(of)456 4274 y(isomorphism)20 b(classes)h(of)h(curv)n(es)e(of)i(gen)n(us)f FJ(g)k FQ(with)d FJ(n)g FQ(mark)n(ed)e(p)r(oin)n(ts,)j(but)g(without)f (tangen)n(t)456 4373 y(v)n(ectors.)38 b(Ho)n(w)n(ev)n(er,)27 b(they)i(are)f(closely)f(related:)39 b(for)28 b(example,)h(if)g FJ(g)s(;)14 b(n)28 b FQ(are)f(suc)n(h)i(that)g(there)456 4473 y(are)35 b(no)h(non-trivial)g(automorphisms)f(of)i(a)f FJ(n)p FQ(-p)r(oin)n(ted)g(gen)n(us)g FJ(g)j FQ(curv)n(e,)f(then)f FL(M)3175 4485 y FI(g)r(;n)3310 4473 y FQ(is)g(a)456 4572 y(\()p FH(C)542 4542 y FE(\002)604 4572 y FQ(\))636 4542 y FI(n)715 4572 y FQ(bundle)d(o)n(v)n(er)e FL(M)1277 4542 y FI(n)1277 4593 y(g)1322 4572 y FQ(,)j(so)e(all)h(the)g(results)f (of)h([)p FK(Kn)p FQ(])f(can)h(b)r(e)g(easily)f(reform)n(ulated)f(for) 456 4672 y FL(M)556 4684 y FI(g)r(;n)655 4672 y FQ(.)51 b(One)33 b(can)f(also)f(de\014ne)i(more)f(general)f(mo)r(duli)i(spaces) e FL(M)2630 4642 y FI(n)2630 4693 y(g)r(;r)2753 4672 y FQ(in)i(an)f(ob)n(vious)g(w)n(a)n(y;)456 4772 y(they)27 b(will)h(not)g(b)r(e)g(used)f(in)h(our)f(w)n(ork.)605 4917 y(Let)e(us)g(de\014ne)g(the)g FO(T)-6 b(eichm)q(\177)-43 b(ul)t(ler)29 b(sp)l(ac)l(e)j FJ(T)1956 4929 y FI(g)r(;n)2080 4917 y FQ(to)25 b(b)r(e)g(the)g(set)g(of)g(all)g(complex)f(structures) 456 5016 y(on)c(a)h(surface)g(\006)g(of)g(gen)n(us)f FJ(g)k FQ(with)e FJ(n)f FQ(mark)n(ed)f(p)r(oin)n(ts)h(and)g(tangen)n(t) g(v)n(ectors)e(up)j(to)f(an)g(isotop)n(y)456 5116 y(whic)n(h)31 b(\014xes)g(the)h(mark)n(ed)e(p)r(oin)n(ts)i(and)f(v)n(ectors.)47 b(This)31 b(space)g(has)g(a)g(natural)g(structure)g(of)456 5216 y(an)c(analytic)g(manifold.)37 b(Then)28 b(the)g(previous)e (results)h(can)g(b)r(e)h(generalized)e(as)h(follo)n(ws:)p eop %%Page: 137 3 137 140 bop 940 238 a FM(6.1.)29 b(MODULI)g(SP)-5 b(A)n(CES)28 b(AND)h(COMPLEX)f(TEICHM)2481 223 y(\177)2473 238 y(ULLER)h(TO)n(WER) 363 b(137)605 425 y FP(Theorem)32 b FQ(6.1.6)p FP(.)40 b FQ(\(i\))30 b FO(The)h(T)-6 b(eichm)q(\177)-43 b(ul)t(ler)31 b(sp)l(ac)l(e)f FJ(T)2260 437 y FI(g)r(;n)2389 425 y FO(is)g(c)l(ontr)l(actible.)605 525 y FQ(\(ii\))38 b FO(L)l(et)f FQ(\000)956 495 y FE(0)956 545 y FI(g)r(;n)1092 525 y FL(\032)g FQ(\000)1246 537 y FI(g)r(;n)1382 525 y FO(b)l(e)h(the)f(gr)l(oup)h(of)g(automorphisms)h(of)f(an)g(extende)l (d)f(top)l(olo)l(gi-)456 624 y(c)l(al)e(surfac)l(e)h(of)f(genus)g FJ(g)j FO(with)d FJ(n)g FO(b)l(oundary)h(c)l(omp)l(onents)f(which)h (act)f(trivial)t(ly)i(on)f(the)f(set)456 724 y(of)i(b)l(oundary)g(c)l (omp)l(onents.)59 b(Then)37 b(this)f(gr)l(oup)h(acts)g(holomorphic)l (al)t(ly)j(on)d FJ(T)3006 736 y FI(g)r(;n)3105 724 y FO(,)h(and)f(the)456 824 y(stabilizer)31 b(of)f(every)h(p)l(oint)f(is)g (\014nite.)605 923 y FQ(\(iii\))24 b FO(As)g(a)f(c)l(omplex)i(variety,) h FL(M)1653 935 y FI(g)r(;n)1775 923 y FL(')d FJ(T)1912 935 y FI(g)r(;n)2011 923 y FJ(=)p FQ(\000)2105 893 y FE(0)2105 944 y FI(g)r(;n)2203 923 y FO(.)37 b(In)23 b(p)l(articular,)k FL(M)2868 935 y FI(g)r(;n)2990 923 y FO(is)d(c)l(onne)l(cte)l(d.)605 1083 y FP(Remark)32 b FQ(6.1.7)p FP(.)39 b FQ(In)25 b(fact,)g(it)g(is)f(sho)n(wn)f(in)h([)p FK(DM)p FQ(],)h([)p FK(Kn)p FQ(],)g(that)g FL(M)2746 1095 y FI(g)r(;n)2869 1083 y FQ(is)f(an)g(irreducible)456 1182 y(quasipro)5 b(jectiv)n(e)25 b(algebraic)h(v)-5 b(ariet)n(y)27 b(o)n(v)n(er)e FH(C)15 b FQ(|t)q(his)33 b(is)28 b(a)f(di\016cult)h(theorem.)605 1339 y(If)33 b(the)h(action)e(of)h(\000)1248 1308 y FE(0)1248 1359 y FI(g)r(;n)1381 1339 y FQ(on)f(the)i(T)-7 b(eic)n(hm)r(\177)-44 b(uller)32 b(space)h FJ(T)2384 1351 y FI(g)r(;n)2515 1339 y FQ(w)n(ere)g(free,)h(then)f FJ(\031)3143 1351 y FM(1)3181 1339 y FQ(\()p FL(M)3313 1351 y FI(g)r(;n)3412 1339 y FQ(\))456 1441 y(w)n(ould)26 b(b)r(e)h(equal)f(to)h(\000)1180 1411 y FE(0)1180 1462 y FI(g)r(;n)1279 1441 y FQ(.)37 b(Unfortunately)-7 b(,)27 b(the)g(action)f(of)h(\000)2428 1411 y FE(0)2428 1462 y FI(g)r(;n)2554 1441 y FQ(is)f(not)h(free:)36 b(the)27 b(stabilizer)456 1541 y(of)c(a)f(p)r(oin)n(t)h(coincides)g (with)g(the)g(group)f(of)h(automorphisms)f(of)h(the)g(corresp)r(onding) e(complex)456 1640 y(curv)n(e.)36 b(Therefore,)26 b(in)i(general)e FJ(\031)1539 1652 y FM(1)1577 1640 y FQ(\()p FL(M)1709 1652 y FI(g)r(;n)1808 1640 y FQ(\))d FL(6)p FQ(=)g(\000)2003 1610 y FE(0)2003 1661 y FI(g)r(;n)2102 1640 y FQ(,)28 b(as)f(can)g(b)r(e)h(seen)f(already)f(for)h FJ(g)f FQ(=)d(1:)36 b(in)456 1743 y(this)27 b(case,)g FJ(\031)863 1755 y FM(1)901 1743 y FQ(\()p FL(M)1033 1755 y FM(1)p FI(;)p FM(0)1123 1743 y FQ(\))c(=)g FL(f)p FQ(1)p FL(g)p FQ(,)j(while)i(\000) 1710 1755 y FM(0)p FI(;)p FM(1)1823 1743 y FL(')23 b FQ(SL)2009 1755 y FM(2)2046 1743 y FQ(\()p FH(Z)o FQ(\).)605 1842 y(No)n(w,)30 b(let)h(us)f(discuss)f(in)h(what)g(sense)g FL(M)1957 1854 y FI(g)r(;n)2086 1842 y FQ(is)g(the)h(mo)r(duli)f(space) f(of)h(curv)n(es.)43 b(Let)31 b(us)456 1942 y(recall)18 b(\(see,)i(e.g.,)h([)p FK(Ha)p FQ(]\))e(that)g(a)g(family)g(of)g(curv)n (es)f FJ(C)25 b FQ(o)n(v)n(er)17 b(a)i(smo)r(oth)g(v)-5 b(ariet)n(y)18 b FJ(U)27 b FQ(b)n(y)19 b(de\014nition)456 2042 y(is)32 b(a)f(v)-5 b(ariet)n(y)32 b FJ(C)956 2054 y FI(U)1044 2042 y FQ(with)h(a)e(prop)r(er)g(\015at)i(morphism)e FJ(\031)12 b FQ(:)30 b FJ(C)2297 2054 y FI(U)2384 2042 y FL(!)g FJ(U)9 b FQ(,)34 b(suc)n(h)d(that)i FJ(\031)3046 2012 y FE(\000)p FM(1)3135 2042 y FQ(\()p FJ(t)p FQ(\))f(=)e FJ(C)3415 2054 y FI(t)456 2141 y FQ(is)35 b(a)f(compact)h(complex)f (curv)n(e)g(\(unless)h(sp)r(eci\014ed)h(otherwise,)g(w)n(e)e(will)h (assume)g(that)g(the)456 2241 y(\014b)r(ers)c(are)g(connected\).)50 b(Note)32 b(that)g FJ(\031)1731 2211 y FE(\000)p FM(1)1821 2241 y FQ(\()p FJ(t)p FQ(\))g(can)g(b)r(e)g(singular)f(ev)n(en)g(if)i FJ(C)2873 2253 y FI(U)2961 2241 y FQ(is)e(smo)r(oth,)i(as)456 2341 y(sho)n(wn)28 b(b)n(y)g(the)i(example)e(of)h(the)g(surface)f FJ(xy)h FQ(=)c FJ(tu)j FQ(in)h FH(P)2280 2310 y FM(4)2316 2341 y FQ(.)41 b(Similarly)-7 b(,)29 b(a)f(family)h(of)g(p)r(oin)n(ted) 456 2440 y(curv)n(es)d(is)i(a)f(family)h FJ(C)1176 2452 y FI(U)1260 2440 y FQ(together)f(with)i FJ(n)e FQ(non-in)n(tersecting)g (sections)g FJ(p)2815 2452 y FI(i)2852 2440 y FQ(:)h FJ(U)k FL(!)23 b FJ(C)3157 2452 y FI(U)3241 2440 y FQ(and)28 b(a)456 2540 y(non-v)-5 b(anishing)26 b(v)n(ertical)h(v)n(ector)f (\014eld)i FJ(v)1749 2552 y FI(i)1804 2540 y FQ(on)g FJ(p)1962 2552 y FI(i)1989 2540 y FQ(\()p FJ(U)9 b FQ(\))28 b(\(v)n(ertical)f(means)g(that)h FJ(\031)2956 2552 y FE(\003)2994 2540 y FQ(\()p FJ(v)3066 2552 y FI(i)3094 2540 y FQ(\))c(=)e(0\).)605 2696 y FP(Theorem)32 b FQ(6.1.8)p FP(.)40 b FL(M)1349 2708 y FI(g)r(;n)1487 2696 y FO(is)f(the)g FQ(coarse)d(mo)r(duli)i(space)g FO(of)i(curves)f(in)g(the)h(sense)f(of) 456 2796 y FQ([)p FK(MFK)o FQ(]:)c FO(for)25 b(every)f(family)h(of)f(p) l(ointe)l(d)g(curves)f FJ(C)2052 2808 y FI(U)2132 2796 y FO(over)h FJ(U)9 b FO(,)25 b(the)e(induc)l(e)l(d)h(map)g FJ(U)32 b FL(!)23 b(M)3320 2808 y FI(g)r(;n)3419 2796 y FO(,)456 2896 y FJ(t)g FL(7!)g FQ([)p FJ(C)697 2908 y FI(t)726 2896 y FQ(])p FO(,)31 b(is)f(analytic.)40 b FQ(\()p FO(Her)l(e)29 b FQ([)p FJ(C)6 b FQ(])31 b FO(denotes)f(the)g (isomorphism)i(class)e(of)h(a)f(curve)g FJ(C)6 b FO(.)p FQ(\))605 3052 y(Unfortunately)-7 b(,)28 b(it)g(is)f(not)h(true)f(that) h(the)g(construction)f(ab)r(o)n(v)n(e)f(giv)n(es)h(a)g(bijection)1018 3201 y FL(f)p FQ(families)g(of)h(curv)n(es)e(o)n(v)n(er)g FJ(U)9 b FL(g)2048 3154 y FE(\030)2019 3201 y FL(\000)-39 b(!)23 b(f)p FQ(maps)k FJ(U)k FL(!)23 b(M)2704 3213 y FI(g)r(;n)2804 3201 y FL(g)p FQ(;)456 3348 y(in)34 b(other)f(w)n(ords,) i FL(M)1152 3360 y FI(g)r(;n)1285 3348 y FQ(is)f(not)g(the)g(\014ne)h (mo)r(duli)f(space.)55 b(The)35 b(reason)d(for)i(the)g(failure)g(is)456 3448 y(that)27 b FL(M)735 3460 y FI(g)r(;n)862 3448 y FQ(carries)f(no)h(information)g(ab)r(out)h(the)g(automorphisms)e(of)i (a)f(curv)n(e.)605 3604 y FP(Exer)n(cise)32 b FQ(6.1.9)p FP(.)40 b FQ(Let)21 b FJ(C)27 b FQ(b)r(e)21 b(a)g(p)r(oin)n(ted)g(curv) n(e,)g(and)g FJ(\033)s FQ(|a)g(non-trivial)f(automorphism)456 3704 y(of)25 b FJ(C)6 b FQ(.)36 b(Construct)25 b(a)g(family)g(of)g (curv)n(es)f FJ(C)1779 3716 y FI(t)1834 3704 y FQ(o)n(v)n(er)f FH(C)2063 3674 y FE(\002)2150 3704 y FQ(suc)n(h)i(that)h FJ(C)2572 3716 y FI(t)2624 3704 y FL(')d FJ(C)32 b FQ(for)24 b(an)n(y)h FJ(t)p FQ(,)g(but)h(this)456 3804 y(family)h(is)h(not)f (isomorphic)g(to)g(the)h(direct)g(pro)r(duct)f FJ(C)e FL(\002)18 b FH(C)2373 3774 y FE(\002)2435 3804 y FQ(.)605 3960 y(It)26 b(turns)g(out)g(that)g(this)h(w)n(as)e(the)h(only)f (problem:)36 b(if)27 b(w)n(e)e(assume)g(that)i(the)f(curv)n(es)f(ha)n (v)n(e)456 4060 y(no)i(non-trivial)f(automorphisms,)h(then)h FL(M)1874 4072 y FI(g)r(;n)2001 4060 y FQ(is)f(the)h(\014ne)g(mo)r (duli)g(space.)605 4216 y FP(Theorem)k FQ(6.1.10)p FP(.)39 b FO(Assume)29 b(that)h FJ(g)25 b(>)e FQ(0)p FJ(;)14 b(n)22 b(>)h FQ(0)29 b FO(or)h FJ(G)24 b FQ(=)e(0)p FJ(;)14 b(n)23 b(>)f FQ(1)p FO(.)38 b(Then:)605 4339 y FQ(1.)j FO(F)-6 b(or)38 b(every)g(c)l(omplex)f(curve)g FJ(C)44 b FO(of)38 b(genus)f FJ(g)i FO(with)f FJ(n)f FO(marke)l(d)h(p)l(oint,)i (the)d(gr)l(oup)h(of)711 4439 y(automorphisms)31 b(is)f(trivial.)605 4539 y FQ(2.)41 b FO(The)28 b(action)g(of)g(the)f(gr)l(oup)g FQ(\000)1635 4509 y FE(0)1635 4559 y FI(g)r(;n)1761 4539 y FO(on)g(the)h(c)l(orr)l(esp)l(onding)g(T)-6 b(eichm)q(\177)-43 b(ul)t(ler)29 b(sp)l(ac)l(e)e(is)h(fr)l(e)l(e,)711 4641 y(so)i FL(M)917 4653 y FI(g)r(;n)1039 4641 y FQ(=)23 b FJ(T)1176 4653 y FI(g)r(;n)1275 4641 y FJ(=)p FQ(\000)1369 4611 y FE(0)1369 4662 y FI(g)r(;n)1497 4641 y FO(is)30 b(smo)l(oth.)605 4741 y FQ(3.)41 b FL(M)811 4753 y FI(g)r(;n)940 4741 y FO(is)30 b(the)g(\014ne)f(mo)l(duli)i(sp)l(ac)l(e:)39 b(for)31 b(every)f(variety)h FJ(S)5 b FO(,)30 b(the)g(functors)926 4888 y FJ(S)d FL(7!)d FO(families)31 b(of)g(curves)f(of)g(genus)g FJ(g)i FO(with)e FJ(n)g FO(marke)l(d)g(p)l(oints)g(over)h FJ(S)711 5035 y FO(and)1607 5182 y FJ(S)c FL(7!)d FQ(Mor)o(\()p FJ(S;)14 b FL(M)2162 5194 y FI(g)r(;n)2261 5182 y FQ(\))p eop %%Page: 138 4 138 141 bop 456 226 a FM(138)445 b(6.)30 b(MODULI)f(SP)-5 b(A)n(CES)28 b(AND)h(COMPLEX)f(MODULAR)h(FUNCTOR)711 425 y FO(ar)l(e)e(c)l(anonic)l(al)t(ly)h(isomorphic.)41 b(In)26 b(other)h(wor)l(ds,)h(every)g(family)g(of)f(curves)g(on)f FJ(S)31 b FO(c)l(an)711 525 y(b)l(e)c(obtaine)l(d)i(as)e(a)h(pul)t(l-b) l(ack)g(of)g(a)g(universal)f(family)i(over)f FL(M)2701 537 y FI(g)r(;n)2827 525 y FO(for)g(a)g(unique)f(map)711 624 y FJ( )12 b FQ(:)28 b FJ(S)g FL(!)23 b(M)1113 636 y FI(g)r(;n)1212 624 y FO(.)605 800 y FQ(It)i(turns)g(out)f(that)h(one) f(can)h(also)e(de\014ne)i(a)f(suitable)h(\\\014ne)f(mo)r(duli)i(space") d FJ(M)3113 812 y FI(g)r(;n)3237 800 y FQ(under)456 899 y(less)k(restrictiv)n(e)f(assumptions)h(that)h(\()p FJ(g)s(;)14 b(n)p FQ(\))27 b(is)h(stable,)f(i.e.)1473 1079 y(\()p FJ(g)s(;)14 b(n)p FQ(\))22 b FL(6)p FQ(=)h(\(0)p FJ(;)14 b FQ(0\))p FJ(;)g FQ(\(0)p FJ(;)g FQ(1\))p FJ(;)g FQ(\(1)p FJ(;)g FQ(0\))p FJ(:)-1973 b FQ(\(6.1.2\))456 1259 y(In)22 b(this)h(case,)g(the)f(group)g(of)g(automorphisms)f(of)i(ev)n(ery)e (curv)n(e)g FJ(C)30 b FL(2)23 b(M)2727 1271 y FI(g)r(;n)2849 1259 y FQ(is)f(\014nite.)36 b(It)23 b(turns)456 1359 y(out)29 b(that)g(under)h(this)f(assumption,)g(it)h(is)f(p)r(ossible)g (to)g(accoun)n(t)g(for)f(these)i(automorphisms)456 1458 y(and)d(de\014ne)h(a)g(\\\014ne")f(mo)r(duli)h(space,)f(if)i(w)n(e)e (allo)n(w)g(the)h(mo)r(duli)g(space)f(to)h(b)r(e)g(not)g(a)f(v)-5 b(ariet)n(y)e(,)456 1558 y(but)31 b(a)g FO(stack)p FQ(,)h(as)e (de\014ned)h(in)g([)p FK(DM)p FQ(],)h([)p FK(Ar)q FQ(].)46 b(In)n(tuitiv)n(ely)-7 b(,)32 b(this)f(means)g(that)g(ev)n(ery)e(p)r (oin)n(t)j(of)456 1658 y FL(M)556 1670 y FI(g)r(;n)688 1658 y FQ(has)i(some)f(additional)g(structure,)i(whic)n(h)e(enco)r(des) g(the)i(group)d(of)i(automorphisms)456 1757 y(of)h(the)h(corresp)r (onding)d(curv)n(e.)59 b(Unfortunately)-7 b(,)37 b(an)e(accurate)f(exp) r(osition)h(of)h(the)f(theory)456 1857 y(of)e(algebraic)e(stac)n(ks)h (go)r(es)g(far)g(b)r(ey)n(ond)h(the)g(scop)r(e)g(of)g(this)g(b)r(o)r (ok;)j(w)n(e)c(can)h(only)g(refer)f(the)456 1957 y(reader)39 b(to)h(the)g(App)r(endix)i(to)e([)p FK(Vi)p FQ(])g(for)g(an)g(in)n(tro) r(duction)g(to)h(this)f(language.)74 b(Another)456 2056 y(approac)n(h,)41 b(whic)n(h)f(applies)g(if)g FJ(g)s(;)14 b(n)40 b FQ(are)f(suc)n(h)g(that)i(the)f(generic)f(curv)n(e)g FJ(C)51 b FL(2)44 b(M)3185 2068 y FI(g)r(;n)3324 2056 y FQ(has)456 2156 y(no)36 b(automorphisms,)i(is)f(to)f(consider)g FL(M)1833 2168 y FI(g)r(;n)1969 2156 y FQ(as)g(an)g FO(orbifold)p FQ(,)42 b(or)36 b FJ(V)19 b FO(-variety)38 b FQ(\(see)e([)p FK(Sat)q FQ(]\).)456 2255 y(Finally)-7 b(,)24 b(the)h(third)f(p)r (ossibilit)n(y)-7 b(,)25 b(used)f(in)g([)p FK(TUY)r FQ(],)h(is)f(to)g (consider)f(\\lo)r(cal)g(univ)n(ersal)g(families)456 2355 y(of)37 b(curv)n(es",)g(whic)n(h)g(can)g(b)r(e)g(view)n(ed)g(as)f (lo)r(cal)g(c)n(harts)g(of)h(the)h(algebraic)d(stac)n(k.)64 b(F)-7 b(or)36 b(our)456 2455 y(purp)r(oses,)41 b(w)n(e)e(can)g(use)g (an)n(y)f(of)h(these)g(approac)n(hes:)58 b(all)39 b(of)g(them)h(will)f (yield)g(the)h(same)456 2554 y(results,)31 b(and)f(eac)n(h)g(has)h(its) g(o)n(wn)f(adv)-5 b(an)n(tages)29 b(and)h(disadv)-5 b(an)n(tages.)45 b(W)-7 b(e)31 b(c)n(hose)f(to)g(use)h(the)456 2654 y(language)26 b(of)h(algebraic)f(stac)n(ks.)605 2754 y(W)-7 b(e)31 b(will)f(denote)g(b)n(y)g FJ(M)1379 2766 y FI(g)r(;n)1508 2754 y FQ(the)h FO(mo)l(duli)i(stack)39 b FQ(of)30 b(p)r(oin)n(ted)h (curv)n(es)e(of)h(gen)n(us)f FJ(g)k FQ(with)e FJ(n)456 2853 y FQ(mark)n(ed)g(p)r(oin)n(ts;)36 b(as)c(w)n(e)h(said,)g(w)n(e)g (will)g(not)g(explain)f(what)h(it)g(is,)i(referring)c(the)i(reader)f (to)456 2953 y([)p FK(DM)p FQ(])f(instead.)46 b(Nev)n(ertheless,)31 b(w)n(e)f(can)h(sa)n(y)f(what)g(are)g(p)r(oin)n(ts,)i(v)n(ector)e (bundles,)i(etc.,)g(on)456 3052 y FJ(M)537 3064 y FI(g)r(;n)636 3052 y FQ(.)45 b(Namely:)d(for)30 b(a)g(complex)g(manifold)g FJ(S)5 b FQ(,)31 b(a)f(morphism)g FJ(S)i FL(!)27 b FJ(M)2769 3064 y FI(g)r(;n)2899 3052 y FQ(is)j(b)n(y)g(de\014nition)456 3152 y(the)i(same)f(as)g(a)g(family)h(of)g(p)r(oin)n(ted)g(complex)f (curv)n(es)f(of)i(gen)n(us)f FJ(g)j FQ(with)e FJ(n)g FQ(mark)n(ed)f(p)r(oin)n(ts)456 3252 y(o)n(v)n(er)c FJ(S)5 b FQ(.)41 b(In)30 b(particular,)e(this)i(implies)f(that)h(the)f(set)g (of)h(\(closed\))f(p)r(oin)n(ts)g(of)g FJ(M)3009 3264 y FI(g)r(;n)3137 3252 y FQ(is)g FL(M)3322 3264 y FI(g)r(;n)3421 3252 y FQ(.)456 3351 y(Similarly)-7 b(,)28 b(a)g(v)n(ector)g(bundle)h FJ(E)34 b FQ(on)28 b FJ(M)1712 3363 y FI(g)r(;n)1840 3351 y FQ(is)h(the)g(same)f(as)g(a)g(collection)g(of)h(v)n(ector)e (bundles)456 3451 y FJ(\036)505 3421 y FE(\003)543 3451 y FJ(E)39 b FQ(for)34 b(ev)n(ery)e(morphism)h FJ(\036)h FQ(:)g FJ(S)k FL(!)c FJ(M)1823 3463 y FI(g)r(;n)1922 3451 y FQ(,)h(suc)n(h)f(that)g(this)g(collection)f(is)h(functorial)f (in)456 3551 y FJ(S)5 b FQ(.)47 b(Lo)r(cal)30 b(systems,)i(\015at)f (connections,)h(etc.,)g(can)f(b)r(e)h(de\014ned)f(in)h(a)e(similar)h(w) n(a)n(y)-7 b(.)46 b(W)-7 b(e)32 b(can)456 3650 y(also)c(de\014ne)i(a)f (divisor)g FJ(D)g FL(\032)d FJ(M)1480 3662 y FI(g)r(;n)1608 3650 y FQ(as)j(a)h(compatible)f(collection)g(of)h(divisors)e FJ(\036)3032 3620 y FE(\003)3071 3650 y FJ(D)g FL(\032)f FJ(S)34 b FQ(for)456 3750 y(ev)n(ery)28 b(\023)-39 b(etale)30 b(\(i.e.,)j(\014nite)g(unrami\014ed)e(co)n(v)n(ering\))f FJ(\036)g FQ(:)g FJ(S)k FL(!)c FJ(M)2484 3762 y FI(g)r(;n)2583 3750 y FQ(.)49 b(Finally)-7 b(,)33 b(w)n(e)e(de\014ne)h(the)456 3849 y(fundamen)n(tal)27 b(group)g(of)g FJ(M)1346 3861 y FI(g)r(;n)1473 3849 y FQ(b)n(y)855 4039 y FJ(\031)902 4051 y FM(1)939 4039 y FQ(\()p FJ(M)1052 4051 y FI(g)r(;n)1152 4039 y FJ(;)14 b(C)6 b FQ(\))23 b(=)g FL(f)p FQ(\()p FJ(C)1530 4051 y FI(t)1559 4039 y FQ(\))1591 4051 y FM(0)p FE(\024)p FI(t)p FE(\024)p FM(1)1791 4039 y FJ(;)14 b(')1882 4051 y FM(0)1942 4039 y FQ(:)23 b FJ(C)2047 4051 y FM(0)2136 3992 y FE(\030)2108 4039 y FL(\000)-39 b(!)23 b FJ(C)q(;)14 b(')2391 4051 y FM(1)2452 4039 y FQ(:)23 b FJ(C)2557 4051 y FM(1)2646 3992 y FE(\030)2618 4039 y FL(\000)-40 b(!)23 b FJ(C)6 b FL(g)p FJ(=)p FQ(homotop)n(y)n FJ(:)456 4219 y FQ(Here)28 b FJ(C)712 4231 y FI(t)769 4219 y FQ(is)h(a)f FJ(C)989 4189 y FE(1)1088 4219 y FQ(family)g(of)g(complex)g(curv)n(es,) f(i.e.,)i(a)f FJ(C)2334 4189 y FE(1)2433 4219 y FQ(real)f(manifold)i (\006)f(with)h(a)f(map)456 4319 y FJ(\031)12 b FQ(:)31 b(\006)36 b FL(!)h FQ([0)p FJ(;)14 b FQ(1])34 b(suc)n(h)i(that)g FJ(d\031)k FL(6)p FQ(=)c(0,)h(and)e(for)h(ev)n(ery)e FJ(t)i FL(2)h FH(R)p FQ(,)44 b FJ(C)2573 4331 y FI(t)2639 4319 y FQ(=)36 b FJ(\031)2790 4289 y FE(\000)p FM(1)2880 4319 y FQ(\()p FJ(t)p FQ(\))g(is)f(a)h(smo)r(oth)456 4419 y(compact)c(orien)n(ted)g(surface,)i(and)f(with)g(a)g(family)g(of) g(complex)f(structures)h FJ(\026)2999 4431 y FI(t)3061 4419 y FQ(in)g FJ(C)3222 4431 y FI(t)3285 4419 y FQ(suc)n(h)456 4518 y(that)23 b FJ(\026)681 4530 y FI(t)733 4518 y FQ(is)g(a)f FJ(C)941 4488 y FE(1)1035 4518 y FQ(function)h(of)g FJ(t)g FQ(\(this)g(should)g(b)r(e)g(mo)r(di\014ed)g(in)g(an)g(ob)n(vious)e(w)n (a)n(y)h(for)g(p)r(oin)n(ted)456 4618 y(curv)n(es\).)36 b(Later)26 b(w)n(e)i(will)f(sho)n(w)g(that)h(in)g(fact,)g FJ(\031)1993 4630 y FM(1)2030 4618 y FQ(\()p FJ(M)2143 4630 y FI(g)r(;n)2242 4618 y FQ(\))c(=)e(\000)2437 4588 y FE(0)2437 4638 y FI(g)r(;n)2537 4618 y FQ(.)605 4717 y(Of)41 b(course,)j(w)n(e)d(are)f(just)i(hiding)g(the)g(real)e (problem:)64 b(wh)n(y)41 b(so)g(de\014ned)g FJ(M)3165 4729 y FI(g)r(;n)3306 4717 y FQ(is)g(a)456 4817 y(reasonable)29 b(geometric)h(ob)5 b(ject,)31 b(i.e.,)i(wh)n(y)d(the)i(standard)e (results)h(ab)r(out,)g(sa)n(y)-7 b(,)31 b(shea)n(v)n(es)f(on)456 4917 y(v)-5 b(arieties)24 b(apply)h(to)h FJ(M)1186 4929 y FI(g)r(;n)1285 4917 y FQ(?)36 b(This)26 b(is)f(indeed)h(a)f (di\016cult)h(question,)g(and)f(the)h(b)r(est)g(w)n(e)f(can)g(do)456 5016 y(here)i(is)g(to)h(refer)f(to)h([)p FK(DM)p FQ(].)37 b(Their)27 b(results)g(sho)n(w)g(that)h(as)f(far)g(as)h(w)n(e)f(are)g (concerned,)f FJ(M)3345 5028 y FI(g)r(;n)456 5116 y FQ(can)j(b)r(e)g (treated)g(in)h(the)g(same)e(w)n(a)n(y)g(as)h(a)g(non-singular)e(v)-5 b(ariet)n(y:)40 b(all)29 b(the)h(standard)e(results)456 5216 y(from)f(algebraic)f(geometry)g(w)n(e)h(will)h(b)r(e)g(using)f (apply)h(to)f FJ(M)2385 5228 y FI(g)r(;n)2484 5216 y FQ(.)p eop %%Page: 139 5 139 142 bop 940 238 a FM(6.1.)29 b(MODULI)g(SP)-5 b(A)n(CES)28 b(AND)h(COMPLEX)f(TEICHM)2481 223 y(\177)2473 238 y(ULLER)h(TO)n(WER) 363 b(139)605 425 y FQ(As)32 b(w)n(as)e(men)n(tioned)h(ab)r(o)n(v)n(e,) g(for)g FJ(g)h(>)d FQ(0)p FJ(;)14 b(n)29 b(>)g FQ(0)i(or)g FJ(g)h FQ(=)d(0)p FJ(;)14 b(n)28 b(>)h FQ(1)p FJ(;)i FL(M)2906 437 y FI(g)r(;n)3037 425 y FQ(itself)h(is)f(the)456 525 y(\014ne)j(mo)r(duli)h(space,)g(so)e(in)i(this)f(case)f(w)n(e)h(ha) n(v)n(e)f FL(M)2147 537 y FI(g)r(;n)2280 525 y FQ(=)h FJ(M)2460 537 y FI(g)r(;n)2559 525 y FQ(.)56 b(In)35 b(general,)f(this)g(is)g(not)456 624 y(true.)605 724 y(F)-7 b(rom)22 b(no)n(w)f(on,)i(w)n(e)e(will)h(use)g FJ(M)1605 736 y FI(g)r(;n)1726 724 y FQ(as)f(the)i(mo)r(duli)f(space,)g (and)g(all)g(geometric)f(construc-)456 824 y(tions)k(will)g(b)r(e)h (understo)r(o)r(d)f(in)g(the)h(stac)n(k)e(sense.)36 b(A)25 b(reader)f(who)h(is)g(not)g(to)r(o)g(exp)r(erienced)g(in)456 923 y(this)i(language)f(can)i(just)g(think)g(of)f FJ(M)1674 935 y FI(g)r(;n)1801 923 y FQ(as)g(a)g(smo)r(oth)h(manifold.)605 1023 y(Sometimes,)40 b(it)d(is)h(con)n(v)n(enien)n(t)e(to)h(de\014ne)g (a)g(sligh)n(tly)g(di\013eren)n(t)g(space.)65 b(Let)38 b FJ(A)f FQ(b)r(e)h(a)456 1123 y(\014nite)g(set.)66 b(Denote)38 b FL(M)1264 1135 y FI(g)r(;A)1411 1123 y FQ(=)h FL(f)p FQ(\()p FJ(C)q(;)14 b(f)9 b FQ(\))p FL(g)p FQ(,)40 b(where)d FJ(C)44 b FQ(is)37 b(a)g(gen)n(us)f FJ(g)41 b FQ(complex)c(curv)n(e)f (with)456 1222 y FJ(n)g FK(unordered)g FQ(mark)n(ed)f(p)r(oin)n(ts)h (and)g(non-zero)e(tangen)n(t)i(v)n(ectors,)h(and)f FJ(f)44 b FQ(is)36 b(a)g(bijection)92 b Fq(?!)456 1324 y FJ(A)p FQ(\()p FJ(C)6 b FQ(\))705 1277 y FE(\030)677 1324 y FL(\000)-40 b(!)29 b FJ(A)p FQ(,)j(where)f FJ(n)d FQ(=)h FL(j)p FJ(A)p FL(j)p FQ(,)j(and)f FJ(A)p FQ(\()p FJ(C)6 b FQ(\))32 b(is)f(the)g(set)h(of)f(mark)n(ed)e(p)r(oin)n(ts)i(of)g FJ(C)6 b FQ(.)48 b(In)31 b(other)456 1424 y(w)n(ords,)23 b FL(M)814 1436 y FI(g)r(;A)947 1424 y FQ(is)h(the)h(mo)r(duli)g(space) f(of)h(curv)n(es)e(of)h(gen)n(us)g FJ(g)j FQ(with)e FJ(n)g FQ(mark)n(ed)e(p)r(oin)n(ts)h(lab)r(eled)456 1524 y(b)n(y)f(elemen)n (ts)h(of)f FJ(A)p FQ(,)i(and)f(with)g(non-zero)e(tangen)n(t)h(v)n (ectors)f(at)i(these)f(p)r(oin)n(ts.)36 b(Ob)n(viously)-7 b(,)23 b(for)456 1623 y FJ(A)29 b FQ(=)g FL(f)p FQ(1)p FJ(;)14 b(:)g(:)g(:)f(;)h(n)p FL(g)p FQ(,)32 b(this)f(coincides)g(with) h(the)g(de\014nition)f(of)h FL(M)2487 1635 y FI(g)r(;n)2586 1623 y FQ(.)48 b(One)31 b(de\014nes)h(the)f(stac)n(k)456 1723 y FJ(M)537 1735 y FI(g)r(;A)672 1723 y FQ(in)d(a)f(similar)g(w)n (a)n(y)-7 b(.)605 1822 y(W)g(e)33 b(will)g(also)e(consider)g(the)i(mo)r (duli)g(space)f FL(M)2173 1834 y FE(\003)p FI(;n)2304 1822 y FQ(of)g(not)h(necessarily)e(connected)h FJ(n)p FQ(-)456 1922 y(p)r(oin)n(ted)g(curv)n(es,)g(and)h(the)f(space)g FL(M)e FQ(=)h FL(t)1865 1934 y FI(n)p FE(\025)p FM(0)1995 1922 y FL(M)2095 1934 y FE(\003)p FI(;n)2194 1922 y FQ(.)51 b(One)32 b(easily)f(sees)h(that)h FL(M)3138 1934 y FE(\003)p FI(;n)3269 1922 y FQ(is)f(an)456 2022 y(unrami\014ed)27 b(\014nite)h(co)n(v)n(er)e(o)n(v)n(er)g(the)i(space)1456 2169 y FL(t)p FQ(\()p FL(M)1643 2181 y FI(g)1675 2189 y FF(1)1708 2181 y FI(;n)1769 2189 y FF(1)1824 2169 y FL(\002)18 b(\001)c(\001)g(\001)k(\002)g(M)2205 2181 y FI(g)2237 2190 y FG(k)2274 2181 y FI(;n)2335 2190 y FG(k)2375 2169 y FQ(\))p FJ(;)456 2322 y FQ(where)43 b(the)i(disjoin)n(t)f(union)g(is)g(tak)n(en)g(o)n(v)n(er)e(all)i (\014nite)h(sequences)e(\()p FJ(g)2794 2334 y FM(1)2831 2322 y FJ(;)14 b(n)2918 2334 y FM(1)2955 2322 y FQ(\))p FJ(;)g(:)g(:)g(:)g(;)g FQ(\()p FJ(g)3244 2334 y FI(k)3285 2322 y FJ(;)g(n)3372 2334 y FI(k)3412 2322 y FQ(\))456 2422 y(suc)n(h)36 b(that)841 2359 y Fy(P)943 2422 y FJ(n)993 2434 y FI(i)1059 2422 y FQ(=)i FJ(n)p FQ(,)h(up)e(to)g(p)r(erm)n (utation.)64 b(Th)n(us,)39 b(w)n(e)e(can)f(easily)g(de\014ne)h(the)h (stac)n(ks)456 2521 y FJ(M)537 2533 y FE(\003)p FI(;n)635 2521 y FJ(;)14 b(M)753 2533 y FE(\003)p FI(;A)860 2521 y FQ(.)43 b(The)30 b(last)f(stac)n(k)g(will)h(b)r(e)g(frequen)n(tly)f (used)g(later,)h(so)f(w)n(e)g(state)g(its)h(de\014nition)456 2621 y(explicitly:)846 2751 y FJ(M)927 2763 y FE(\003)p FI(;A)1057 2751 y FQ(=mo)r(duli)e(stac)n(k)f(of)g(stable)g(smo)r(oth)h (p)r(ossibly)f(disconnected)g(curv)n(es)1122 2876 y(with)h(unordered)e (mark)n(ed)h(p)r(oin)n(ts)g(and)h(non-zero)e(tangen)n(t)h(v)n(ectors) 1122 3021 y(and)g(a)g(bijection)h(\(mark)n(ed)f(p)r(oin)n(ts\))2302 2974 y FE(\030)2273 3021 y FL(\000)-39 b(!)23 b FJ(A)456 2895 y FQ(\(6.1.3\))605 3169 y(Recall)k(that)g(w)n(e)g(ha)n(v)n(e)f (de\014ned)h(the)h(notion)f(of)g(a)f(to)n(w)n(er)g(of)h(group)r(oids,)f (whic)n(h)h(is)g(just)h(a)456 3269 y(group)r(oid)22 b(\000)i(with)g(a)f (functor)h FJ(A)9 b FQ(:)28 b(\000)23 b FL(!)g(S)6 b FJ(ets)24 b FQ(and)g(with)g(the)g(functors)f(of)h(disjoin)n(t)g(union)g (and)456 3368 y(gluing)k(\(see)h(De\014nition)h(5.6.1\).)41 b(In)29 b(particular,)f(w)n(e)h(ha)n(v)n(e)f(de\014ned)i(the)f(T)-7 b(eic)n(hm)r(\177)-44 b(uller)29 b(to)n(w)n(er)456 3468 y FL(T)7 b FJ(eich)p FQ(,)21 b(in)g(whic)n(h)f(the)h(ob)5 b(jects)20 b(are)g(\(top)r(ological\))f(surfaces)g(with)i(b)r(oundary) -7 b(,)22 b(morphisms)d(are)456 3568 y(homeomorphisms)j(of)i(surfaces,) g(and)g(gluing)f(is)h(the)h(gluing)e(of)h(t)n(w)n(o)g(b)r(oundary)f (comp)r(onen)n(ts)456 3667 y(\(see)40 b(De\014nition)h(5.1.7,)h (Section)e(5.6\).)74 b(The)41 b(form)n(ula)e FJ(\031)2358 3679 y FM(1)2395 3667 y FQ(\()p FJ(M)2508 3679 y FI(g)r(;n)2608 3667 y FQ(\))44 b(=)g(\000)2845 3637 y FE(0)2845 3688 y FI(g)r(;n)2944 3667 y FQ(,)f(whic)n(h)e(is)f(an)456 3770 y(ob)n(vious)31 b(corollary)f(of)j(Theorem)f(6.1.6)f(if)j(the)f (action)f(of)h(\000)2405 3740 y FE(0)2405 3790 y FI(g)r(;n)2537 3770 y FQ(is)g(free,)h(suggests)d(that)i(the)456 3869 y(same)27 b(to)n(w)n(er)f(can)h(b)r(e)h(de\014ned)g(in)g(terms)f(of)h (the)g(mo)r(duli)g(spaces)e FJ(M)2621 3881 y FI(g)r(;n)2720 3869 y FQ(.)605 3969 y(Recall)34 b(that)h(for)e(a)h(top)r(ological)f (space)g FJ(M)9 b FQ(,)36 b(its)f FO(Poinc)l(ar)n(\023)-40 b(e)37 b(gr)l(oup)l(oid)f FQ(\(also)d(called)h(the)456 4069 y FO(fundamental)g(gr)l(oup)l(oid)p FQ(\))g(is)f(de\014ned)f(as)g (the)h(group)r(oid)e(with)i(ob)5 b(jects:)46 b(p)r(oin)n(ts)32 b(of)g FJ(M)9 b FQ(,)34 b(and)456 4168 y(morphisms:)i(homotop)n(y)26 b(classes)g(of)i(paths)g(in)f FJ(M)37 b FQ(connecting)27 b(t)n(w)n(o)g(p)r(oin)n(ts.)605 4325 y FP(Definition)32 b FQ(6.1.11)p FP(.)39 b FQ(The)24 b FO(c)l(omplex)j(T)-6 b(eichm)q(\177)-43 b(ul)t(ler)27 b(tower)g(of)g(gr)l(oup)l(oids)32 b FL(T)7 b FJ(eich)3180 4295 y Fz(C)3249 4325 y FQ(is)24 b(the)456 4425 y(fundamen)n(tal)j(group)r(oid)g(of)g(the)h(stac)n(k)f FJ(M)9 b FQ(,)27 b(i.e.,)1282 4573 y(Ob)13 b FL(T)7 b FJ(eich)1610 4538 y Fz(C)1679 4573 y FQ(=)23 b(p)r(oin)n(ted)k(complex) h(curv)n(es)456 4721 y(and)814 4868 y(Mor)o(\()p FJ(C)1061 4834 y FE(0)1085 4868 y FJ(;)14 b(C)1187 4834 y FE(00)1229 4868 y FQ(\))24 b(=)e FL(f)p FQ(\()p FJ(C)1505 4880 y FI(t)1535 4868 y FQ(\))1567 4880 y FM(0)p FE(\024)p FI(t)p FE(\024)p FM(1)1766 4868 y FJ(;)14 b(')1857 4880 y FM(0)1904 4868 y FQ(:)28 b FJ(C)2014 4880 y FM(0)2103 4821 y FE(\030)2074 4868 y FL(\000)-39 b(!)23 b FJ(C)2271 4834 y FE(0)2294 4868 y FJ(;)14 b(')2385 4880 y FM(1)2432 4868 y FQ(:)28 b FJ(C)2542 4880 y FM(1)2631 4821 y FE(\030)2602 4868 y FL(\000)-39 b(!)23 b FJ(C)2799 4834 y FE(0)q(0)2842 4868 y FL(g)p FJ(=)p FQ(homotop)n(y)n FJ(;)456 5016 y FQ(where,)29 b(as)g(b)r(efore,)g FJ(C)1158 5028 y FI(t)1217 5016 y FQ(is)h(a)f FJ(C)1439 4986 y FE(1)1539 5016 y FQ(family)g(of)g(p)r(oin)n(ted)h(curv)n(es.)41 b(W)-7 b(e)30 b(also)e(de\014ne)i(the)g(functor)456 5116 y FJ(A)9 b FQ(:)28 b FL(T)7 b FJ(eich)782 5086 y Fz(C)854 5116 y FL(!)25 b(S)6 b FJ(ets)29 b FQ(b)n(y)g FJ(A)p FQ(\()p FJ(C)q(;)14 b(y)1504 5128 y FI(i)1533 5116 y FJ(;)g(v)1610 5128 y FI(i)1637 5116 y FQ(\))26 b(=)f FL(f)p FJ(y)1868 5128 y FI(i)1895 5116 y FL(g)p FQ(,)k(and)g(the)h(disjoin)n(t)f(union)g (and)g(empt)n(y)g(set)g(in)456 5216 y(an)e(ob)n(vious)f(w)n(a)n(y)-7 b(.)p eop %%Page: 140 6 140 143 bop 456 226 a FM(140)445 b(6.)30 b(MODULI)f(SP)-5 b(A)n(CES)28 b(AND)h(COMPLEX)f(MODULAR)h(FUNCTOR)605 425 y FQ(In)d(a)f(similar)f(w)n(a)n(y)-7 b(,)25 b(for)g(a)g(\014nite)g (set)h FJ(A)g FQ(w)n(e)e(de\014ne)i(the)g(group)r(oid)e FL(T)7 b FJ(eich)2892 395 y Fz(C)2892 448 y FI(A)2971 425 y FQ(as)25 b(the)g(funda-)456 525 y(men)n(tal)i(group)r(oid)g(of)g (the)h(stac)n(k)f FJ(M)1610 537 y FE(\003)p FI(;A)1717 525 y FQ(.)605 624 y(Note)j(that)h(in)f(particular,)f(for)h(ev)n(ery)f (curv)n(e)g FJ(C)36 b FQ(w)n(e)30 b(ha)n(v)n(e)f(a)h(canonical)e(map)i (Aut)15 b FJ(C)33 b FL(!)456 724 y FQ(Mor)o(\()p FJ(C)q(;)14 b(C)6 b FQ(\),)32 b(whic)n(h)e(assigns)f(to)h FJ(\033)h FL(2)d FQ(Aut)15 b FJ(C)36 b FQ(the)31 b(data)f FJ(C)2320 736 y FI(t)2377 724 y FQ(=)d FJ(C)g FL(\002)19 b FQ([0)p FJ(;)14 b FQ(1])p FJ(;)g(')2897 736 y FM(0)2962 724 y FQ(=)27 b(id)p FJ(;)14 b(')3214 736 y FM(1)3279 724 y FQ(=)27 b FJ(\033)s FQ(,)456 824 y(whic)n(h)g(explains)g(wh)n(y)g(w)n (e)h(in)n(tro)r(duced)f FJ(')1781 836 y FM(0)1819 824 y FJ(;)14 b(')1910 836 y FM(1)1975 824 y FQ(in)28 b(the)g (de\014nition.)605 923 y(T)-7 b(o)26 b(complete)f(the)h(de\014nition,)h (w)n(e)f(also)e(ha)n(v)n(e)h(to)h(de\014ne)g(the)g(gluing)f(functor.)36 b(This)26 b(will)456 1023 y(b)r(e)i(done)f(in)h(the)g(next)g(section.) 605 1123 y(No)n(w)37 b(w)n(e)h(can)f(compare)g(this)h(complex)g(T)-7 b(eic)n(hm)r(\177)-44 b(uller)37 b(group)r(oid)g(with)h(the)h(group)r (oid)456 1222 y FL(T)7 b FJ(eich)27 b FQ(de\014ned)h(in)g(the)g (previous)f(c)n(hapter)f(in)i(terms)g(of)f(top)r(ological)g(surfaces)f (with)i(b)r(ound-)456 1322 y(ary)-7 b(.)67 b(Note,)41 b(ho)n(w)n(ev)n(er,)d(that)h(since)e(w)n(e)h(ha)n(v)n(e)f(imp)r(osed)h (the)g(stabilit)n(y)g(condition)g(\(6.1.2\))o(,)456 1421 y(it)33 b(only)g(mak)n(es)g(sense)g(to)g(compare)f FL(T)7 b FJ(eich)1856 1391 y Fz(C)1935 1421 y FQ(with)34 b(the)g(subgroup)r (oid)e FL(T)7 b FJ(eich)2962 1391 y FI(stab)3120 1421 y FL(\032)33 b(T)7 b FJ(eich)p FQ(,)456 1521 y(formed)30 b(b)n(y)g(top)r(ological)f(surfaces)g(all)h(connected)h(comp)r(onen)n (ts)e(of)i(whic)n(h)f(satisfy)g(the)h(sta-)456 1621 y(bilit)n(y)c (condition)h(\(6.1.2\))o(.)605 1759 y FP(Theorem)k FQ(6.1.12)p FP(.)39 b FO(The)23 b(towers)e(of)i(gr)l(oup)l(oids)g FL(T)7 b FJ(eich)2359 1729 y FI(stab)2505 1759 y FO(and)23 b FL(T)7 b FJ(eich)2863 1729 y Fz(C)2930 1759 y FO(ar)l(e)21 b(e)l(quivalent.)456 1858 y(In)29 b(p)l(articular,)i FJ(\031)1016 1870 y FM(1)1054 1858 y FQ(\()p FJ(M)1167 1870 y FI(g)r(;n)1266 1858 y FQ(\))23 b(=)g(\000)1461 1828 y FE(0)1461 1879 y FI(g)r(;n)1560 1858 y FO(.)605 1996 y FP(Pr)n(oof.)41 b FQ(The)30 b(pro)r(of)g(essen)n(tially)g(rep)r (eats)f(the)i(pro)r(of)f(of)h(the)f(fact)h(that)g(for)f(a)g(simply-)456 2096 y(connected)d FJ(T)12 b FQ(,)27 b(one)g(has)g FJ(\031)1301 2108 y FM(1)1339 2096 y FQ(\()p FJ(T)9 b(=)p FQ(\000\))22 b(=)h(\000.)605 2196 y(First)h(w)n(e)g(construct)f(a)h(functor)g FL(T)7 b FJ(eich)1839 2165 y FI(stab)1988 2196 y FL(!)23 b(T)7 b FJ(eich)2298 2165 y Fz(C)2368 2196 y FQ(as)23 b(follo)n(ws.)35 b(Let)24 b(\006)g(b)r(e)h(an)f(ob)5 b(ject)456 2295 y(of)33 b FL(T)7 b FJ(eich)760 2265 y FI(stab)886 2295 y FQ(,)35 b(i.e.,)g(an)f(extended)g(surface.)54 b(W)-7 b(e)34 b(will)g(use)f(De\014nition)i(5.1.10)d(of)h(extended)456 2395 y(surface;)d(th)n(us,)h(\006)f(is)g(a)g(top)r(ological)e(surface)h (with)i(mark)n(ed)e(p)r(oin)n(ts)h(and)g(non-zero)e(tangen)n(t)456 2495 y(v)n(ectors)j(at)j(these)f(p)r(oin)n(ts.)54 b(Fix)34 b(a)f(complex)g(structure)g FJ(\026)g FQ(on)g(the)h(surface)f(\006.)54 b(Let)34 b FJ(C)3281 2507 y FI(\026)3359 2495 y FQ(b)r(e)456 2594 y(the)c(complex)f(curv)n(e)g(obtained)h(from)f(\006)h(with)g(the)h (complex)e(structure)g FJ(\026)p FQ(.)44 b(It)30 b(is)g(a)f(p)r(oin)n (ted)456 2694 y(curv)n(e,)22 b(with)g(the)h(same)e(mark)n(ed)g(p)r(oin) n(ts)h(and)g(tangen)n(t)g(v)n(ectors)e(as)h(\006)i(\(recall)e(that)h(a) g(complex)456 2793 y(structure)31 b(de\014nes)g(an)g FH(R)p FQ(-linear)37 b(isomorphism)30 b(of)i(the)g(real)e(tangen)n(t)h (space)g FJ(T)3027 2763 y Fz(R)3015 2814 y FI(p)3072 2793 y FQ(\006)h(and)f(the)456 2904 y(complex)g(tangen)n(t)h(space)g FJ(T)1380 2874 y Fz(C)1368 2924 y FI(p)1425 2904 y FJ(C)6 b FQ(;)35 b(for)d(example,)h(for)f(\006)f(=)f FH(R)2406 2874 y FM(2)2482 2904 y FQ(and)i(the)h(standard)e(complex)456 3006 y(structure,)f(w)n(e)f(get)h FJ(@)1147 3018 y FI(x)1216 3006 y FL(7!)d FJ(@)1370 3018 y FI(z)1409 3006 y FJ(;)14 b(@)1490 3018 y FI(y)1557 3006 y FL(7!)27 b FQ(i)p FJ(@)1734 3018 y FI(z)1772 3006 y FQ(\).)45 b(This)30 b(construction)f(dep)r (ends)h(on)g(the)g(c)n(hoice)g(of)456 3106 y FJ(\026)p FQ(.)42 b(By)30 b(Theorem)e(6.1.6,)h(the)h(set)g FJ(T)12 b FQ(\(\006\))29 b(of)g(all)g(complex)h(structures)e(on)h(\006)h(is)f (con)n(tractible.)456 3205 y(Therefore,)c(ev)n(ery)g(t)n(w)n(o)h (complex)g(structures)g FJ(\026)1990 3217 y FM(0)2027 3205 y FJ(;)14 b(\026)2114 3217 y FM(1)2178 3205 y FQ(can)26 b(b)r(e)h(connected)f(b)n(y)h(a)f(unique)g(path)456 3305 y FJ(\026)506 3317 y FI(t)564 3305 y FQ(in)j FJ(T)12 b FQ(\(\006\).)42 b(This)29 b(giv)n(es)f(a)g(canonical)g(family)i(of)f (curv)n(es)f FJ(C)2411 3317 y FI(\026)2451 3325 y FG(t)2512 3305 y FQ(connecting)h FJ(C)2988 3317 y FI(\026)3028 3325 y FF(0)3094 3305 y FQ(with)h FJ(C)3344 3317 y FI(\026)3384 3325 y FF(1)3421 3305 y FQ(,)456 3407 y(or)38 b(a)i(canonical)e (morphism)h FJ(C)1486 3419 y FI(\026)1526 3427 y FF(0)1606 3407 y FL(!)k FJ(C)1791 3419 y FI(\026)1831 3427 y FF(1)1908 3407 y FQ(in)d FL(T)7 b FJ(eich)2221 3377 y Fz(C)2267 3407 y FQ(.)73 b(Th)n(us,)42 b(w)n(e)e(ha)n(v)n(e)e(assigned)g(to)i(a) 456 3509 y(top)r(ological)29 b(surface)i FJ(C)38 b FQ(a)31 b(collection)f(of)i(ob)5 b(jects)31 b FJ(C)2153 3521 y FI(\026)2227 3509 y FL(2)f(T)7 b FJ(eich)2516 3479 y Fz(C)2561 3509 y FQ(,)33 b(canonically)d(isomorphic)456 3609 y(to)c(eac)n(h)g(other.)36 b(As)27 b(w)n(as)f(discussed)h(b)r (efore)f(\(see)h(De\014nition)g(1.1.11,)f(Lemma)g(1.1.12\),)g(suc)n(h) 456 3709 y(a)h(collection)g(can)g(b)r(e)h(view)n(ed)f(as)g(an)g(ob)5 b(ject)28 b(of)f FL(T)7 b FJ(eich)2198 3679 y Fz(C)2244 3709 y FQ(.)605 3808 y(This)33 b(de\014nes)h(the)g(functor)f FL(T)7 b FJ(eich)1726 3778 y FI(stab)1884 3808 y FL(!)33 b(T)7 b FJ(eich)2204 3778 y Fz(C)2283 3808 y FQ(on)33 b(the)h(ob)5 b(jects)33 b(of)g FL(T)7 b FJ(eich)3147 3778 y FI(stab)3272 3808 y FQ(.)55 b(T)-7 b(o)456 3908 y(de\014ne)37 b(it)h(on)e(morphisms,)j(w)n(e)e(note)g(that)h(an)n(y)e (homeomorphism)g(of)h(extended)h(surfaces)456 4010 y FJ(f)17 b FQ(:)30 b(\006)687 3963 y FE(\030)659 4010 y FL(\000)-40 b(!)32 b FQ(\006)859 3980 y FE(0)915 4010 y FQ(giv)n(es)g(an)h(iden)n(ti\014cation)f(of)h(the)h(T)-7 b(eic)n(hm)r(\177)-44 b(uller)32 b(spaces)g FJ(f)2757 4022 y FE(\003)2804 4010 y FQ(:)e FJ(T)12 b FQ(\(\006\))3101 3963 y FE(\030)3073 4010 y FL(\000)-39 b(!)31 b FJ(T)12 b FQ(\(\006)3366 3980 y FE(0)3389 4010 y FQ(\).)456 4110 y(Th)n(us,)29 b(for)f(an)n(y)g FJ(\026)e FL(2)f FJ(T)12 b FQ(\(\006\))p FJ(;)i(\026)1405 4080 y FE(0)1453 4110 y FL(2)26 b FJ(T)12 b FQ(\(\006)1687 4080 y FE(0)1710 4110 y FQ(\))29 b(there)g(is)g(a)f(unique)h(path)g(connecting)g FJ(f)3064 4122 y FE(\003)3102 4110 y FJ(\026)g FQ(with)g FJ(\026)3421 4080 y FE(0)456 4209 y FQ(in)e FJ(T)12 b FQ(\(\006)705 4179 y FE(0)728 4209 y FQ(\),)28 b(and)f(th)n(us,)h(a)f (unique)h(path)g(connecting)f FJ(C)2185 4221 y FI(\026)2257 4209 y FQ(with)h FJ(C)2505 4221 y FI(\026)2545 4205 y Fx(0)2600 4209 y FQ(in)g(the)g(mo)r(duli)g(stac)n(k)f FJ(M)9 b FQ(.)605 4312 y(The)29 b(in)n(v)n(erse)e(functor)h FL(T)7 b FJ(eich)1546 4282 y Fz(C)1616 4312 y FL(!)25 b(T)7 b FJ(eich)1928 4282 y FI(stab)2082 4312 y FQ(is)28 b(constructed)g(as)g(follo)n(ws.)39 b(On)28 b(ob)5 b(jects,)456 4411 y(it)29 b(is)f(just)i(the)f(forgetful)g(functor,)g(whic)n(h)f (assigns)g(to)g(a)h(complex)f(curv)n(e)g FJ(C)35 b FQ(the)29 b(underlying)456 4511 y(top)r(ological)36 b(surface.)66 b(T)-7 b(o)37 b(de\014ne)h(it)g(on)g(morphisms,)h(let)f FJ(C)2476 4523 y FI(t)2506 4511 y FJ(;)14 b(t)39 b FL(2)i FQ([0)p FJ(;)14 b FQ(1])36 b(b)r(e)j(a)e(family)h(of)456 4611 y(curv)n(es.)51 b(As)33 b(b)r(efore,)h(let)f(us)g(forget)f(the)h (complex)g(structure)f(and)h(view)g(it)g(as)f(a)h(family)g(of)456 4710 y(extended)e(surfaces.)46 b(Then)31 b(eac)n(h)g(of)g(the)g (surfaces)f(\006)2198 4722 y FI(t)2258 4710 y FQ(is)h(homeomorphic)f (to)h(\006)3072 4722 y FM(0)3109 4710 y FQ(,)h(and)f(the)456 4810 y(homeomorphism)f(is)h(unique)g(if)h(w)n(e)f(additionally)g (require)f(that)h(it)h(dep)r(ends)g(con)n(tin)n(uously)456 4910 y(on)e FJ(t)h FQ(\(this)g(follo)n(ws)e(from)i(the)g(discreteness)e (of)i(the)g(mapping)f(class)g(group\).)45 b(This)30 b(giv)n(es)g(a)456 5016 y(family)d(of)g(homeomorphisms)f FJ(')1513 5028 y FI(t)1552 5016 y FQ(:)h(\006)1662 5028 y FM(0)1751 4969 y FE(\030)1723 5016 y FL(\000)-40 b(!)23 b FQ(\006)1914 5028 y FI(t)1943 5016 y FQ(.)37 b(In)28 b(particular,)e(this)h (de\014nes)g FJ(')3000 5028 y FM(1)3047 5016 y FQ(:)h(\006)3158 5028 y FM(0)3218 5016 y FL(!)23 b FQ(\006)3384 5028 y FM(1)3421 5016 y FQ(.)605 5116 y(It)33 b(is)f(easy)f(to)h(c)n(hec)n(k)f (that)i(the)f(ab)r(o)n(v)n(e)f(t)n(w)n(o)h(functors)g(are)f(in)n(v)n (erse)g(to)h(eac)n(h)f(other)h(and)456 5216 y(are)26 b(compatible)i(with)g(gluing)f(\(see)g(the)h(next)g(section\).)p 3384 5216 4 57 v 3388 5163 50 4 v 3388 5216 V 3437 5216 4 57 v eop %%Page: 141 7 141 144 bop 892 226 a FM(6.2.)29 b(COMP)-5 b(A)n(CTIFICA)g(TION)30 b(OF)f(THE)g(MODULI)f(SP)-5 b(A)n(CE)29 b(AND)g(GLUING)315 b(141)605 425 y FP(Remark)32 b FQ(6.1.13)p FP(.)39 b FQ(Sometimes)g(w)n(e)h(will)f(use)h(an)f(alternativ)n(e)f(de\014nition) i(of)f(p)r(oin)n(ted)456 525 y(curv)n(e.)54 b(Recall)34 b(that)g(extended)g(surface)f(can)h(b)r(e)g(de\014ned)g(in)h(an)n(y)e (of)h(the)g(follo)n(wing)f(three)456 624 y(w)n(a)n(ys:)e(1\))20 b(as)g(a)f(surface)h(with)g(b)r(oundary)f(and)h(a)g(p)r(oin)n(t)g(on)g (eac)n(h)f(b)r(oundary)h(comp)r(onen)n(t;)i(2\))e(as)456 724 y(a)h(surface)g(with)h(b)r(oundary)f(and)g(a)g(parametrization)f (of)i(ev)n(ery)e(b)r(oundary)h(comp)r(onen)n(t;)j(3\))d(as)456 824 y(a)26 b(surface)g(without)i(b)r(oundary)e(but)h(with)h(mark)n(ed)e (p)r(oin)n(ts)g(and)h(non-zero)f(tangen)n(t)g(v)n(ectors.)456 923 y(All)i(these)f(de\014nitions)h(giv)n(e)f(rise)g(to)g(equiv)-5 b(alen)n(t)27 b(group)r(oids)g(\(see)g(Prop)r(osition)f(5.1.8\).)605 1023 y(Similarly)-7 b(,)21 b(in)g(the)f(complex)g(situation)g(w)n(e)f (can)h(use)g(the)h(follo)n(wing)e(de\014nition)h(of)g(p)r(oin)n(ted)456 1123 y(curv)n(e:)45 b(a)31 b(p)r(oin)n(ted)i(curv)n(e)e(is)h(a)g (complex)f(curv)n(e)h(with)g(mark)n(ed)f(p)r(oin)n(ts)h FJ(y)2829 1135 y FI(i)2889 1123 y FQ(and)g(a)g(lo)r(cal)f(pa-)456 1222 y(rameter)40 b FJ(z)818 1234 y FI(i)886 1222 y FQ(near)h(eac)n(h)f (of)h(these)h(p)r(oin)n(ts.)77 b(The)42 b(corresp)r(onding)d(mo)r(duli) j(space)e(\(whic)n(h)456 1338 y(is)35 b(in\014nite-dimensional\))h (will)h(b)r(e)f(denoted)g FL(M)2035 1295 y FM(\()p FE(1)p FM(\))2035 1348 y FI(g)r(;n)2157 1338 y FQ(;)k(similarly)-7 b(,)37 b(one)f(de\014nes)g FL(M)3128 1308 y FM(\()p FE(1)p FM(\))3250 1338 y FQ(,)i(and)456 1441 y(the)k(group)r(oid)g FL(T)7 b FJ(eich)1181 1411 y Fz(C)s FM(\()p FE(1)p FM(\))1344 1441 y FQ(.)82 b(One)42 b(has)g(an)g(ob)n(vious)f(forgetting)h(functor) g FL(T)7 b FJ(eich)3150 1411 y Fz(C)s FM(\()p FE(1)q FM(\))3361 1441 y FL(!)456 1541 y(T)g FJ(eich)660 1511 y Fz(C)714 1541 y FQ(:)32 b(\()p FJ(C)q(;)14 b(y)939 1553 y FI(i)967 1541 y FJ(;)g(z)1043 1553 y FI(i)1070 1541 y FQ(\))41 b FL(7!)g FQ(\()p FJ(C)q(;)14 b(y)1437 1553 y FI(i)1466 1541 y FJ(;)g(v)1543 1553 y FI(i)1570 1541 y FQ(\),)42 b(where)c(the)g(v)n(ector)f FJ(v)2371 1553 y FI(i)2438 1541 y FQ(is)h(de\014ned)g(b)n(y)h FL(h)p FJ(v)3027 1553 y FI(i)3055 1541 y FJ(;)14 b(dz)3174 1553 y FI(i)3201 1541 y FL(i)41 b FQ(=)g(1.)456 1640 y(Since)35 b(the)h(set)f FL(f)p FJ(f)44 b FQ(=)35 b FJ(z)27 b FQ(+)1349 1578 y Fy(P)1436 1665 y FI(n>)p FM(1)1580 1640 y FJ(a)1624 1652 y FI(n)1670 1640 y FJ(z)1713 1610 y FI(n)1793 1640 y FL(j)36 b FJ(f)g FQ(con)n(v)n(erges)25 b(in)j(a)f(neigh)n(b)r(orho)r (o)r(d)f(of)i(0)p FL(g)34 b FQ(is)h(con-)456 1740 y(tractible,)29 b(this)h(functor)f(is)g(an)g(equiv)-5 b(alence.)42 b(Therefore,)29 b(w)n(e)f(can)h(use)h(either)f(de\014nition)h(of)456 1840 y(the)e(T)-7 b(eic)n(hm)r(\177)-44 b(uller)27 b(group)r(oid.)605 1939 y(Finally)-7 b(,)34 b(there)e(exists)f(y)n(et)i(one)e(more)h (de\014nition:)46 b(a)32 b(p)r(oin)n(ted)h(curv)n(e)e(is)h(a)g(top)r (ological)456 2039 y(surface)26 b(\006)h(with)h(a)f(b)r(oundary)-7 b(,)26 b(with)i(a)f(complex)g(structure)g FJ(\026)g FQ(and)g(with)h (parametrizations)456 2138 y FJ(\031)503 2150 y FI(i)540 2138 y FQ(:)i(\()p FJ(@)5 b FQ(\006\))766 2150 y FI(i)828 2138 y FL(!)35 b FJ(S)1002 2108 y FM(1)1074 2138 y FQ(of)f(the)h(b)r (oundary)f(comp)r(onen)n(ts)g(whic)n(h)h(are)e(analytic)h(with)h(resp)r (ect)g(to)456 2238 y(the)30 b(complex)f(structure)g FJ(\026)p FQ(.)42 b(W)-7 b(e)30 b(lea)n(v)n(e)f(it)h(to)f(the)h(reader)e(to)h(c)n (hec)n(k)g(that)h(this)g(de\014nition)g(is)456 2338 y(equiv)-5 b(alen)n(t)27 b(to)g(the)h(t)n(w)n(o)f(previous)g(ones.)837 2507 y FK(6.2.)46 b(Compacti\014cation)31 b(of)h(the)g(mo)s(duli)d (space)j(and)g(gluing)605 2656 y FQ(In)g(this)h(section,)f(w)n(e)g (will)g(de\014ne)h(the)f(gluing)f(functor)h(for)g(the)g(complex)g(T)-7 b(eic)n(hm)r(\177)-44 b(uller)456 2756 y(group)r(oid.)55 b(A)35 b(straigh)n(tforw)n(ard)c(approac)n(h)i(w)n(ould)g(b)r(e)i(to)f (cut)h(from)f(a)g(curv)n(e)f(small)h(disks)456 2856 y(around)e(the)h (mark)n(ed)f(p)r(oin)n(ts,)j(and)e(glue)f(the)i(b)r(oundary)e(circles)h (together)f(\(this)i(w)n(as)e(\014rst)456 2955 y(suggested)27 b(b)n(y)h(V)-7 b(afa,)28 b(see)f([)p FK(V1)p FQ(]\).)39 b(Ho)n(w)n(ev)n(er,)27 b(there)h(is)g(a)f(m)n(uc)n(h)h(b)r(etter)h(w)n (a)n(y)e(of)h(de\014ning)g(the)456 3055 y(gluing,)j(whic)n(h)h(uses)e (the)i(so-called)e(Deligne{Mumford)h(compacti\014cation)f(of)h(the)h (mo)r(duli)456 3155 y(space.)605 3254 y(F)-7 b(ollo)n(wing)20 b([)p FK(DM)p FQ(],)i(let)g(us)f(call)f(a)h(p)r(ossibly)g(singular)e (complex)i(curv)n(e)f FJ(C)27 b FO(stable)22 b FQ(if)g(its)f(only)456 3354 y(singularities)29 b(are)g(ordinary)g(double)h(p)r(oin)n(ts,)i 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FQ(=)e(0,)h(w)n(e)e(get)h(the)h(follo)n(wing) d(result:)50 b(the)34 b(irreducible)456 525 y(comp)r(onen)n(ts)27 b(of)g FJ(D)e FL(\032)p 1189 458 90 4 v 23 w FJ(M)1279 537 y FM(0)p FI(;A)1413 525 y FQ(are)i(giv)n(en)g(b)n(y)970 666 y FJ(I)7 b(r)r(r)31 b(D)25 b FL($)e(f)p FQ(\()p FJ(A)1456 632 y FE(0)1480 666 y FJ(;)14 b(A)1579 632 y FE(00)1621 666 y FQ(\))24 b FL(j)f FJ(A)1785 632 y FE(0)1808 666 y FJ(;)14 b(A)1907 632 y FE(0)q(0)1973 666 y FL(\032)23 b FJ(A;)14 b(A)23 b FQ(=)g FJ(A)2395 632 y FE(0)2437 666 y FL(t)c FJ(A)2573 632 y FE(00)2615 666 y FJ(;)14 b FL(j)p FJ(A)2737 632 y FE(0)2761 666 y FL(j)p FJ(;)g FL(j)p FJ(A)2906 632 y FE(00)2948 666 y FL(j)24 b(\025)e FQ(1)p FL(g)-2710 b FQ(\(6.2.3\))456 813 y(\(here)24 b(\()p FJ(A)759 783 y FE(0)783 813 y FJ(;)14 b(A)882 783 y FE(00)924 813 y FQ(\))26 b(is)e(an)g(unordered)g(pair\).)35 b(The)25 b(corresp)r(onding)e(comp)r(onen)n(t)h(of)h FJ(D)i FQ(is)d(de\014ned)456 913 y(b)n(y)1425 1019 y FJ(D)1496 985 y FM(0)1494 1040 y FI(A)1544 1023 y Fx(0)1566 1040 y FI(;A)1636 1023 y Fx(00)1704 1019 y FQ(=)f FJ(S)5 b FQ(\()p FJ(M)1970 985 y FM(1)1961 1040 y(0)p FI(;A)2064 1023 y Fx(0)2108 1019 y FL(\002)18 b FJ(M)2281 985 y FM(1)2272 1040 y(0)p FI(;A)2375 1023 y Fx(00)2419 1019 y FQ(\);)456 1143 y(in)32 b(other)f(w)n(ords,)h(these)g(are)f(curv)n (es)g(whic)n(h)h(can)f(b)r(e)i(obtained)e(b)n(y)h(iden)n(tifying)g(a)g (p)r(oin)n(t)g(on)456 1243 y FJ(C)521 1213 y FM(\(1\))633 1243 y FL(2)23 b FJ(M)792 1255 y FM(0)p FI(;A)895 1239 y Fx(0)949 1243 y FQ(with)28 b(a)f(p)r(oin)n(t)h(on)f FJ(C)1604 1213 y FM(\(2\))1717 1243 y FL(2)c FJ(M)1876 1255 y FM(0)p FI(;A)1979 1239 y Fx(00)2024 1243 y FQ(,)k(as)g(in)h (Example)f(6.2.3.)605 1343 y(Let)32 b(us)h(sho)n(w)e(ho)n(w)g(suc)n(h)h (a)g(curv)n(e)f(can)h(b)r(e)h(obtained)e(as)h(a)g(limit)h(of)f(a)g (family)g(of)g(non-)456 1442 y(singular)26 b(curv)n(es.)605 1596 y FP(Example)31 b FQ(6.2.4)p FP(.)40 b FQ(Let)d FJ(z)1433 1608 y FM(1)1470 1596 y FJ(;)14 b(:)g(:)g(:)g(;)g(z)1694 1608 y FI(n)1778 1596 y FL(2)39 b FH(C)15 b FQ(,)46 b FJ(v)2035 1608 y FM(1)2072 1596 y FJ(;)14 b(:)g(:)g(:)g(;)g(v)2297 1608 y FI(n)2381 1596 y FL(2)40 b FH(C)2530 1566 y FE(\002)2629 1596 y FQ(b)r(e)e(suc)n(h)f(that)h FJ(z)3178 1608 y FI(i)3244 1596 y FL(6)p FQ(=)h FJ(z)3387 1608 y FI(j)3421 1596 y FQ(.)456 1695 y(Denote)856 1817 y(\()p FH(P)940 1783 y FM(1)976 1817 y FQ(;)14 b FJ(z)1052 1829 y FM(1)1089 1817 y FJ(;)g(:)g(:)g(:)f(;)h(z)1312 1829 y FI(n)1357 1817 y FQ(;)p FJ(v)1420 1829 y FM(1)1457 1817 y FJ(;)g(:)g(:)g(:)g(;)g (v)1682 1829 y FI(n)1727 1817 y FQ(\))24 b(=)e(the)28 b(pro)5 b(jectiv)n(e)27 b(line)g FH(P)2607 1783 y FM(1)1380 1942 y FQ(with)h(the)g(standard)f(co)r(ordinate)f FJ(z)g FL(2)e FH(C)39 b FL([)19 b(1)p FQ(,)1380 2067 y(with)28 b(mark)n(ed)e(p)r(oin)n(ts)i FJ(z)e FQ(=)d FJ(z)2306 2079 y FI(i)2361 2067 y FQ(and)k(tangen)n(t)g(v)n(ectors)f FJ(v)3146 2079 y FI(i)3174 2067 y FJ(@)3218 2079 y FI(z)3257 2067 y FQ(.)456 1944 y(\(6.2.4\))456 2209 y(Ob)n(viously)-7 b(,)26 b(an)n(y)h(curv)n(e)f FJ(C)k FL(2)23 b FJ(M)1488 2221 y FM(0)p FI(;n)1613 2209 y FQ(can)28 b(b)r(e)g(written)f(in)h(suc) n(h)f(form.)605 2309 y(No)n(w,)h(c)n(ho)r(ose)g FJ(z)1126 2278 y FE(0)1122 2329 y FM(1)1172 2309 y FJ(:)14 b(:)g(:)g(;)g(z)1363 2278 y FE(0)1359 2332 y FI(k)1399 2309 y FJ(;)g(a;)g(z)1560 2278 y FE(00)1556 2329 y FM(1)1602 2309 y FJ(;)g(:)g(:)g(:)f(;)h(z)1829 2278 y FE(00)1825 2329 y FI(m)1917 2309 y FQ(suc)n(h)28 b(that)h FJ(z)2329 2278 y FE(0)2325 2330 y FI(i)2377 2309 y FL(6)p FQ(=)24 b FJ(z)2509 2278 y FE(0)2505 2330 y FI(j)2539 2309 y FJ(;)14 b(z)2619 2278 y FE(0)2615 2330 y FI(i)2667 2309 y FL(6)p FQ(=)24 b FJ(a;)14 b(z)2880 2278 y FE(00)2876 2330 y FI(i)2947 2309 y FL(6)p FQ(=)24 b FJ(z)3079 2278 y FE(0)o(0)3075 2330 y FI(j)3121 2309 y FQ(.)39 b(Cho)r(ose)456 2416 y FJ(q)26 b FL(2)d FH(C)651 2386 y FE(\002)741 2416 y FQ(small)k(enough)g(and)g(de\014ne)h(the)g (curv)n(e)f FJ(C)2069 2428 y FI(q)2134 2416 y FQ(b)n(y)867 2657 y FJ(C)926 2669 y FI(t)979 2657 y FQ(=)c(\()p FH(P)1151 2623 y FM(1)1187 2657 y FQ(;)14 b FJ(z)1267 2623 y FE(0)1263 2678 y FM(1)1300 2657 y FJ(;)g(:)g(:)g(:)f(;)h(z)1527 2623 y FE(0)1523 2678 y FI(k)1564 2657 y FJ(;)g(a)k FQ(+)g FJ(q)s(z)1829 2623 y FE(0)o(0)1825 2678 y FM(1)1871 2657 y FJ(;)c(:)g(:)g(:)f(;)h(a)k FQ(+)g FJ(q)s(z)2283 2623 y FE(00)2279 2678 y FI(m)2342 2657 y FQ(;)c FJ(v)2422 2623 y FE(0)2419 2678 y FM(1)2457 2657 y FJ(;)g(:)g(:)g(:)f(;)h(v)2684 2623 y FE(0)2681 2678 y FI(k)2722 2657 y FJ(;)g(q)s(v)2842 2623 y FE(00)2839 2678 y FM(1)2885 2657 y FJ(;)g(:)g(:)g(:)f(;)h(q)s(v) 3152 2623 y FE(0)q(0)3149 2678 y FI(m)3213 2657 y FQ(\))p FJ(:)605 2815 y FQ(Then)32 b(w)n(e)e(claim)i(that)f(the)h(limit)g FJ(C)1768 2827 y FM(0)1835 2815 y FQ(=)c(lim)2044 2827 y FI(q)r FE(!)p FM(0)2194 2815 y FJ(C)2253 2827 y FI(q)2321 2815 y FQ(exists)j(in)p 2654 2748 V 31 w FJ(M)2744 2827 y FM(0)p FI(;k)q FM(+)p FI(m)2948 2815 y FQ(,)h(and)f(is)g(giv)n(en)456 2920 y(b)n(y)h(the)h(singular)e(curv)n(e)h(obtained)g(b)n(y)g(iden)n (tifying)h(the)g(p)r(oin)n(t)f FJ(a)f FL(2)h FJ(C)2750 2890 y FM(\(1\))2872 2920 y FQ(with)h FL(1)e(2)h FJ(C)3332 2890 y FM(\(2\))3421 2920 y FQ(,)456 3020 y(where)1332 3162 y FJ(C)1397 3127 y FM(\(1\))1510 3162 y FQ(=)o(\()p FH(P)1658 3127 y FM(1)1694 3162 y FQ(;)14 b FJ(z)1774 3127 y FE(0)1770 3182 y FM(1)1807 3162 y FJ(;)g(:)g(:)g(:)g(;)g(z)2035 3127 y FE(0)2031 3182 y FI(k)2071 3162 y FQ(;)g FJ(v)2151 3127 y FE(0)2148 3182 y FM(1)2186 3162 y FJ(;)g(:)g(:)g(:)f(;)h(v)2413 3127 y FE(0)2410 3182 y FI(k)2451 3162 y FQ(\))p FJ(;)1332 3307 y(C)1397 3272 y FM(\(2\))1510 3307 y FQ(=)o(\()p FH(P)1658 3272 y FM(1)1694 3307 y FQ(;)g FJ(z)1774 3272 y FE(00)1770 3327 y FM(1)1816 3307 y FJ(;)g(:)g(:)g(:)g(;)g(z)2044 3272 y FE(0)o(0)2040 3327 y FI(m)2102 3307 y FQ(;)g FJ(v)2182 3272 y FE(00)2179 3327 y FM(1)2225 3307 y FJ(;)g(:)g(:)g(:)f(;)h(v)2452 3272 y FE(0)q(0)2449 3327 y FI(m)2513 3307 y FQ(\))p FJ(:)456 3449 y FQ(Sp)r(eaking)30 b(informally)-7 b(,)31 b(one)f(can)g(sa)n(y)g(that)h(as)f FJ(q)h FL(!)d FQ(0,)j(the)g(curv)n (e)f FJ(C)2681 3461 y FI(q)2749 3449 y FQ(lo)r(oks)f(to)i(a)f(bare)g (ey)n(e)456 3554 y(as)36 b FJ(C)632 3524 y FM(\(1\))759 3554 y FQ(and)h FJ(m)h FQ(p)r(oin)n(ts)f FJ(a)25 b FQ(+)f FJ(q)s(z)1541 3524 y FE(00)1537 3576 y FI(i)1621 3554 y FQ(all)37 b(collapsed)f(at)h FJ(a)p FQ(;)43 b(lo)r(oking)36 b(at)h(the)h(p)r(oin)n(t)g FJ(a)f FQ(with)h(a)456 3654 y(microscop)r(e,)26 b(one)h(can)h(separate)e(these)i FJ(m)f FQ(p)r(oin)n(ts)h(and)g(see)f(that)h(their)g(relativ)n(e)e(p)r (osition)i(is)456 3754 y(describ)r(ed)f(b)n(y)g FJ(C)1003 3723 y FM(\(2\))1093 3754 y FQ(.)605 3907 y FP(Sketch)k(of)h(pr)n(oof.) 41 b FQ(F)-7 b(or)30 b(simplicit)n(y)-7 b(,)32 b(w)n(e)e(will)g (disregard)f(the)i(tangen)n(t)f(v)n(ectors)f(and)456 4007 y(will)34 b(tak)n(e)g FJ(a)g FQ(=)g(0.)56 b(First)34 b(of)h(all,)g(recall)f(that)g(the)h(top)r(ology)e(in)p 2577 3940 V 34 w FJ(M)2667 4019 y FI(g)r(;n)2801 4007 y FQ(is)h(de\014ned)h(so)e(that)456 4106 y(ev)n(ery)e(map)i FJ(S)j FL(!)c FJ(M)42 b FQ(is)32 b(con)n(tin)n(uous.)52 b(In)33 b(particular,)g(if)g(one)g(can)f(construct)h(an)f(analytic)456 4206 y(family)27 b FJ(C)768 4218 y FI(q)805 4206 y FJ(;)14 b(q)26 b FL(2)e FJ(U;)j FQ(of)g(curv)n(es)g(o)n(v)n(er)f(a)h(disk)g FJ(U)37 b FQ(in)28 b(the)g FJ(q)s FQ(-plane,)f(then)h(lim)2810 4218 y FI(q)r FE(!)p FM(0)2960 4206 y FJ(C)3019 4218 y FI(q)3079 4206 y FQ(=)23 b FJ(C)3226 4218 y FM(0)3263 4206 y FQ(.)605 4305 y(Let)28 b(us)f(construct)h(suc)n(h)f(a)g(family)h (with)g FJ(C)1984 4317 y FM(0)2049 4305 y FQ(de\014ned)g(ab)r(o)n(v)n (e.)36 b(Let)28 b(\006)f(b)r(e)h(the)g(surface)f(in)456 4405 y FH(P)508 4375 y FM(2)562 4405 y FL(\002)18 b FJ(U)36 b FQ(giv)n(en)27 b(b)n(y)h(the)g(equation)1325 4547 y FJ(uv)e FQ(=)d FJ(q)s(w)1628 4512 y FM(2)1666 4547 y FJ(;)97 b FQ(\()p FJ(u)23 b FQ(:)g FJ(v)j FQ(:)d FJ(w)r FQ(\))h FL(2)g FH(P)2295 4512 y FM(2)2330 4547 y FJ(;)14 b(q)26 b FL(2)e FJ(U)-2110 b FQ(\(6.2.5\))456 4698 y(and)32 b(the)g(mark)n(ed)f(p)r(oin)n(ts)i(giv)n(en)e(b)n(y)h FL(f)p FQ(\(\()p FJ(z)1814 4668 y FE(0)1810 4720 y FI(i)1837 4698 y FQ(\))1869 4668 y FM(2)1938 4698 y FQ(:)f FJ(q)i FQ(:)e FJ(z)2159 4668 y FE(0)2155 4720 y FI(i)2183 4698 y FQ(\))p FJ(;)14 b FQ(\()p FJ(q)s FQ(\()p FJ(z)2399 4668 y FE(00)2395 4720 y FI(i)2441 4698 y FQ(\))2473 4668 y FM(2)2541 4698 y FQ(:)31 b(1)g(:)g FJ(z)2765 4668 y FE(0)o(0)2761 4720 y FI(i)2806 4698 y FQ(\))p FL(g)2880 4668 y FI(n)2880 4720 y(i)p FM(=1)2992 4698 y FQ(.)51 b(Ob)n(viously)-7 b(,)456 4798 y(this)27 b(is)g(a)g(smo)r(oth)g(family) g(of)h(p)r(oin)n(ted)f(curv)n(es)f(o)n(v)n(er)f FJ(U)9 b FQ(,)28 b(with)f(an)g(ob)n(vious)f(pro)5 b(jection)26 b(to)i FJ(U)9 b FQ(;)456 4899 y(de\014ne)31 b FJ(\031)749 4869 y FE(\000)p FM(1)838 4899 y FQ(\()p FJ(q)s FQ(\))f(=)1084 4878 y(~)1065 4899 y FJ(C)1124 4911 y FI(q)1161 4899 y FQ(.)47 b(W)-7 b(e)31 b(claim)g(that)g(for)g FJ(q)g FL(6)p FQ(=)e(0,)i(the)g(curv)n(e)2564 4878 y(~)2545 4899 y FJ(C)2604 4911 y FI(q)2670 4899 y FL(')d FJ(C)2822 4911 y FI(q)2859 4899 y FQ(.)47 b(Explicitly)-7 b(,)32 b(the)456 5007 y(isomorphism)27 b FJ(C)1001 5019 y FI(q)1064 5007 y FL(!)1191 4986 y FQ(~)1172 5007 y FJ(C)1231 5019 y FI(q)1297 5007 y FQ(is)i(giv)n(en)f(b)n(y)h(\()p FJ(z)g FQ(:)c FJ(s)p FQ(\))h FL(7!)f FQ(\()p FJ(u)g FQ(:)h FJ(v)i FQ(:)e FJ(w)r FQ(\))g(=)f(\()p FJ(z)2625 4977 y FM(2)2687 5007 y FQ(:)g FJ(q)s(s)2814 4977 y FM(2)2877 5007 y FQ(:)g FJ(z)t(s)p FQ(\).)41 b(Similarly)-7 b(,)456 5116 y(for)26 b FJ(q)g FQ(=)c(0,)843 5095 y(~)824 5116 y FJ(C)883 5128 y FM(0)947 5116 y FQ(can)k(b)r(e)h(iden)n(ti\014ed)g(with)g FJ(C)1818 5128 y FM(0)1882 5116 y FQ(b)n(y)f FJ( )2050 5128 y FM(1)2111 5116 y FQ(:)d FJ(C)2222 5086 y FM(\(1\))2334 5116 y FL(!)2459 5095 y FQ(~)2440 5116 y FJ(C)2499 5128 y FM(0)2537 5116 y FJ(;)14 b( )2628 5128 y FM(2)2688 5116 y FQ(:)23 b FJ(C)2799 5086 y FM(\(2\))2912 5116 y FL(!)3037 5095 y FQ(~)3018 5116 y FJ(C)3077 5128 y FM(0)3141 5116 y FQ(giv)n(en)j(b)n(y)456 5216 y FJ( )510 5228 y FM(1)556 5216 y FQ(:)j(\()p FJ(z)k FQ(:)c FJ(s)p FQ(\))g FL(7!)g FQ(\()p FJ(z)k FQ(:)c(0)f(:)i FJ(s)p FQ(\))p FJ(;)14 b( )1417 5228 y FM(2)1463 5216 y FQ(:)29 b(\()p FJ(w)j FQ(:)d FJ(s)p FQ(\))h FL(7!)f FQ(\(0)f(:)i FJ(s)f FQ(:)g FJ(w)r FQ(\).)48 b(This)32 b(completes)e(the)i(pro)r(of.) 47 b(W)-7 b(e)p eop %%Page: 144 10 144 147 bop 456 226 a FM(144)445 b(6.)30 b(MODULI)f(SP)-5 b(A)n(CES)28 b(AND)h(COMPLEX)f(MODULAR)h(FUNCTOR)456 425 y FQ(lea)n(v)n(e)23 b(it)j(to)f(the)g(reader)f(to)h(c)n(hec)n(k)f (that)i(the)f(ab)r(o)n(v)n(e)f(construction)g(in)i(fact)f(also)f(giv)n (es)g(correct)456 525 y(tangen)n(t)j(v)n(ectors.)p 3384 525 4 57 v 3388 472 50 4 v 3388 525 V 3437 525 4 57 v 605 694 a(Let)f(us)f(denote)g(b)n(y)g FJ(N)9 b FQ(\()p FJ(D)r FQ(\))26 b(the)g(normal)e(bundle)i(to)f FJ(D)j FQ(in)p 2451 627 90 4 v 25 w FJ(M)2541 706 y FI(g)r(;n)2640 694 y FQ(:)36 b(for)25 b FJ(C)k FL(2)24 b FJ(D)r(;)14 b(N)3166 706 y FI(C)3221 694 y FQ(\()p FJ(D)r FQ(\))24 b(=)456 799 y FJ(T)505 811 y FI(C)p 560 732 V 560 799 a FJ(M)650 811 y FI(g)r(;n)749 799 y FJ(=T)840 811 y FI(C)895 799 y FJ(D)r FQ(;)33 b(and)d(let)i FJ(N)1386 769 y FE(\002)1441 799 y FQ(\()p FJ(D)r FQ(\))g(b)r(e)f(the)h (complemen)n(t)f(to)f(the)i(zero)e(section:)43 b FJ(N)3160 764 y FE(\002)3151 824 y FI(C)3216 799 y FQ(\()p FJ(D)r FQ(\))29 b(=)456 899 y FJ(N)523 911 y FI(C)578 899 y FQ(\()p FJ(D)r FQ(\))19 b FL(n)f(f)p FQ(zero)26 b(section)p FL(g)o FQ(.)605 1056 y FP(Lemma)31 b FQ(6.2.5)p FP(.)40 b FO(If)25 b FJ(C)31 b FO(is)26 b(a)f(stable)g(singular)g(curve)g(with) h(only)f(one)g(double)h(p)l(oint)f FJ(a)p FO(,)i(then)456 1167 y FJ(N)523 1179 y FI(C)578 1167 y FJ(D)40 b FO(is)f (one-dimensional,)j(and)d(c)l(an)f(b)l(e)g(c)l(anonic)l(al)t(ly)j (identi\014e)l(d)e(with)f FJ(T)2937 1124 y FM(\(1\))2925 1177 y FI(a)3026 1167 y FJ(C)30 b FL(\012)24 b FJ(T)3265 1124 y FM(\(2\))3253 1177 y FI(a)3354 1167 y FJ(C)6 b FO(,)456 1278 y(wher)l(e)30 b FJ(T)751 1235 y FM(\(1\))739 1288 y FI(a)839 1278 y FJ(C)6 b FO(,)31 b FJ(T)1021 1235 y FM(\(2\))1009 1288 y FI(a)1109 1278 y FJ(C)36 b FO(ar)l(e)30 b(the)g(tangent)f(sp)l(ac)l(es)h(to)g(the)g(two)g(c)l(omp)l(onents)f (of)i FJ(C)36 b FO(at)30 b FJ(a)p FO(.)605 1436 y FP(Sketch)h(of)h(pr)n (oof.)41 b FQ(By)d(de\014nition,)k(the)c(space)g FJ(N)2376 1448 y FI(C)2431 1436 y FJ(D)j FQ(is)d(the)g(set)h(of)f(equiv)-5 b(alence)456 1535 y(classes)21 b(of)h(one-parameter)e(families)i(of)g (curv)n(es)f FJ(C)2042 1547 y FI(U)2098 1535 y FQ(,)j(de\014ned)e(o)n (v)n(er)f(a)h(disk)g FJ(U)31 b FQ(in)22 b(the)h(complex)456 1635 y(plane,)33 b(suc)n(h)f(that)g FJ(C)1141 1647 y FM(0)1209 1635 y FQ(=)e FJ(C)39 b FQ(and)32 b FJ(C)1627 1647 y FI(q)1696 1635 y FQ(is)g(non-singular)e(for)i FJ(q)h FL(6)p FQ(=)e(0.)50 b(A)32 b(t)n(ypical)g(example)g(of)456 1734 y(suc)n(h)27 b(a)g(family)h(is)f(giv)n(en)g(b)n(y)h(\(6.2.5\).)605 1834 y(Let)19 b FJ(a)k FL(2)g FJ(C)949 1846 y FM(0)1010 1834 y FL(\032)g FJ(C)1157 1846 y FI(U)1232 1834 y FQ(b)r(e)c(the)g (double)g(p)r(oin)n(t.)34 b(Then)19 b(it)g(can)g(b)r(e)g(sho)n(wn)f (that)h(one)g(can)f(alw)n(a)n(ys)456 1934 y(in)n(tro)r(duce)26 b(lo)r(cal)h(co)r(ordinates)e FJ(x)1507 1946 y FM(1)1545 1934 y FJ(;)14 b(x)1629 1946 y FM(2)1693 1934 y FQ(on)27 b(\006)g(near)f FJ(a)h FQ(suc)n(h)f(that)i FJ(x)2563 1946 y FM(1)2600 1934 y FJ(x)2647 1946 y FM(2)2708 1934 y FQ(=)23 b FJ(q)s FQ(;)k(when)g(restricted)456 2033 y(to)21 b FJ(C)610 2045 y FM(0)648 2033 y FQ(,)h(these)g(co)r (ordinates)e(b)r(ecome)h(the)h(lo)r(cal)f(co)r(ordinates)f(on)i(the)g (t)n(w)n(o)e(comp)r(onen)n(ts)h(of)h FJ(C)3384 2045 y FM(0)3421 2033 y FQ(.)456 2144 y(No)n(w,)j(de\014ne)g(the)h(map)f FJ(N)1293 2156 y FI(C)1349 2144 y FJ(D)g FL(!)e FJ(T)1610 2101 y FM(\(1\))1598 2154 y FI(a)1698 2144 y FJ(C)d FL(\012)14 b FJ(T)1917 2101 y FM(\(2\))1905 2154 y FI(a)2005 2144 y FJ(C)31 b FQ(b)n(y)25 b FJ(@)2252 2156 y FI(q)2312 2144 y FL(7!)e FJ(@)2462 2156 y FI(x)2500 2164 y FF(1)2550 2144 y FL(\012)14 b FJ(@)2673 2156 y FI(x)2711 2164 y FF(2)2747 2144 y FQ(.)36 b(W)-7 b(e)25 b(lea)n(v)n(e)f(it)i(to)f(the) 456 2244 y(reader)k(to)i(c)n(hec)n(k)g(that)g(this)h(map)f(do)r(es)g (not)g(dep)r(end)h(on)f(the)h(c)n(hoice)e(of)h(lo)r(cal)g(co)r (ordinates)456 2344 y FJ(t)486 2356 y FM(1)523 2344 y FJ(;)14 b(t)590 2356 y FM(2)627 2344 y FQ(.)p 3384 2344 4 57 v 3388 2291 50 4 v 3388 2344 V 3437 2344 4 57 v 456 2513 a(Informally)-7 b(,)30 b(the)g(family)g FJ(C)1339 2525 y FI(q)1406 2513 y FQ(corresp)r(onding)f(to)h(the)g(v)n(ector)f FJ(v)h FL(2)e FJ(N)2673 2477 y FE(\002)2664 2537 y FI(C)2729 2513 y FJ(D)k FQ(can)e(b)r(e)g(presen)n(ted)456 2612 y(as)d(\\thic)n(k)n(ening")f(of)h(the)h(double)g(p)r(oin)n(t,)f(as)g (sho)n(wn)g(in)h(the)g(\014gure)f(b)r(elo)n(w.)-3223 b Fq(?!)1252 2804 y FQ(THERE)27 b(WILL)g(BE)g(A)h(FIGURE)g(HERE)682 3004 y FP(Figure)k(6.1.)41 b FQ(F)-7 b(amily)27 b(of)h(smo)r(oth)f (curv)n(es)g(con)n(v)n(erging)e(to)i(a)g(singular)g(curv)n(e.)605 3222 y(More)g(generally)-7 b(,)27 b(if)i FJ(C)i FL(2)p 1436 3156 90 4 v 24 w FJ(M)1526 3234 y FI(g)r(;n)1653 3222 y FQ(is)e(a)e(curv)n(e)h(with)h FJ(k)i FQ(double)d(p)r(oin)n(ts)g FJ(a)2856 3234 y FM(1)2893 3222 y FJ(;)14 b(:)g(:)g(:)g(;)g(a)3122 3234 y FI(k)3191 3222 y FQ(\(equiv-)456 3322 y(alen)n(tly)-7 b(,)28 b FJ(C)35 b FQ(lies)29 b(in)f(the)h(in)n(tersection)f(of)h FJ(k)i FQ(comp)r(onen)n(ts)d(of)h FJ(D)r FQ(:)39 b FJ(C)31 b FL(2)25 b FJ(D)2768 3334 y FM(1)2824 3322 y FL(\\)20 b(\001)14 b(\001)g(\001)19 b(\\)g FJ(D)3158 3334 y FI(k)3199 3322 y FQ(\),)29 b(then)456 3424 y FJ(N)523 3436 y FI(C)578 3424 y FJ(D)c FQ(=)e FJ(T)809 3436 y FI(C)p 864 3358 V 864 3424 a FJ(M)954 3436 y FI(g)r(;n)1053 3424 y FJ(=)18 b FL(\\)h FJ(T)1236 3436 y FI(C)1291 3424 y FJ(D)1360 3436 y FI(i)1415 3424 y FQ(is)28 b FJ(k)s FQ(-dimensional,)f(and)g(w)n (e)g(ha)n(v)n(e)g(a)g(canonical)f(isomorphism)1454 3657 y FJ(N)1521 3669 y FI(C)1576 3657 y FJ(D)f FL(')1802 3554 y FI(k)1758 3578 y Fy(M)1767 3755 y FI(i)p FM(=1)1897 3657 y FJ(T)1958 3623 y FM(\(1\))1946 3678 y FI(a)1982 3686 y FG(i)2047 3657 y FJ(C)f FL(\012)18 b FJ(T)2274 3623 y FM(\(2\))2262 3678 y FI(a)2298 3686 y FG(i)2362 3657 y FJ(C)q(:)-1989 b FQ(\(6.2.6\))605 3878 y(Using)33 b(this)g(lemma,)i(w)n(e)d(can)h(no)n(w)g(de\014ne)g(the)g(gluing)g (functor)g(for)g(the)g(complex)g(T)-7 b(e-)456 3978 y(ic)n(hm)r(\177) -44 b(uller)27 b(group)r(oid.)36 b(This)27 b(is)h(done)f(in)h(t)n(w)n (o)f(steps.)605 4077 y(First,)32 b(let)f FJ(A)g FQ(b)r(e)h(a)e (\014nite)i(set,)g FJ(\013;)14 b(\014)33 b FL(2)c FJ(A)p FQ(|an)i(unordered)f(pair.)46 b(Then)31 b(w)n(e)g(de\014ne)g(the)456 4177 y(\\clutc)n(hing")26 b(map)1185 4323 y FJ(S)1236 4335 y FI(\013\014)1333 4323 y FQ(:)i FJ(M)1465 4335 y FE(\003)p FI(;A)1595 4323 y FL(!)23 b FJ(N)9 b FQ(\()p FJ(D)1880 4289 y FM(0)1918 4323 y FQ(\))p FJ(;)97 b(D)2141 4289 y FM(0)2201 4323 y FL(\032)p 2289 4256 V 23 w FJ(M)2378 4338 y FE(\003)p FI(;A)p FE(nf)p FI(\013;\014)s FE(g)2692 4323 y FJ(;)1458 4457 y(C)1523 4423 y FE(_)1595 4457 y FL(7!)23 b FQ(\()p FJ(C)q(;)14 b(v)s FQ(\))p FJ(;)456 4391 y FQ(\(6.2.7\))456 4615 y(where)25 b FJ(C)30 b FL(2)23 b FJ(D)932 4585 y FM(0)996 4615 y FQ(is)j(the)h(singular)e(curv)n(e)g (obtained)h(b)n(y)g(iden)n(tifying)h(the)f(mark)n(ed)f(p)r(oin)n(ts)i FJ(\013;)14 b(\014)456 4715 y FQ(of)23 b FJ(C)611 4685 y FE(_)660 4715 y FQ(,)i(and)f FJ(v)i FQ(=)d FJ(v)1060 4727 y FI(\013)1118 4715 y FL(\012)11 b FJ(v)1234 4727 y FI(\014)1301 4715 y FL(2)24 b FJ(T)1429 4727 y FI(\013)1475 4715 y FJ(C)1540 4685 y FE(_)1600 4715 y FL(\012)11 b FJ(T)1725 4727 y FI(\014)1769 4715 y FJ(C)1834 4685 y FE(_)1906 4715 y FL(')23 b FJ(N)2061 4727 y FI(C)2116 4715 y FQ(\()p FJ(D)r FQ(\).)37 b(The)23 b(map)h FJ(S)k FQ(is)c(a)f FH(C)2936 4685 y FE(\002)2998 4715 y FQ(-bundle)h(o)n(v)n (er)456 4815 y FJ(N)9 b FQ(\()p FJ(D)635 4785 y FM(0)672 4815 y FQ(\).)34 b(W)-7 b(e)19 b(will)g(also)f(denote)g(b)n(y)g FJ(S)1616 4827 y FI(\013;\014)1743 4815 y FQ(the)h(corresp)r(onding)d (functor)j(b)r(et)n(w)n(een)f(fundamen)n(tal)456 4914 y(group)r(oids:)1453 5062 y FJ(S)1504 5074 y FI(\013;\014)1621 5062 y FQ(:)27 b FL(T)7 b FJ(eich)1875 5028 y Fz(C)1875 5083 y FI(A)1952 5062 y FL(!)23 b FJ(\031)2105 5074 y FM(1)2143 5062 y FQ(\()p FJ(N)2251 5028 y FE(\002)2307 5062 y FJ(D)r FQ(\))p FJ(;)456 5216 y FQ(where)k FJ(\031)743 5228 y FM(1)780 5216 y FQ(\()p FJ(X)7 b FQ(\))28 b(denotes)f(the)h (fundamen)n(tal)g(group)r(oid)e(of)i FJ(X)7 b FQ(.)p eop %%Page: 145 11 145 148 bop 892 226 a FM(6.2.)29 b(COMP)-5 b(A)n(CTIFICA)g(TION)30 b(OF)f(THE)g(MODULI)f(SP)-5 b(A)n(CE)29 b(AND)g(GLUING)315 b(145)605 425 y FQ(The)40 b(second)g(step)g(is)g(to)g(pass)f(from)h FJ(N)1945 395 y FE(\002)2001 425 y FJ(D)46 b FL(\032)p 2224 359 90 4 v 43 w FJ(M)j FQ(to)40 b FJ(M)9 b FQ(.)74 b(Cho)r(ose)39 b(some)h(tubular)456 525 y(neigh)n(b)r(orho)r(o)r(d)26 b FJ(N)1044 537 y FI(")1107 525 y FQ(of)h FJ(D)j FQ(in)e FJ(N)9 b FQ(\()p FJ(D)r FQ(\),)28 b(and)g(a)f FJ(C)1955 495 y FE(1)2053 525 y FQ(em)n(b)r(edding)1745 662 y FJ(i)9 b FQ(:)28 b FJ(N)1901 674 y FI(")1959 662 y FL(!)p 2065 595 V 23 w FJ(M)-1690 b FQ(\(6.2.8\))456 802 y(suc)n(h)39 b(that)g FJ(i)g FQ(is)g(iden)n(tit)n(y)h(on)f(the)g(normal)f(bundles)i (\(note)f(that)h(the)g(normal)e(bundle)i(to)456 901 y FJ(D)f FQ(in)e FJ(N)737 913 y FI(")810 901 y FQ(is)g(canonically)f (iden)n(ti\014ed)h(with)h FJ(N)9 b FQ(\()p FJ(D)r FQ(\)\).)66 b(Suc)n(h)38 b(a)e(map)h(exists;)42 b(moreo)n(v)n(er,)37 b(it)456 1001 y(can)26 b(b)r(e)h(sho)n(wn)f(that)h(the)g(set)g(of)g (all)f(suc)n(h)h(maps)f(is)h(con)n(tractible.)35 b(Restricting)27 b(this)g(map)g(to)456 1100 y FJ(N)532 1070 y FE(\002)523 1121 y FI(")587 1100 y FQ(\()p FJ(D)r FQ(\))d(=)f FJ(N)901 1112 y FI(")941 1100 y FL(n)5 b FJ(D)r FQ(,)23 b(w)n(e)d(get)h(a)g(w)n (ell-de\014ned)f(functor)h(b)r(et)n(w)n(een)g(the)h(fundamen)n(tal)f (group)r(oids:)1322 1239 y FJ(i)9 b FQ(:)27 b FJ(\031)1457 1251 y FM(1)1495 1239 y FQ(\()p FJ(N)1603 1204 y FE(\002)1594 1259 y FI(")1659 1239 y FQ(\()p FJ(D)r FQ(\)\))d FL(!)f FJ(\031)2003 1251 y FM(1)2041 1239 y FQ(\()p FJ(M)9 b FQ(\))23 b(=)g FL(T)7 b FJ(eich)2510 1204 y Fz(C)2555 1239 y FJ(:)456 1382 y FQ(Since)33 b(the)h(em)n(b)r(edding)f FJ(N)1328 1352 y FE(\002)1319 1403 y FI(")1384 1382 y FQ(\()p FJ(D)r FQ(\))h FL(!)e FJ(N)1744 1352 y FE(\002)1800 1382 y FJ(D)k FQ(is)d(a)g(homotop)n(y)f(equiv)-5 b(alence,)35 b FJ(\031)2978 1394 y FM(1)3015 1382 y FQ(\()p FJ(N)3123 1352 y FE(\002)3114 1403 y FI(")3179 1382 y FQ(\()p FJ(D)r FQ(\)\))f FL(')456 1482 y FJ(\031)503 1494 y FM(1)540 1482 y FQ(\()p FJ(N)648 1451 y FE(\002)704 1482 y FJ(D)r FQ(\).)k(Th)n(us,)27 b(w)n(e)g(can)g(view)h FJ(i)f FQ(as)g(a)g(functor) 1529 1620 y FJ(i)9 b FQ(:)27 b FJ(\031)1664 1632 y FM(1)1702 1620 y FQ(\()p FJ(N)1810 1586 y FE(\002)1866 1620 y FJ(D)r FQ(\))c FL(!)g(T)7 b FJ(eich)2302 1586 y Fz(C)2348 1620 y FJ(:)-1915 b FQ(\(6.2.9\))605 1755 y(No)n(w,)28 b(let)h(us)f (de\014ne)h(the)g(gluing)f(functor)g(for)g(the)h(complex)f(T)-7 b(eic)n(hm)r(\177)-44 b(uller)28 b(group)r(oid)f(as)456 1854 y(the)h(comp)r(osition)1103 2018 y FJ(F)1156 2030 y FI(\013;\014)1273 2018 y FQ(:)g FL(T)7 b FJ(eich)1528 1984 y Fz(C)1528 2039 y FI(A)1633 1962 y(S)1674 1971 y FG(\013;\014)1604 2018 y FL(\000)-32 b(\000)-19 b(\000)-33 b(!)23 b FJ(\031)1868 2030 y FM(1)1905 2018 y FQ(\()p FJ(N)2013 1984 y FE(\002)2069 2018 y FJ(D)r FQ(\))2230 1971 y FI(i)2196 2018 y FL(\000)-56 b(!)23 b(T)7 b FJ(eich)2515 1984 y Fz(C)2515 2041 y FI(A)p FE(nf)p FI(\013;\014)s FE(g)2774 2018 y FJ(:)-2341 b FQ(\(6.2.10\))605 2161 y(Note)34 b(that)h(it)f(is)g(de\014ned)h(only)e(for)h(those)g(curv)n (es)e FJ(C)41 b FL(2)34 b FJ(M)2547 2173 y FE(\003)p FI(;A)2688 2161 y FQ(for)g(whic)n(h)g FJ(S)3117 2173 y FI(\013;\014)3225 2161 y FQ(\()p FJ(C)6 b FQ(\))35 b(is)456 2260 y(stable.)605 2409 y FP(Example)c FQ(6.2.6)p FP(.)40 b FQ(Let)g(us)g(describ)r(e)g(the)g(gluing)g(map)f(for)h(gen)n (us)f(zero.)73 b(Let)40 b FJ(A)3312 2378 y FE(0)3380 2409 y FQ(=)456 2508 y FL(f1)581 2478 y FE(0)603 2508 y FJ(;)14 b FQ(1)682 2478 y FE(0)705 2508 y FJ(;)g(:)g(:)g(:)g(;)g(k) 936 2478 y FE(0)959 2508 y FJ(;)g(a)p FL(g)p FJ(;)g(A)1181 2478 y FE(0)o(0)1246 2508 y FQ(=)22 b FL(f1)1458 2478 y FE(00)1500 2508 y FJ(;)14 b FQ(1)1579 2478 y FE(00)1621 2508 y FJ(;)g(:)g(:)g(:)g(;)g(m)1879 2478 y FE(00)1921 2508 y FL(g)p FQ(.)36 b(Then)28 b(the)g(gluing)f(map)1380 2643 y FJ(F)1433 2655 y FI(a;)p FE(1)1555 2639 y Fx(0)q(0)1610 2643 y FQ(:)g FJ(M)1741 2655 y FM(0)p FI(;A)1844 2639 y Fx(0)1889 2643 y FL(\002)18 b FJ(M)2053 2655 y FM(0)p FI(;A)2156 2639 y Fx(0)o(0)2223 2643 y FL(!)23 b FJ(M)2410 2655 y FM(0)p FI(;B)456 2777 y FQ(where)36 b FJ(B)42 b FQ(=)c(\()p FJ(A)1007 2747 y FE(0)1056 2777 y FL(t)25 b FJ(A)1198 2747 y FE(00)1240 2777 y FQ(\))g FL(n)f(f)p FJ(a;)14 b FL(1)1569 2747 y FE(00)1611 2777 y FL(g)38 b FQ(=)g FL(f1)1919 2747 y FE(0)1941 2777 y FJ(;)14 b FQ(1)2020 2747 y FE(0)2043 2777 y FJ(;)g(:)g(:)g(:)g(;)g(k)2274 2747 y FE(0)2297 2777 y FJ(;)g FQ(1)2376 2747 y FE(0)o(0)2418 2777 y FJ(;)g(:)g(:)g(:)f(;)h(m)2675 2747 y FE(00)2717 2777 y FL(g)p FQ(,)39 b(can)e(b)r(e)g(describ)r(ed)456 2880 y(explicitly)27 b(as)g(follo)n(ws.)36 b(Cho)r(ose)27 b(for)g(an)n(y)g FJ(C)1860 2850 y FM(\(1\))1972 2880 y FL(2)d FJ(M)2132 2892 y FM(0)p FI(;A)2235 2876 y Fx(0)2288 2880 y FQ(a)j(presen)n(tation)g(in)h(the)g(form)1155 3025 y FJ(C)1220 2990 y FM(\(1\))1333 3025 y FQ(=)22 b(\()p FH(P)1504 2990 y FM(1)1541 3025 y FQ(;)14 b FL(1)p FJ(;)g(z)1741 2990 y FE(0)1737 3045 y FM(1)1773 3025 y FJ(;)g(:)g(:)g(:)g(;)g(z)2001 2990 y FE(0)1997 3045 y FI(k)2037 3025 y FJ(;)g(a)p FQ(;)g FJ(v)2195 3037 y FE(1)2266 3025 y FJ(;)g(v)2346 2990 y FE(0)2343 3045 y FM(1)2380 3025 y FJ(;)g(:)g(:)g(:)g(;)g(v)2608 2990 y FE(0)2605 3045 y FI(k)2646 3025 y FJ(;)g(t)p FQ(\))456 3159 y(as)39 b(in)h(\(6.2.4\))o(,)j(where)c(the)i(tangen)n(t)e(v)n (ector)f(at)i FL(1)g FQ(is)g(giv)n(en)f(b)n(y)g FJ(v)2692 3171 y FE(1)2806 3159 y FQ(=)k FL(\000)p FJ(@)3023 3174 y FM(1)p FI(=z)3128 3159 y FQ(.)74 b(\(More)456 3268 y(formally)-7 b(,)26 b(c)n(ho)r(ose)g(a)h(section)h(of)f(the)h(pro)5 b(jection)26 b FJ(X)2109 3280 y FI(k)q FM(+1)2257 3268 y FL(!)d FJ(M)2444 3280 y FM(0)p FI(;A)2547 3263 y Fx(0)2573 3268 y FQ(,)k(where)g FJ(X)2932 3280 y FI(k)q FM(+1)3080 3268 y FQ(=)c(\()p FH(C)3254 3237 y FI(k)r FM(+1)3403 3268 y FL(n)456 3370 y FQ(diagonals)n(\))f FL(\002)f FQ(\()p FH(C)1020 3340 y FE(\002)1082 3370 y FQ(\))1114 3340 y FI(k)q FM(+1)1239 3370 y FQ(.\))53 b(Do)32 b(the)h(same)g(for)f FJ(M)2059 3382 y FM(0)p FI(;A)2162 3366 y Fx(0)o(0)2206 3370 y FQ(.)53 b(Then)32 b(a)h(simple)g(generalization)e(of)456 3470 y(the)d(argumen)n(ts)e(of)h(Example)g(6.2.4)g(sho)n(ws)f(that)i (the)g(gluing)f(functor)g FJ(F)2792 3482 y FI(a;)p FE(1)2914 3466 y Fx(0)q(0)2987 3470 y FQ(is)h(giv)n(en)f(b)n(y)1066 3614 y FJ(C)1131 3580 y FM(\(1\))1239 3614 y FL(t)19 b FJ(C)1378 3580 y FM(\(2\))1491 3614 y FL(7!)k FQ(\()p FH(P)1681 3580 y FM(1)1717 3614 y FQ(;)p FL(1)p FJ(;)14 b(z)1903 3580 y FE(0)1899 3635 y FM(1)1936 3614 y FJ(;)g(:)g(:)g(:)f(;) h(z)2163 3580 y FE(0)2159 3635 y FI(k)2200 3614 y FJ(;)g(a)k FQ(+)g FJ(tz)2455 3580 y FE(0)o(0)2451 3635 y FM(1)2497 3614 y FJ(;)c(a)k FQ(+)g FJ(tz)2752 3580 y FE(0)o(0)2748 3635 y FI(m)2811 3614 y FQ(;)1740 3748 y FJ(v)1783 3714 y FE(0)1780 3769 y(1)1851 3748 y FJ(;)c(v)1931 3714 y FE(0)1928 3769 y FM(1)1965 3748 y FJ(;)g(:)g(:)g(:)g(;)g(v)2193 3714 y FE(0)2190 3769 y FI(k)2231 3748 y FJ(;)g(tv)2341 3714 y FE(00)2338 3769 y FM(1)2383 3748 y FJ(;)g(:)g(:)g(:)g(;)g(tv) 2641 3714 y FE(00)2638 3769 y FI(m)2701 3748 y FQ(\))p FJ(:)456 3680 y FQ(\(6.2.11\))456 3883 y(This)25 b(map)f(is)h(w)n(ell)g (de\014ned)g(only)g(for)f(small)h(enough)f FJ(t)p FQ(,)i(and)f(dep)r (ends)g(on)g(the)g(c)n(hoice)f(of)h(pre-)456 3983 y(sen)n(tation)i(of)h FJ(C)974 3953 y FM(\(2\))1091 3983 y FQ(in)g(the)g(form)g(\(6.2.4\))o (;)g(ho)n(w)n(ev)n(er,)e(the)i(induced)h(functor)f(of)f(fundamen)n(tal) 456 4082 y(group)r(oids)f(is)h(w)n(ell)h(de\014ned)g(up)g(to)f(a)g (unique)h(isomorphism.)605 4231 y(The)i(gluing)g(op)r(eration)f (satis\014es)h(the)g(asso)r(ciativit)n(y)f(prop)r(ert)n(y)g(form)n (ulated)h(in)g(De\014ni-)456 4330 y(tion)21 b(5.6.1,)g(i.e.,)i(for)d (distinct)i FJ(\013;)14 b(\014)t(;)g(\015)5 b(;)14 b(\016)26 b FL(2)e FJ(A)p FQ(,)e(the)g(functors)f FJ(F)2419 4342 y FI(\013;\014)2527 4330 y FJ(F)2580 4342 y FI(\015)t(;\016)2675 4330 y FJ(;)14 b(F)2765 4342 y FI(\015)t(;\016)2860 4330 y FJ(F)2913 4342 y FI(\013;\014)3030 4330 y FQ(:)28 b FL(T)7 b FJ(eich)3285 4300 y Fz(C)3285 4353 y FI(A)3361 4330 y FL(!)456 4432 y(T)g FJ(eich)660 4402 y Fz(C)660 4455 y FI(A)710 4439 y Fx(0)736 4432 y FQ(,)28 b(where)f FJ(A)1089 4402 y FE(0)1136 4432 y FQ(=)c FJ(A)c FL(n)g(f)p FJ(\013;)14 b(\014)t(;)g(\015)5 b(;)14 b(\016)s FL(g)p FQ(,)27 b(are)g(canonically)f(isomorphic.)37 b(The)28 b(pro)r(of)f(of)h(this)456 4532 y(fact)21 b(can)f(b)r(e)h(obtained)g (from)g(noting)f(that)h(eac)n(h)f(of)h(them)h(is)e(isomorphic)g(to)h (the)g(comp)r(osition)1031 4667 y FJ(\031)1078 4679 y FM(1)1116 4667 y FQ(\()p FJ(M)1229 4679 y FE(\003)p FI(;A)1337 4667 y FQ(\))i FL(!)g FJ(\031)1545 4679 y FM(1)1583 4667 y FQ(\()p FJ(N)1691 4632 y FE(\002)1747 4667 y FQ(\()p FJ(D)1850 4632 y FM(1)1887 4667 y FQ(\)\))h FL(!)f FJ(\031)2128 4679 y FM(1)2166 4667 y FQ(\()p FJ(M)2279 4679 y FE(\003)p FI(;A)2383 4662 y Fx(0)2409 4667 y FQ(\))g(=)g FL(T)7 b FJ(eich)2756 4632 y Fz(C)2756 4687 y FI(A)2806 4670 y Fx(0)2832 4667 y FJ(;)456 4817 y FQ(where)34 b FJ(D)774 4787 y FM(1)846 4817 y FQ(is)g(the)h(strata)f(of)h(the)g(b)r(oundary)f FJ(D)j FQ(=)p 2172 4750 V 34 w FJ(M)2262 4829 y FE(\003)p FI(;A)2366 4813 y Fx(0)2415 4817 y FL(n)23 b FJ(M)2561 4829 y FE(\003)p FI(;A)2665 4813 y Fx(0)2725 4817 y FQ(consisitng)34 b(of)h(curv)n(es)456 4917 y(with)29 b(t)n(w)n(o)e(double)i(p)r(oin)n (ts,)f(and)g(the)h(\014rst)f(arro)n(w)f(is)h(giv)n(en)g(b)n(y)g(iden)n (tifying)g(the)h(p)r(oin)n(ts)g FJ(\013)24 b FL($)456 5016 y FJ(\014)t(;)14 b(\015)44 b FL($)39 b FJ(\016)h FQ(of)e FJ(C)6 b FQ(,)40 b(th)n(us)d(pro)r(ducing)g(a)g(curv)n(e)f (with)i(t)n(w)n(o)e(double)i(p)r(oin)n(ts,)h(and)e(taking)g(the)456 5116 y(normal)26 b(v)n(ector)g(to)h(b)r(e)g(\()p FJ(v)1271 5128 y FI(\013)1337 5116 y FL(\012)17 b FJ(v)1459 5128 y FI(\014)1504 5116 y FQ(\))h FL(\012)f FQ(\()p FJ(v)1708 5128 y FI(\015)1769 5116 y FL(\012)g FJ(v)1891 5128 y FI(\016)1928 5116 y FQ(\))28 b(\(see)f(\(6.2.6\))o(\).)37 b(The)27 b(second)g(map)g(is)g(de\014ned)456 5216 y(as)g(in)g (\(6.2.9\).)37 b(The)27 b(details)h(are)e(left)i(to)g(the)g(reader.)p eop %%Page: 146 12 146 149 bop 456 226 a FM(146)445 b(6.)30 b(MODULI)f(SP)-5 b(A)n(CES)28 b(AND)h(COMPLEX)f(MODULAR)h(FUNCTOR)605 425 y FQ(It)24 b(is)f(easy)f(to)i(c)n(hec)n(k)e(that)i(the)f(gluing)g (op)r(eration)f(is)i(also)e(compatible)h(with)h(the)f(disjoin)n(t)456 525 y(union)d(and)g(empt)n(y)h(set.)34 b(Th)n(us,)22 b(the)f(group)r(oid)e FL(T)7 b FJ(eich)2148 495 y Fz(C)2214 525 y FQ(is)21 b(a)f(to)n(w)n(er)f(of)h(group)r(oids)f(in)i(the)g (sense)456 624 y(of)g(De\014nition)h(5.6.1.)34 b(It)21 b(is)h(also)e(easy)h(to)g(v)n(erify)g(that)g(the)h(equiv)-5 b(alence)21 b FL(T)7 b FJ(eich)2917 594 y FI(stab)3066 624 y FL(!)23 b(T)7 b FJ(eich)3376 594 y Fz(C)3421 624 y FQ(,)456 724 y(constructed)31 b(in)h(Theorem)f(6.1.12,)h(iden)n (ti\014es)g(this)g(gluing)f(op)r(eration)g(with)i(the)f(gluing)f(in)456 826 y FL(T)7 b FJ(eich)p FQ(.)36 b(Th)n(us,)27 b FL(T)7 b FJ(eich)1156 796 y FI(stab)1333 779 y FE(\030)1305 826 y FL(\000)-40 b(!)23 b(T)7 b FJ(eich)1640 796 y Fz(C)1714 826 y FQ(as)27 b(to)n(w)n(ers)f(of)h(group)r(oids.)605 926 y(The)f(construction)f(of)h(gluing)f(ab)r(o)n(v)n(e)g(requires)f (that)i(all)g(the)g(curv)n(es)f(w)n(e)g(use)h(\(including)456 1025 y(the)c(singular)e(ones\))h(b)r(e)h(stable;)h(otherwise,)f(the)g (mo)r(duli)g(spaces)f(of)g(curv)n(es)g(are)f(not)i(stac)n(ks)e(in)456 1125 y(the)26 b(sense)g(of)h([)p FK(DM)o FQ(],)g(whic)n(h)g(mak)n(es)e (life)i(m)n(uc)n(h)f(more)f(di\016cult.)38 b(In)26 b(particular,)g(w)n (e)g(can)f(not)456 1225 y(de\014ne)32 b(the)g(gluing)f FJ(M)1181 1237 y FM(0)p FI(;)p FM(1)1292 1225 y FL(\002)21 b FJ(M)1459 1237 y FI(g)r(;n)1588 1225 y FL(!)30 b FJ(M)1782 1237 y FI(g)r(;n)p FE(\000)p FM(1)1997 1225 y FQ(b)r(ecause)i FJ(M)2390 1237 y FM(0)p FI(;)p FM(1)2511 1225 y FQ(is)g(not)f(a)h (DM-stac)n(k.)48 b(Note,)456 1324 y(ho)n(w)n(ev)n(er,)41 b(that)g(in)f(the)h(top)r(ological)e(approac)n(h)f(the)j(group)r(oid)e FL(T)7 b FJ(eich)2797 1336 y FM(0)p FI(;)p FM(1)2927 1324 y FQ(is)40 b(trivial)g(\(i.e.,)456 1424 y(equiv)-5 b(alen)n(t)35 b(to)h(the)g(group)e(with)j(one)e(elemen)n(t\),)j(and)e (the)g(op)r(eration)f(of)g(gluing)g FL(T)7 b FJ(eich)24 b FL(\002)456 1524 y(T)7 b FJ(eich)660 1536 y FM(0)p FI(;)p FM(1)772 1524 y FQ(coincides)23 b(with)h(the)f(op)r(eration)f (of)h(erasing)f(a)h(mark)n(ed)f(p)r(oin)n(t)h(\(or)f(patc)n(hing)h(a)g (hole,)456 1623 y(dep)r(ending)36 b(on)g(what)f(de\014nition)i(of)f(an) f(extended)h(surface)g(w)n(as)e(used\).)63 b(This)36 b(op)r(eration)456 1723 y(is)e(also)f(w)n(ell-de\014ned)g(as)h(a)f (functor)h FL(T)7 b FJ(eich)1862 1693 y Fz(C)1862 1746 y FI(A)1950 1723 y FL(!)34 b(T)7 b FJ(eich)2271 1693 y Fz(C)2271 1750 y FI(A)p FE(n)p FI(\013)2435 1723 y FQ(in)35 b(the)f(complex)g(T)-7 b(eic)n(hm)r(\177)-44 b(uller)456 1830 y(group)r(oid.)1057 2024 y FK(6.3.)46 b(Connections)31 b(with)g(regular)h(singularities)605 2173 y FQ(In)g(this)h(section,)g(w)n(e)f(brie\015y)g(giv)n(e)f(the)i (main)f(de\014nitions)h(and)f(results)f(regarding)f(\015at)456 2273 y(connections)19 b(with)i(regular)d(singularities.)33 b(This)21 b(will)f(b)r(e)h(used)f(in)g(the)h(next)f(section)g(to)g (de\014ne)456 2372 y(the)28 b(mo)r(dular)f(functor)g(in)h(terms)g(of)f (connections)g(on)g(the)i(mo)r(duli)f(spaces)e(of)i(curv)n(es.)36 b(Most)456 2472 y(of)22 b(these)h(results)f(are)g(due)h(to)g(Deligne)g (and)f(can)h(b)r(e)g(found)g(in)g([)p FK(De1)p FQ(])g(or)f(in)h(the)g (review)f([)p FK(Ma)p FQ(].)456 2572 y(W)-7 b(e)29 b(assume)f(that)h (the)g(reader)e(is)h(familiar)g(with)i(basic)e(geometric)f(notions)h (suc)n(h)h(as)f(v)n(ector)456 2671 y(bundles,)35 b(shea)n(v)n(es,)f (and)g FL(O)r FQ(-mo)r(dules)g(\(as)f(usual,)j(w)n(e)d(denote)h(b)n(y)g FL(O)i FQ(the)e(structure)g(sheaf,)456 2771 y(i.e.,)23 b(the)g(sheaf)g(of)f(germs)g(of)g(analytic)g(functions)h(on)g FJ(M)9 b FQ(\).)35 b(As)23 b(b)r(efore,)g(the)g(w)n(ord)e(\\manifold") 456 2870 y(stands)31 b(for)g(complex)h(analytic)f(manifold,)i(and)f(v)n (ector)e(bundles)i(are)f(holomorphic)g(v)n(ector)456 2970 y(bundles,)d(etc.)37 b(The)27 b(notation)g FJ(s)c FL(2)h(F)35 b FQ(means)27 b(that)h FJ(s)g FQ(is)f(a)h(lo)r(cal)f (section)g(of)g(the)h(sheaf)g FL(F)8 b FQ(.)605 3070 y(Let)32 b FJ(M)40 b FQ(b)r(e)32 b(a)g(manifold.)49 b(By)32 b(de\014nition,)h(a)e FO(lo)l(c)l(al)k(system)c FQ(on)h FJ(M)40 b FQ(is)32 b(a)f(represen)n(tation)456 3169 y(of)e(the)h(P)n (oincar)n(\023)-39 b(e)26 b(group)r(oid)i(of)h FJ(M)9 b FQ(.)42 b(It)30 b(is)f(w)n(ell-kno)n(wn)f(that)h(this)h(is)f(the)h (same)f(as)f(a)h(lo)r(cally)456 3269 y(constan)n(t)d(sheaf)i(of)f(v)n (ector)g(spaces)f(on)i FJ(M)9 b FQ(.)605 3369 y(A)23 b(con)n(v)n(enien)n(t)e(w)n(a)n(y)g(of)h(constructing)g(lo)r(cal)f (systems)h(on)g(a)g(manifold)g(is)g(giv)n(en)g(b)n(y)g(v)n(ector)456 3468 y(bundles)29 b(with)h(\015at)f(connections.)41 b(Recall)29 b(that)h(a)e FO(c)l(onne)l(ction)i FQ(in)f(a)g(v)n(ector)f(bundle)i FJ(E)k FQ(o)n(v)n(er)456 3568 y FJ(M)i FQ(is)27 b(a)h(morphism)f(of)g (shea)n(v)n(es)1652 3711 y FL(r)9 b FQ(:)28 b FL(E)i(!)23 b(E)j(\012)18 b FQ(\012)2174 3676 y FM(1)2211 3711 y FJ(;)-1778 b FQ(\(6.3.1\))456 3854 y(suc)n(h)27 b(that)1093 3997 y FL(r)p FQ(\()p FJ(sf)9 b FQ(\))23 b(=)g(\()p FL(r)p FJ(s)p FQ(\))p FJ(f)28 b FQ(+)18 b FJ(s)g FL(\012)g FJ(d)-14 b(f)t(;)264 b(s)23 b FL(2)g(E)7 b FJ(;)14 b(f)32 b FL(2)23 b(O)2696 4009 y FI(M)2770 3997 y FJ(;)456 4140 y FQ(where)31 b FL(E)40 b FQ(is)32 b(the)h(sheaf)f(of)g(sections)f(of)i FJ(E)5 b FQ(,)33 b(and)f(\012)2096 4110 y FI(n)2174 4140 y FQ(is)g(the)h(sheaf)f(of)g(di\013eren)n(tial)g(forms)g(of)456 4240 y(degree)26 b FJ(n)p FQ(,)605 4340 y(W)-7 b(e)37 b(can)f(extend)h FL(r)f FQ(to)h(a)f(map)g(from)g FL(E)c(\012)23 b FQ(\012)2113 4309 y FI(n)2195 4340 y FQ(to)36 b FL(E)c(\012)24 b FQ(\012)2530 4309 y FI(n)p FM(+1)2659 4340 y FQ(,)39 b FJ(n)f FQ(=)f(0)p FJ(;)14 b FQ(1)p FJ(;)g(:)g(:)g(:)49 b FQ(.)63 b(The)456 4439 y(connection)27 b FL(r)h FQ(is)f(called)g FO(\015at)36 b FQ(if)28 b(the)g(resulting)f FL(E)e(\012)18 b FQ(\012)2214 4409 y FE(\017)2280 4439 y FQ(is)28 b(a)f(complex,)g (i.e.,)h(if)g FL(r)3083 4409 y FM(2)3144 4439 y FQ(=)22 b(0.)605 4539 y(F)-7 b(or)27 b(an)n(y)g(v)n(ector)f(\014eld)i FJ(X)34 b FQ(on)27 b FJ(M)9 b FQ(,)28 b(\(6.3.1\))f(giv)n(es)f(a)h (linear)g(morphism)1738 4682 y FL(r)1807 4694 y FI(X)1880 4682 y FQ(:)g FL(E)k(!)23 b(E)-1699 b FQ(\(6.3.2\))456 4825 y(suc)n(h)27 b(that)1426 4968 y FL(r)1495 4980 y FI(X)1558 4968 y FQ(\()p FJ(sf)9 b FQ(\))23 b(=)g(\()p 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148 151 bop 456 226 a FM(148)445 b(6.)30 b(MODULI)f(SP)-5 b(A)n(CES)28 b(AND)h(COMPLEX)f(MODULAR)h(FUNCTOR)605 425 y FQ(The)36 b(pro)r(of)f(of)g(this)h(lemma)g(is)f(not)h(to)r(o)f (di\016cult)h(and)g(essen)n(tially)e(follo)n(ws)h(from)g(the)456 525 y(one-dimensional)26 b(case.)36 b(W)-7 b(e)28 b(refer)f(the)h (reader)e(to)h([)p FK(Ma)p FQ(],)h([)p FK(De1)p FQ(])g(for)f(details.) 605 624 y(Note)19 b(that)h(part)f(\(iii\))h(of)f(the)h(lemma)f(ma)n(y)f (fail)i(if)f(w)n(e)g(do)g(not)h(imp)r(ose)f(the)g(non-in)n(tegralit)n (y)456 724 y(condition.)605 877 y FP(Example)31 b FQ(6.3.3)p FP(.)40 b FQ(Let)p 1394 811 90 4 v 37 w FJ(M)48 b FQ(=)39 b FH(C)15 b FJ(;)f(D)47 b FQ(=)39 b FL(f)p FQ(0)p FL(g)p FQ(.)65 b(Let)37 b FJ(E)43 b FQ(b)r(e)37 b(the)h(trivial)f (2-dimensional)456 990 y(v)n(ector)26 b(bundle)i(o)n(v)n(er)e FH(C)48 b FQ(with)29 b(the)f(connection)f(giv)n(en)f(b)n(y)i FL(r)23 b FQ(=)g FJ(d)18 b FL(\000)2656 950 y FI(A)p FM(\()p FI(z)r FM(\))p 2656 971 136 4 v 2707 1019 a FI(z)2802 990 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FJ(F)r(;)14 b FL(r)1134 2453 y FI(F)1190 2441 y FQ(\))28 b(if)h(there)e(exists)g(an)g(in)n(v)n (ertible)g(meromorphic)g(morphism)g FJ(E)h FL(!)23 b FJ(F)12 b FQ(.)605 2597 y FP(Exer)n(cise)32 b FQ(6.3.4)p FP(.)40 b FQ(Let)p 1386 2530 V 24 w FJ(M)31 b FQ(=)23 b FH(C)15 b FJ(;)f(D)31 b FQ(=)23 b FL(f)p FQ(0)p FL(g)p FQ(,)g(and)h(let)g FL(r)2380 2567 y FI(s)2439 2597 y FQ(=)e FJ(d)11 b FQ(+)g FJ(s)2705 2564 y FI(dz)p 2706 2578 69 4 v 2723 2626 a(z)2808 2597 y FQ(b)r(e)25 b(the)f(connection) 456 2697 y(in)37 b(the)h(trivial)f(one-dimensional)f(v)n(ector)g (bundle.)67 b(Sho)n(w)36 b(that)i FL(r)2671 2667 y FI(s)2744 2697 y FQ(is)f(meromorphically)456 2796 y(equiv)-5 b(alen)n(t)27 b(to)g FL(r)1020 2766 y FI(t)1078 2796 y FQ(i\013)h FJ(s)18 b FL(\000)g FJ(t)23 b FL(2)h FH(Z)o FQ(.)605 2950 y(Let)k FJ(E)33 b FQ(b)r(e)28 b(a)f(v)n(ector)f(bundle)i(on)p 1666 2883 90 4 v 27 w FJ(M)37 b FQ(with)28 b(a)f(\015at)h(connection)f FL(r)h FQ(de\014ned)g(on)f FJ(M)9 b FQ(.)605 3103 y FP(Definition)32 b FQ(6.3.5)p FP(.)40 b FQ(The)32 b(connection)h FL(r)g FQ(has)f FO(r)l(e)l(gular)j(singularities)g(at)g FJ(D)f FQ(if)g(\()p FJ(E)5 b(;)14 b FL(r)p FQ(\))456 3202 y(is)39 b(meromorphically)e(equiv)-5 b(alen)n(t)39 b(to)g(a)g(bundle)h(with)g (a)f FJ(l)r(og)16 b(D)41 b FQ(connection)e(\(see)g(De\014ni-)456 3302 y(tion)27 b(6.3.1\).)605 3455 y FP(Exer)n(cise)32 b FQ(6.3.6)p FP(.)40 b FQ(Sho)n(w)29 b(that)i(if)g(dim)14 b FJ(M)36 b FQ(=)27 b(1,)k(then)g FL(r)f FQ(has)g(regular)f (singularities)g(i\013)456 3555 y FL(r)e FQ(is)h(a)f FJ(l)r(og)16 b(D)30 b FQ(connection.)36 b(\(F)-7 b(or)28 b(dim)14 b FJ(M)31 b(>)23 b FQ(1,)k(this)h(is)g(not)f(true.\))605 3708 y(F)-7 b(or)37 b(brevit)n(y)-7 b(,)40 b(w)n(e)e(will)g(refer)f(to) h(the)g(pair)f(\()p FJ(E)5 b(;)14 b FL(r)p FQ(\))39 b(in)f(the)g (de\014nition)g(as)f(a)h(\\connec-)456 3810 y(tion)h(on)p 766 3744 V 40 w FJ(M)48 b FQ(with)41 b(regular)d(singularities)g(at)i FJ(D)r FQ(".)73 b(The)40 b(category)e(of)i(suc)n(h)f(connections)456 3910 y(with)c(resp)r(ect)h(to)f(meromorphic)f(morphisms)g(will)i(b)r(e) g(denoted)f(b)n(y)g FL(RS)7 b FQ(\()p 2882 3843 V FJ(M)i(;)14 b(M)9 b FQ(\))35 b(\(or)g(just)456 4010 y FL(RS)6 b FQ(\()p FJ(M)j FQ(\))45 b(when)g(there)g(is)f(no)h(am)n(biguit)n(y\).)88 b(Note)44 b(that)h(meromorphic)f(morphisms)g(do)456 4112 y(not)36 b(c)n(hange)g(mono)r(drom)n(y)-7 b(,)37 b(and)g(th)n(us)g(w)n (e)f(ha)n(v)n(e)f(a)i(w)n(ell-de\014ned)f(functor)h FL(RS)7 b FQ(\()p 3075 4045 V FJ(M)h(;)14 b(M)9 b FQ(\))38 b FL(!)456 4212 y(f)p FQ(lo)r(cal)26 b(systems)h(on)h FJ(M)8 b FL(g)p FJ(:)605 4365 y FP(Remark)32 b FQ(6.3.7)p FP(.)39 b FQ(If)j(w)n(e)f(considered)g(algebraic)e(theory)i(rather)f(than)i (analytic)f(one,)456 4465 y(then)26 b(an)n(y)f(v)n(ector)g(bundle)h(on) g FJ(M)34 b FQ(admits)26 b(a)g(unique)g(meromorphic)e(con)n(tin)n (uation)h(to)p 3231 4398 V 26 w FJ(M)9 b FQ(,)26 b(so)456 4564 y(the)33 b(category)e(of)i(shea)n(v)n(es)e(on)p 1464 4498 V 33 w FJ(M)42 b FQ(up)33 b(to)g(meromorphic)e(equiv)-5 b(alence)33 b(is)g(the)g(same)g(as)f(the)456 4664 y(category)f(of)j (shea)n(v)n(es)e(on)h FJ(M)9 b FQ(.)55 b(Moreo)n(v)n(er,)33 b(in)h(this)g(case)f(it)h(w)n(as)f(pro)n(v)n(ed)f(b)n(y)h(Deligne)h (that)456 4763 y(the)c(notion)g(of)h(connection)e(with)i(regular)e (singularities)g(on)h FJ(M)39 b FQ(can)30 b(b)r(e)h(de\014ned)f(purely) g(in)456 4863 y(terms)d(of)g FJ(M)9 b FQ(,)28 b(without)g(using)p 1448 4796 V 27 w FJ(M)36 b FQ(at)28 b(all.)36 b(In)28 b(analytic)f(situation,)h(it)g(is)f(not)h(true.)605 5016 y(W)-7 b(e)30 b(quote)e(here)h(without)h(pro)r(ofs)e(sev)n(eral)f(imp)r (ortan)n(t)i(results)g(of)g(Deligne)g(ab)r(out)g(con-)456 5116 y(nections)f(with)g(regular)f(singularities.)38 b(Pro)r(ofs)27 b(and)h(details)g(can)g(b)r(e)g(found)h(in)g([)p FK(De1)o FQ(])g(or)e(in)456 5216 y([)p FK(Ma)p FQ(].)p eop %%Page: 149 15 149 152 bop 1081 226 a FM(6.3.)29 b(CONNECTIONS)h(WITH)f(REGULAR)h (SINGULARITIES)503 b(149)605 425 y FP(Theorem)32 b FQ(6.3.8)p FP(.)40 b FO(L)l(et)31 b FQ(\()p FJ(E)5 b(;)14 b FL(r)p FQ(\))28 b FL(2)f(RS)7 b FQ(\()p 1899 359 90 4 v FJ(M)i(;)14 b(M)9 b FQ(\))p FO(.)45 b(F)-6 b(or)32 b(every)h FJ(z)d FL(2)e FH(C)15 b FJ(=)p FH(Z)p FO(,)32 b(cho)l(ose)h(a)f(r)l(ep-)456 525 y(r)l(esentative)d FJ(\034)9 b FQ(\()p FJ(z)t FQ(\))24 b FL(2)f FH(C)50 b FQ(\()p FJ(\034)39 b FO(ne)l(e)l(ds)29 b(not)g(to)h(b)l(e)f(c)l(ontinuous)7 b FQ(\))p FO(.)38 b(Then)30 b(ther)l(e)f(is)h(a)f(unique)g(ve)l(ctor)456 630 y(bund)t(le)735 609 y FQ(~)716 630 y FJ(E)h FO(with)c(a)f(\015at)g FJ(l)r(og)16 b(D)27 b FO(c)l(onne)l(ction)1836 609 y FQ(~)1822 630 y FL(r)e FO(such)g(that)g FQ(\()2317 609 y(~)2297 630 y FJ(E)5 b(;)2414 609 y FQ(~)2400 630 y FL(r)p FQ(\))26 b FO(is)f(mer)l(omorphic)l(al)t(ly)j(e)l(quiv-)456 737 y(alent)h(to)h FQ(\()p FJ(E)5 b(;)14 b FL(r)p FQ(\))p FO(,)31 b(and)f(al)t(l)h(eigenvalues)1771 715 y FQ(~)1768 737 y FJ(\025)1816 707 y FI(a)1816 758 y(i)1886 737 y FQ(\()p FO(se)l(e)f(L)l(emma)g FQ(6.3.2\))f FO(lie)i(in)e(the)h(image)h (of)g FJ(\034)9 b 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FQ(In)k(other)f(w)n(ords,)h(it)g(su\016ces)f(to) h(c)n(hec)n(k)f(the)h(regularit)n(y)e(condition)i(only)f(for)g(the)h (op)r(en)456 1685 y(part)h(of)h FJ(D)r FQ(.)56 b(\(Note:)50 b(the)35 b(pro)r(of)e(of)h(this)g(theorem)g(in)g([)p FK(De1)p FQ(])g(con)n(tains)f(a)h(mistak)n(e,)h(whic)n(h)456 1785 y(Deligne)27 b(later)g(corrected.\))605 1935 y FP(Theorem)32 b FQ(6.3.11)p FP(.)39 b FO(In)31 b(the)h(notation)g(of)h(De\014nition)e FQ(6.3.1)p FO(,)g(any)h(holomorphic)j(ve)l(ctor)456 2035 y(bund)t(le)30 b(on)h FJ(M)38 b FO(with)31 b(a)g(\015at)f(c)l(onne)l (ction)g(c)l(an)g(b)l(e)h(extende)l(d)f(to)g(a)g(ve)l(ctor)h(bund)t(le) g(on)p 3174 1968 V 30 w FJ(M)39 b FO(with)456 2134 y(a)33 b(c)l(onne)l(ction)f(which)i(has)f(r)l(e)l(gular)g(singularities)h(at)e FJ(D)r FO(.)47 b(This)34 b(extension)e(is)h(unique)f(up)g(to)456 2234 y(a)e(mer)l(omorphic)i(isomorphism.)605 2384 y FP(Cor)n(ollar)-6 b(y)33 b FQ(6.3.12)c(\(The)f(Riemann{Hilb)r(ert)g(corresp)r(ondence\))p FP(.)39 b FO(The)22 b(natur)l(al)f(func-)456 2484 y(tor)1327 2584 y FL(RS)7 b FQ(\()p 1486 2518 V FJ(M)i(;)14 b(M)9 b FQ(\))23 b FL(!)g FQ(lo)r(cal)k(systems)g(on)g FJ(M)456 2703 y FO(is)j(an)g(e)l(quivalenc)l(e.)605 2853 y FQ(In)c(practical)f (applications,)h(it)g(is)g(con)n(v)n(enien)n(t)f(to)h(use)g(the)h (follo)n(wing)e(criterion)g(of)h(regu-)456 2952 y(larit)n(y)-7 b(,)26 b(whic)n(h)i(is)f(easy)g(to)h(pro)n(v)n(e.)605 3102 y FP(Lemma)j FQ(6.3.13)p FP(.)40 b FO(L)l(et)c FJ(E)5 b(;)14 b FL(r)37 b FO(b)l(e)g(as)g(in)g(De\014nition)g FQ(6.3.5)p FO(.)59 b(Then)38 b FL(r)f FO(has)h(r)l(e)l(gular)f(sin-)456 3202 y(gularities)45 b(i\013)h(for)f(every)h(holomorphic)i(map)e FJ(u)9 b FQ(:)33 b FJ(U)59 b FL(!)p 2383 3135 V 50 w FJ(M)9 b FO(,)49 b(wher)l(e)d FJ(U)53 b FO(is)46 b(a)f(disk,)50 b(and)456 3302 y FJ(u)504 3272 y FE(\000)p FM(1)592 3302 y FQ(\()p FJ(D)r FQ(\))24 b(=)f FL(f)p FQ(0)p FL(g)p FO(,)28 b(the)i(induc)l(e)l(d)g(c)l(onne)l(ction)g FJ(u)1917 3272 y FE(\003)1955 3302 y FL(r)g FO(on)f FJ(U)39 b FO(has)30 b(r)l(e)l(gular)g(singularities)h(at)f FQ(0)p FO(.)605 3452 y FQ(In)24 b(fact,)g(due)g(to)f(Theorem)f(6.3.10,)h(it)h (su\016ces)f(to)g(c)n(hec)n(k)f(this)i(condition)f(for)g FJ(u)p FQ(\(0\))g FL(2)g FJ(D)3384 3422 y FM(0)3421 3452 y FQ(.)605 3551 y(It)30 b(will)g(b)r(e)g(con)n(v)n(enien)n(t)f(to)g (rewrite)g(the)h(notion)f(of)h(\015at)g(connection)f(with)h(regular)e (sin-)456 3651 y(gularities)e(in)i(terms)f(of)h FL(D)r FQ(-mo)r(dules.)37 b(As)27 b(w)n(as)g(noted)g(b)r(efore,)h(a)f (connection)g FL(r)h FQ(in)g(a)f(v)n(ector)456 3751 y(bundle)34 b FJ(E)k FQ(is)c(\015at)f(if)h(it)g(de\014nes)g(for)f(ev)n(ery)f(op)r (en)i FJ(U)41 b FL(\032)33 b FJ(M)42 b FQ(an)33 b(action)g(of)h(the)g (Lie)f(algebra)456 3850 y(\002\()p FJ(U)9 b FQ(\))26 b(of)g(v)n(ector)e(\014elds)j(on)e FJ(U)35 b FQ(on)26 b(the)g(space)g(of)g(sections)f FL(E)7 b FQ(\()p FJ(U)i FQ(\),)27 b(whic)n(h)f(is)g(compatible)g(with)456 3950 y(m)n(ultiplication)32 b(b)n(y)g(functions.)52 b(Suc)n(h)32 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4070 74 3 v 1124 4118 a FI(M)1198 4081 y FJ(=I)1283 4051 y FI(n)p FM(+1)1412 4081 y FL(D)1478 4051 y FM(0)p 1476 4070 V 1476 4118 a FI(M)605 4184 y FQ(As)33 b(w)n(as)f(men)n(tioned)h(ab)r(o)n(v)n(e,)g (a)f(\015at)h(connection)g(with)g(\014rst)g(order)f(p)r(oles)g(at)h FJ(D)i FQ(is)e(the)456 4283 y(same)g(as)h(a)f FL(O)920 4295 y FI(M)994 4283 y FQ(-coheren)n(t)g FL(D)1429 4253 y FM(0)p 1427 4272 V 1427 4320 a FI(M)1500 4283 y FQ(-mo)r(dule)i FL(E)7 b FQ(.)56 b(Suc)n(h)34 b(a)g(mo)r(dule)g(is)g(also)f(naturally)g (\014ltered,)456 4396 y(and)27 b(action)g(of)h FL(D)1027 4366 y FM(0)p 1025 4385 V 1025 4433 a FI(M)1126 4396 y FQ(preserv)n(es)e(this)h(\014ltration.)37 b(No)n(w)27 b(de\014ne)1384 4618 y FJ(S)5 b(p)1482 4630 y FI(D)1542 4618 y FQ(\()p FL(E)i FQ(\))24 b(=)1771 4566 y Fy(d)1768 4539 y(M)1894 4668 y FI(n)p FE(\025)p FM(0)2024 4618 y FJ(I)2067 4584 y FI(n)2112 4618 y FL(E)7 b FJ(=)p FQ(\()p FJ(I)2280 4584 y FI(n)p FM(+1)2409 4618 y FL(E)g FQ(\))p FJ(:)456 4817 y FQ(This)25 b(is)g(naturally)f(a)h FL(D)1212 4787 y FM(0)1210 4840 y FI(N)6 b(D)1354 4817 y FQ(mo)r(dule,)26 b(and)f(th)n(us)g(a)g(sheaf)g(of)g(sections)f(of)i(a)e(v)n(ector)g (bundle)i(on)456 4917 y FJ(N)9 b(D)29 b FQ(with)f(a)f(\015at)h (connection)f(whic)n(h)h(has)f(\014rst)g(order)f(p)r(oles)i(at)f FJ(D)r FQ(.)605 5016 y(Here)g(is)f(a)h(more)f(explicit)h(construction.) 36 b(Cho)r(ose)26 b(co)r(ordinates)f FJ(z)2726 5028 y FM(1)2763 5016 y FJ(;)14 b(:)g(:)g(:)g(;)g(z)2987 5028 y FI(n)3058 5016 y FQ(in)28 b(a)e(neigh-)456 5116 y(b)r(orho)r(o)r(d)33 b(of)h(the)h(p)r(oin)n(t)f FJ(p)g FL(2)g FJ(D)j FQ(suc)n(h)d(that)g FJ(D)i FQ(is)e(giv)n(en)g(b)n(y)g(the)g(equation)g FJ(z)2992 5128 y FM(1)3063 5116 y FQ(=)f(0.)57 b(This)456 5216 y(also)22 b(giv)n(es)h(co)r(ordinates)g FJ(t;)14 b(z)1362 5228 y FM(2)1399 5216 y FJ(;)g(:)g(:)g(:)f(;)h(z)1622 5228 y FI(n)1691 5216 y FQ(in)24 b FJ(N)9 b(D)r FQ(,)25 b(where)e FJ(t)p FQ(\()p FJ(a;)14 b(v)s FQ(\))24 b(=)e FL(h)p FJ(v)s(;)14 b(dz)2738 5228 y FM(1)2776 5216 y FL(i)p FJ(;)g(a)23 b FL(2)g FJ(D)r(;)14 b(v)27 b FL(2)c FJ(T)3292 5228 y FI(a)3332 5216 y FJ(M)9 b FQ(.)p eop %%Page: 151 17 151 154 bop 1081 226 a FM(6.3.)29 b(CONNECTIONS)h(WITH)f(REGULAR)h (SINGULARITIES)503 b(151)456 425 y FQ(Cho)r(ose)27 b(a)g (trivialization)g(of)h FJ(E)33 b FQ(near)27 b FJ(p)p FQ(;)i(then)f FL(r)g FQ(is)g(giv)n(en)f(b)n(y)i(\(6.3.3\),)f(with)g FJ(k)f FQ(=)c(1.)38 b(De\014ne)456 525 y(the)28 b(connection)f FJ(S)5 b(p)p FL(r)22 b FQ(=)h FJ(S)5 b(p)1389 537 y FI(D)1449 525 y FQ(\()p FL(r)p FQ(\))28 b(in)g FJ(N)1783 495 y FE(\002)1839 525 y FJ(D)i FQ(b)n(y)1170 663 y(\()p FJ(S)5 b(p)p FL(r)p FQ(\))1401 675 y FI(t)1453 663 y FQ(=)23 b FJ(@)1585 675 y FI(t)1633 663 y FQ(+)18 b FJ(A)1778 675 y FM(1)1815 663 y FQ(\(0)p FJ(;)c(z)1965 675 y FM(2)2002 663 y FJ(;)g(:)g(:)g(:)f FQ(\))p FJ(=t;)1137 792 y FQ(\()p FJ(S)5 b(p)p FL(r)p FQ(\))1368 804 y FI(z)1400 812 y FG(i)1453 792 y FQ(=)23 b FJ(@)1585 804 y FI(z)1617 812 y FG(i)1666 792 y FQ(+)18 b FJ(A)1811 804 y FI(i)1839 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b(a)g(solution)g(of)h(the)g(system)f FL(r)1564 2138 y FI(i)1592 2126 y FJ(f)49 b FQ(=)39 b(0.)67 b(Let)37 b(us)h(restrict)f(this)h(solution)f(to)g(the)h(curv)n(e)456 2225 y FJ(z)t FQ(\()p FJ(t)p FQ(\))22 b(=)h FJ(p)p FQ(+)p FJ(tv)s(;)14 b(t)23 b FL(2)g FH(R)1105 2237 y FI(>)p FM(0)1200 2225 y FJ(;)14 b FQ(\()p FJ(p;)g(v)s FQ(\))24 b FL(2)f FJ(N)1601 2195 y FE(\002)1657 2225 y FQ(\()p FJ(D)r FQ(\).)34 b(By)19 b(the)f(classical)f(theory)h(of)h(ODE's)e (with)i(regular)456 2325 y(singularities)33 b(\(see,)j(e.g.,)g([)p FK(CL)q FQ(]\),)h(there)d(exists)h(a)f(v)n(ector)f FJ(g)s FQ(\()p FJ(p;)14 b(v)s FQ(\))35 b(suc)n(h)f(that)h FJ(f)9 b FQ(\()p FJ(p)23 b FQ(+)g FJ(tv)s FQ(\))35 b(=)456 2425 y FJ(F)12 b FQ(\()p FJ(t)p FQ(\))p FJ(g)s FQ(\()p FJ(p;)i(v)s FQ(\),)36 b(where)e FJ(F)12 b FQ(\()p FJ(t)p FQ(\))36 b(is)e(the)h(fundamen)n(tal)f(\(matrix\))h(solution;)i(usually)-7 b(,)36 b FJ(g)h FQ(is)e(called)456 2524 y(asymptotics)29 b(of)i FJ(f)39 b FQ(along)29 b(this)h(curv)n(e.)44 b(Then)31 b FJ(g)s FQ(\()p FJ(p;)14 b(v)s FQ(\))30 b(is)h(a)f(\015at)g(section)g (of)g(the)h(connection)456 2624 y FJ(S)5 b(p)554 2636 y FI(D)613 2624 y FQ(\()p FL(r)p FQ(\).)605 2724 y(T)-7 b(o)19 b(pro)n(v)n(e)e(\(ii\),)22 b(w)n(e)d(need)g(to)g(c)n(hec)n(k)f (that)i FJ(S)5 b(p)1966 2736 y FI(D)2020 2744 y FG(i)2050 2724 y FQ(\()p FL(r)p FQ(\))20 b(has)e(regular)g(singularities)g(at)h FJ(D)3232 2736 y FI(i)3261 2724 y FL(\\)r FJ(D)3387 2736 y FI(j)3421 2724 y FQ(,)456 2823 y(whic)n(h)27 b(can)g(b)r(e)h(done)g (explicitly)-7 b(.)p 3384 2823 4 57 v 3388 2770 50 4 v 3388 2823 V 3437 2823 4 57 v 605 2977 a FP(Remarks)32 b FQ(6.3.16)p FP(.)39 b FQ(\(i\))19 b(The)g(sp)r(ecialization)e (functor)h(can)g(b)r(e)h(easily)f(describ)r(ed)g(in)g(terms)456 3077 y(of)27 b(the)h(functor)g(of)f(nearb)n(y)g(cycles)g(for)g FL(D)r FQ(-mo)r(dules)g(\(see)h([)p FK(KasS)p FQ(].)605 3177 y(\(ii\))37 b(Note)f(that)g(the)g(sp)r(ecialization)f(functor)h (is)g(de\014ned)g(ev)n(en)g(if)g(the)h(eigen)n(v)-5 b(alues)35 b(of)456 3276 y FJ(A)518 3288 y FI(i)546 3276 y FQ(\()p FJ(z)617 3288 y FI(i)667 3276 y FQ(=)23 b(0\))k(di\013er)g(b)n(y)h(a)f (non-zero)e(in)n(teger.)36 b(Ho)n(w)n(ev)n(er,)26 b(in)h(this)h(case)f (this)g(functor)g(is)h(not)f(so)456 3376 y(easy)20 b(to)h(describ)r(e:) 33 b(one)21 b(\014rst)g(needs)g(to)h(replace)e(the)i(\015at)f (connection)g(b)n(y)g(a)f(meromorphically)456 3475 y(equiv)-5 b(alen)n(t)40 b(one)g(whic)n(h)g(satis\014es)g(the)h(non-in)n(tegralit) n(y)d(condition.)75 b(Suc)n(h)41 b(a)f(connection)456 3575 y(exists)27 b(b)n(y)g(Corollary)e(6.3.9,)i(but)h(explicitly)g (constructing)e(it)i(can)g(b)r(e)g(di\016cult.)605 3726 y(Finally)-7 b(,)38 b(it)f(is)f(natural)f(to)h(consider)f(the)i(follo)n (wing)e(question.)62 b(Supp)r(ose)36 b(w)n(e)g(ha)n(v)n(e)f(a)456 3826 y(divisor)c(with)i(normal)f(crossings,)f(and)i FJ(D)1824 3838 y FM(1)1861 3826 y FQ(,)h FJ(D)1987 3838 y FM(2)2056 3826 y FQ(are)e(t)n(w)n(o)g(of)g(the)h(comp)r(onen)n(ts.)51 b(Is)33 b(it)g(true)456 3926 y(that)27 b FJ(S)5 b(p)733 3938 y FI(D)787 3946 y FF(1)851 3926 y FQ(and)28 b FJ(S)5 b(p)1111 3938 y FI(D)1165 3946 y FF(2)1229 3926 y FQ(comm)n(ute?)605 4025 y(In)28 b(order)d(for)i(this)h(question)f(to)g(mak)n(e)g(sense,)f (one)h(m)n(ust)h(\014rst)f(de\014ne)g(the)h(comp)r(osition)456 4125 y FJ(S)5 b(p)554 4137 y FI(D)608 4145 y FF(1)644 4125 y FJ(S)g(p)742 4137 y FI(D)796 4145 y FF(2)832 4125 y FQ(.)58 b(F)-7 b(or)34 b(simplicit)n(y)-7 b(,)37 b(let)d(us)h(assume) f(that)h(dim)p 2346 4058 90 4 v 14 w FJ(M)43 b FQ(=)34 b(2,)i FJ(D)2739 4137 y FM(1)2799 4125 y FL(\\)24 b FJ(D)2947 4137 y FM(2)3018 4125 y FQ(=)35 b FL(f)p FJ(p)p FL(g)p FQ(.)56 b(Let)456 4225 y FJ(N)523 4237 y FM(1)583 4225 y FQ(=)22 b FJ(N)9 b FQ(\()p FJ(D)847 4237 y FM(1)884 4225 y FQ(\);)27 b(it)e(con)n(tains)f(the)h(divisor)f FJ(D)1846 4237 y FM(12)1939 4225 y FQ(=)f FJ(N)2094 4237 y FI(p)2132 4225 y FQ(\()p FJ(D)2233 4237 y FM(1)2270 4225 y FQ(\).)37 b(Let)25 b FJ(N)2575 4237 y FM(12)2670 4225 y FQ(b)r(e)h(the)f(normal)f(bundle)456 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FL([)h FJ(T)2981 4698 y FI(p)3019 4686 y FJ(D)3088 4698 y FM(2)3125 4686 y FQ(\))p FO(,)40 b(as)e(the)456 4786 y(c)l(omp)l(osition)851 4924 y FL(RS)7 b FQ(\()p 1010 4857 V FJ(M)i(;)14 b(M)9 b FQ(\))23 b FL(!)g(RS)7 b FQ(\()p FJ(N)1614 4936 y FM(1)1651 4924 y FJ(;)14 b(N)1764 4888 y FE(\002)1755 4946 y FM(1)1820 4924 y FQ(\))23 b FL(!)g(RS)7 b FQ(\()p FJ(N)2207 4936 y FM(12)2277 4924 y FJ(;)14 b(N)2390 4888 y FE(\002)2381 4946 y FM(12)2451 4924 y FQ(\))24 b FL(')e(RS)7 b FQ(\()p FJ(T)2802 4936 y FI(p)2840 4924 y FJ(;)14 b(T)2938 4889 y FE(\002)2926 4944 y FI(p)2993 4924 y FQ(\))p FJ(:)605 5064 y FQ(\(ii\))28 b FO(The)g(functors)f FJ(S)5 b(p)1329 5076 y FM(12)1399 5064 y FJ(;)14 b(S)5 b(p)1534 5076 y FM(21)1604 5064 y FO(,)28 b(de\014ne)l(d)f(as)h(in)f(p) l(art)36 b FQ(\(i\))p FO(,)28 b(ar)l(e)g(c)l(anonic)l(al)t(ly)h (isomorphic.)605 5216 y FQ(The)f(pro)r(of)f(of)g(this)h(lemma)g(is)f (not)h(di\016cult)g(and)g(is)f(left)h(as)f(an)h(exercise.)p eop %%Page: 152 18 152 155 bop 456 226 a FM(152)445 b(6.)30 b(MODULI)f(SP)-5 b(A)n(CES)28 b(AND)h(COMPLEX)f(MODULAR)h(FUNCTOR)1130 425 y FK(6.4.)46 b(Complex)30 b(analytic)i(mo)s(dular)e(functor)605 575 y FQ(In)d(this)g(section,)f(w)n(e)h(giv)n(e)e(a)i(de\014nition)g (of)f(mo)r(dular)g(functor)h(in)g(terms)f(of)h(\015at)f(connec-)456 674 y(tions)h(on)g(the)h(mo)r(duli)g(spaces.)605 774 y(Let)k FL(C)37 b FQ(b)r(e)32 b(a)g(semisimple)g(ab)r(elian)f(category) f(o)n(v)n(er)h FH(C)15 b FQ(,)39 b(and)32 b FJ(R)f FL(2)f FQ(ind)p FL(\000C)2959 744 y Fv(\002)p FM(2)3080 774 y FQ(b)r(e)i(a)g(sym-)456 874 y(metric)d(ob)5 b(ject,)31 b(as)e(in)h(Section)g(2.4.)43 b(Recalling)29 b(the)h(de\014nition)g(of) g(a)g FL(C)5 b FQ(-extended)29 b(mo)r(dular)456 973 y(functor)i (\(De\014nition)h(5.1.12\))e(and)h(the)g(results)g(of)g(the)h(previous) e(section,)i(w)n(e)e(can)h(rewrite)456 1073 y(the)d(de\014nition)g(of)f (mo)r(dular)g(functor)h(as)f(follo)n(ws.)605 1235 y FP(Definition)32 b FQ(6.4.1)p FP(.)40 b FQ(A)26 b FO(c)l(omplex)j FL(C)5 b FO(-extende)l(d)27 b(mo)l(dular)i(functor)34 b FQ(is)26 b(the)g(follo)n(wing)f(col-)456 1334 y(lection)i(of)h(data:)605 1434 y(\(i\))d(F)-7 b(or)23 b(ev)n(ery)f(\014nite)j(set)f FJ(A)g FQ(and)g FJ(W)35 b FL(2)23 b(C)1896 1404 y Fv(\002)p FI(A)2002 1434 y FQ(,)h(a)g(\014nite-dimensional)f(v)n(ector)g(bundle)h (o)n(v)n(er)p 456 1470 90 4 v 456 1536 a FJ(M)545 1548 y FE(\003)p FI(;A)687 1536 y FQ(with)33 b(a)g(\015at)h(connection)f (with)h(regular)d(singularities.)53 b(This)34 b(bundle)g(is)f(called)g (the)456 1642 y FO(bund)t(le)27 b(of)h(c)l(onformal)h(blo)l(cks)p FQ(;)d(its)f(\014b)r(er)g(at)g(a)f(p)r(oin)n(t)h FJ(C)k FL(2)p 2302 1575 V 24 w FJ(M)2391 1654 y FE(\003)p FI(;A)2524 1642 y FQ(will)c(b)r(e)g(denoted)g(b)n(y)f FL(h)p FJ(W)12 b FL(i)3365 1654 y FI(C)3421 1642 y FQ(.)605 1741 y(\(ii\))28 b(Isomorphisms)f FL(h)p FJ(X)7 b FL(i)1409 1753 y FI(C)1461 1737 y Fx(0)1505 1741 y FL(\012)18 b(h)p FJ(Y)h FL(i)1719 1753 y FI(C)1771 1737 y Fx(00)1839 1741 y FL(')k(h)p FJ(X)i Fw(\002)18 b FJ(Y)g FL(i)2234 1753 y FI(C)2286 1737 y Fx(0)2309 1753 y FE(t)p FI(C)2406 1737 y Fx(00)2450 1741 y FQ(.)605 1841 y(\(iii\))34 b FK(Gluing)i(isomorphisms.)47 b FQ(Let)33 b FJ(A)g FQ(b)r(e)g(a)f(\014nite)i(set,)g FJ(\013;)14 b(\014)36 b FL(2)c FJ(A)p FQ(|an)h(unordered)456 1948 y(pair,)c FJ(A)714 1918 y FE(0)763 1948 y FQ(=)d FJ(A)20 b FL(n)f(f)p FJ(\013;)14 b(\014)t FL(g)p FJ(;)g(W)38 b FL(2)26 b(C)1505 1918 y Fv(\002)p FI(A)1607 1893 y Fx(0)1633 1948 y FQ(.)42 b(F)-7 b(or)29 b(ev)n(ery)f(suc)n(h)h (collection,)h(w)n(e)f(require)f(an)h(isomor-)456 2048 y(phism)f(of)f(v)n(ector)f(bundles)i(with)g(connections)f(on)h FJ(M)2183 2060 y FE(\003)p FI(;A)2290 2048 y FQ(:)1343 2214 y FJ(G)1408 2226 y FI(\013;\014)1525 2214 y FQ(:)f FL(h)p FJ(W)k Fw(\002)18 b FJ(R)q FL(i)1947 2167 y FE(\030)1918 2214 y FL(\000)-39 b(!)23 b FJ(S)2106 2180 y FE(\003)2101 2235 y FI(\013;\014)2208 2214 y FJ(S)5 b(p)2306 2226 y FI(D)2366 2214 y FL(h)p FJ(W)12 b FL(i)p FJ(;)-2087 b FQ(\(6.4.1\))456 2381 y(where)39 b 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FJ(eich)3368 1198 y Fz(C)3368 1251 y FI(A)3418 1234 y Fx(0)456 1328 y FQ(w)n(as)26 b(de\014ned)i(using)f(the)h(comp)r(osition)1414 1492 y FJ(M)1495 1504 y FE(\003)p FI(;A)1654 1436 y(S)1695 1445 y FG(\013;\014)1625 1492 y FL(\000)-32 b(\000)-19 b(\000)-33 b(!)23 b FJ(N)1918 1458 y FE(\002)1974 1492 y FQ(\()p FJ(D)r FQ(\))g FL(!)g FJ(M)2319 1504 y FE(\003)p FI(;A)2423 1488 y Fx(0)2449 1492 y FJ(;)456 1633 y FQ(see)i(\(6.2.10\)) o(.)36 b(In)27 b(the)f(language)e(of)i(lo)r(cal)g(systems,)g(the)g (gluing)g(isomorphism)f(should)h(iden-)456 1732 y(tify)31 b(the)f(v)n(ector)f(spaces)h FL(h)p FJ(W)i Fw(\002)20 b FJ(R)q FL(i)1589 1744 y FI(C)1673 1732 y FL(')27 b(h)p FJ(W)12 b FL(i)1919 1744 y FI(C)1971 1728 y Fx(0)2028 1732 y FQ(in)31 b(suc)n(h)f(a)g(w)n(a)n(y)f(that)h(it)h(agrees)e(with)i (mor-)456 1832 y(phisms)e(in)g FL(T)7 b FJ(eich)1037 1844 y FI(A)1091 1832 y FQ(.)41 b(This)29 b(is)g(equiv)-5 b(alen)n(t)29 b(to)g(sa)n(ying)f(that)h(it)h(m)n(ust)f(b)r(e)h(an)f (isomorphism)f(of)456 1932 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FJ(;)g(:)g(:)g(:)f(;)h(v)3025 2277 y FI(n)3071 2265 y FQ(\))p FJ(;)456 2412 y FQ(where,)21 b(as)f(b)r(efore,)i FJ(v)1115 2424 y FE(1)1208 2412 y FQ(=)h FL(\000)p FJ(@)1405 2427 y FM(1)p FI(=z)1510 2412 y FQ(,)f(giv)n(es)e(an)g(iden)n(ti\014cation)g FJ(M)2434 2424 y FM(0)p FI(;n)p FM(+1)2639 2412 y FQ(=)j FJ(X)2803 2382 y FM(0)2796 2433 y FI(n)2841 2412 y FJ(=)p FH(C)14 b FQ(,)28 b(where)20 b FH(C)42 b FQ(acts)456 2517 y(on)18 b FJ(X)638 2487 y FM(0)631 2538 y FI(n)694 2517 y FQ(b)n(y)h FJ(z)840 2529 y FI(i)890 2517 y FL(7!)k FJ(z)1035 2529 y FI(i)1063 2517 y FQ(+)p FJ(a)p FQ(.)33 b(Let)19 b(us)g(\014x)f FJ(W)1652 2529 y FE(1)1723 2517 y FJ(;)c(W)1838 2529 y FM(1)1876 2517 y FJ(;)g(:)g(:)g(:)f(;)h(W)2138 2529 y FI(n)2202 2517 y FQ(and)19 b(consider)e(the)i(connection)f(\(6.5.3\)) 456 2617 y(in)24 b FJ(W)627 2629 y FM(1)675 2617 y FL(\012)11 b(\001)j(\001)g(\001)c(\012)h FJ(W)1012 2629 y FI(n)1057 2617 y FQ(.)36 b(This)23 b(connection)h(induces)g(a)f(connection)g(in)h (\()p FJ(W)2684 2629 y FE(1)2766 2617 y FL(\012)11 b FJ(W)2920 2629 y FM(1)2968 2617 y FL(\012)g(\001)j(\001)g(\001)c(\012)h FJ(W)3305 2629 y FI(n)3350 2617 y FQ(\))3382 2587 y Fk(g)3421 2617 y FQ(,)456 2717 y(whic)n(h)30 b(is)g(ob)n(viously)f(translation)g (in)n(v)-5 b(arian)n(t)29 b(and)h(th)n(us)h(de\014nes)f(a)g(connection) g(on)g FJ(M)3240 2729 y FM(0)p FI(;n)p FM(+1)3421 2717 y FQ(.)456 2816 y(One)j(easily)h(c)n(hec)n(ks)f(that)h(this)h (connection)e(coincides)h(with)g(the)h(KZ)e(connection)h(de\014ned)456 2916 y(ab)r(o)n(v)n(e.)605 3067 y FP(Theorem)e FQ(6.5.3)p FP(.)40 b FO(The)28 b(KZ)f(c)l(onne)l(ction,)h(c)l(onsider)l(e)l(d)h (as)e(a)h(c)l(onne)l(ction)f(on)h(the)f(trivial)456 3167 y(ve)l(ctor)h(bund)t(le)h(with)g(\014b)l(er)f FQ(\()p FJ(W)1434 3179 y FM(1)1488 3167 y FL(\012)15 b(\001)f(\001)g(\001)h (\012)g FJ(W)1838 3179 y FI(n)1884 3167 y FQ(\))1916 3136 y Fk(g)1984 3167 y FO(over)29 b(the)g(c)l(omp)l(acti\014c)l(ation) p 2912 3100 90 4 v 29 w FJ(M)3002 3179 y FM(0)p FI(;n)3100 3167 y 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3835 y FM(1)1019 3823 y FL(\012)g FJ(:)c(:)g(:)k FL(\012)g FJ(w)1359 3835 y FI(k)1419 3823 y FL(\012)g FJ(v)s FQ(\))h FL(\012)f FQ(\()p FJ(v)1754 3789 y FE(\003)1811 3823 y FL(\012)g FJ(w)1953 3835 y FI(k)q FM(+1)2097 3823 y FL(\012)g FJ(:)c(:)g(:)k FL(\012)g FJ(w)2437 3835 y FI(n)2483 3823 y FQ(\))23 b FL(7!)g FQ(\()p FJ(v)s(;)14 b(v)2799 3789 y FE(\003)2838 3823 y FQ(\))p FJ(w)2929 3835 y FM(1)2986 3823 y FL(\012)k FJ(:)c(:)g(:)k FL(\012)g FJ(w)3326 3835 y FI(n)3371 3823 y FJ(:)456 3490 y FQ(\(6.5.4\))605 3975 y FP(Pr)n(oof.)41 b FQ(By)36 b(Theorem)g(6.3.10,)h(it)g(su\016ces)f(to)g(c)n(hec)n(k)g (the)h(regularit)n(y)e(condition)h(for)456 4074 y(the)f(op)r(en)g (strata)g(of)g FJ(D)j FQ(=)p 1372 4008 V 35 w FJ(M)1462 4086 y FM(0)p FI(;n)1583 4074 y FL(n)23 b FJ(M)1729 4086 y FM(0)p FI(;n)1827 4074 y FQ(,)37 b(i.e.,)g(for)e(the)h(curv)n(es)e (with)h(one)g(double)g(p)r(oin)n(t.)456 4174 y(Th)n(us,)25 b(it)g(su\016ces)g(to)g(c)n(hec)n(k)f(the)h(regularit)n(y)e(and)i(the)g 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FM(156)445 b(6.)30 b(MODULI)f(SP)-5 b(A)n(CES)28 b(AND)h(COMPLEX)f(MODULAR)h(FUNCTOR)605 425 y FQ(Then)f(to)f(pro)n(v)n(e)f(the)i(theorem)f(it)h(su\016ces)g(to) f(c)n(hec)n(k)g(that)539 838 y FJ(S)5 b(p)637 850 y FI(t)p FM(=0)749 838 y FJ(F)814 803 y FE(\003)802 858 y FI(a;)p FE(1)924 841 y Fx(0)q(0)970 838 y FL(h)p FJ(W)1092 803 y FE(0)1080 858 y(1)1151 838 y FJ(;)14 b(W)1278 803 y FE(0)1266 858 y FM(1)1303 838 y FJ(;)g(:)g(:)g(:)g(;)g(W)1578 803 y FE(0)1566 858 y FI(k)1607 838 y FJ(;)g(W)1734 803 y FE(00)1722 858 y FM(1)1776 838 y FJ(;)g(:)g(:)g(:)g(;)g(W)2051 803 y FE(00)2039 858 y FI(m)2102 838 y FL(i)1599 992 y FQ(=)1687 913 y Fy(M)1738 1090 y FI(i)1812 992 y FL(h)p FJ(W)1934 958 y FE(0)1922 1012 y(1)1993 992 y FJ(;)g(W)2120 958 y FE(0)2108 1012 y FM(1)2146 992 y FJ(;)g(:)g(:)g(:)g(;)g(W)2421 958 y FE(0)2409 1012 y FI(k)2450 992 y FJ(;)g(V)2535 1004 y FI(i)2563 992 y FL(i)k(\012)h(h)p FJ(V)2796 958 y FE(\003)2777 1012 y FI(i)2834 992 y FJ(;)14 b(W)2961 958 y FE(00)2949 1012 y FM(1)3003 992 y FJ(;)g(:)g(:)g(:)g(;)g(W)3278 958 y FE(00)3266 1012 y FI(m)3329 992 y FL(i)456 1452 y FQ(as)40 b(v)n(ector)g(bundles)i(with)f(connections.)77 b(T)-7 b(o)41 b(obtain)g(the)h(left)g(hand)f(side,)j(w)n(e)d(need)h(to) 456 1552 y(substitute)28 b(in)g(\(6.5.3\))977 1939 y FJ(z)1016 1951 y FI(i)1066 1939 y FQ(=)23 b FJ(z)1197 1905 y FE(0)1193 1959 y FI(i)1220 1939 y FJ(;)716 b(v)1999 1951 y FI(i)2050 1939 y FQ(=)22 b FJ(v)2180 1905 y FE(0)2177 1959 y FI(i)2205 1939 y FJ(;)547 b(i)23 b FL(\024)g FJ(k)s FQ(;)890 2073 y FJ(z)929 2085 y FI(k)q FM(+)p FI(i)1066 2073 y FQ(=)g FJ(a)18 b FQ(+)g FJ(tz)1372 2038 y FE(00)1368 2093 y FI(i)1414 2073 y FJ(;)434 b(v)1911 2085 y FI(k)q FM(+)p FI(i)2050 2073 y FQ(=)22 b FJ(tv)2210 2038 y FE(0)2207 2093 y FI(i)2235 2073 y FJ(;)517 b(i)23 b FL(\024)g FJ(m;)456 2461 y FQ(and)k(then)h(sp)r(ecialize)f(to)h FJ(t)23 b FQ(=)f(0.)605 2560 y(Explicit)j(calculation)g(sho)n(ws)f(that)i(this)g (substitution)g(giv)n(es)e(the)i(follo)n(wing)e(connection)456 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FI(p)2540 4591 y FL(\000)g FJ(z)2666 4567 y FE(0)o(0)2662 4611 y FI(q)2708 4591 y FQ(\))2781 4534 y(+)2874 4478 y FJ(D)2943 4490 y FI(k)q FM(+)p FI(p)p 2874 4515 195 4 v 2936 4591 a FQ(2)p FJ(t)3079 4367 y Fy(1)3079 4516 y(A)3165 4534 y FJ(;)456 5016 y FQ(where)27 b(the)h(indices)f FJ(i;)14 b(j)28 b FQ(=)22 b(1)p FJ(;)14 b(:)g(:)g(:)f(;)h(k)31 b FQ(and)c FJ(p;)14 b(q)26 b FQ(=)d(1)p FJ(;)14 b(:)g(:)g(:)f(;)h(m)p FQ(.)605 5116 y(It)24 b(is)g(ob)n(vious)f(that)h(this)g(is)g(a)g(connection)f (with)i(regular)d(singularities)h(at)g FJ(t)h FQ(=)e(0.)35 b(Let)25 b(us)456 5216 y(sp)r(ecialize)c(it)h(to)f FJ(t)i FQ(=)g(0)e(as)g(describ)r(ed)g(in)h(Lemma)f(6.3.15;)h(w)n(e)f(will)h(c) n(hec)n(k)f(the)g(non-in)n(tegralit)n(y)p eop %%Page: 157 23 157 160 bop 1299 226 a FM(6.5.)29 b(EXAMPLE:)e(DRINFELD'S)i(CA)-5 b(TEGOR)g(Y)721 b(157)456 425 y FQ(condition)27 b(later.)36 b(This)28 b(giv)n(es:)1212 657 y FL(r)1281 671 y FI(z)1315 651 y Fx(0)1313 690 y FG(i)1366 657 y FQ(=)1497 601 y FJ(@)p 1464 638 115 4 v 1464 714 a(@)5 b(z)1556 685 y FE(0)1552 737 y FI(i)1607 657 y FL(\000)1709 601 y FQ(1)p 1700 638 59 4 v 1700 714 a FH({)1782 490 y Fy(0)1782 640 y(@)1855 578 y(X)1862 757 y FI(j)s FE(6)p FM(=)p FI(i)2060 601 y FQ(\012)2120 613 y FI(ij)p 1999 638 242 4 v 1999 714 a FJ(z)2042 685 y FE(0)2038 737 y FI(i)2083 714 y FL(\000)19 b FJ(z)2210 685 y FE(0)2206 737 y FI(j)2269 657 y FQ(+)2371 595 y(\012)2431 610 y FI(i;)p FM(\(2\))p 2362 638 212 4 v 2362 714 a FJ(z)2405 685 y FE(0)2401 737 y FI(i)2446 714 y FL(\000)f FJ(a)2583 490 y Fy(1)2583 640 y(A)2670 657 y FJ(;)1210 933 y FL(r)1279 947 y FI(v)1314 927 y Fx(0)1312 966 y FG(i)1366 933 y FQ(=)1498 877 y FJ(@)p 1464 914 117 4 v 1464 990 a(@)5 b(v)1556 961 y FE(0)1553 1013 y FI(i)1609 933 y FL(\000)1731 877 y FQ(1)p 1702 914 101 4 v 1702 990 a(2)p FH({)1822 877 y FJ(D)1891 889 y FI(i)p 1822 914 97 4 v 1836 990 a FJ(v)1879 961 y FE(0)1876 1013 y FI(i)1928 933 y FJ(;)1195 1216 y FL(r)1264 1228 y FI(z)1298 1212 y Fx(00)1296 1245 y FG(p)1366 1216 y FQ(=)1506 1160 y FJ(@)p 1464 1197 134 4 v 1464 1273 a(@)g(z)1556 1249 y FE(0)o(0)1552 1294 y FI(p)1625 1216 y FL(\000)1727 1160 y FQ(1)p 1718 1197 59 4 v 1718 1273 a FH({)1801 1049 y Fy(0)1801 1199 y(@)1874 1137 y(X)1875 1316 y FI(q)r FE(6)p FM(=)p FI(p)2017 1160 y FQ(\012)2077 1172 y FI(k)q FM(+)p FI(pk)q FM(+)p FI(q)p 2017 1197 307 4 v 2035 1273 a FJ(z)2078 1249 y FE(0)o(0)2074 1294 y FI(p)2138 1273 y FL(\000)18 b FJ(z)2264 1249 y FE(0)o(0)2260 1294 y FI(q)2333 1049 y Fy(1)2333 1199 y(A)2420 1216 y FJ(;)1194 1492 y FL(r)1263 1504 y FI(v)1298 1488 y Fx(00)1296 1521 y FG(p)1366 1492 y FQ(=)1506 1436 y FJ(@)p 1464 1473 135 4 v 1464 1549 a(@)5 b(v)1556 1525 y FE(00)1553 1569 y FI(p)1626 1492 y FL(\000)1749 1436 y FQ(1)p 1719 1473 101 4 v 1719 1549 a(2)p FH({)1839 1436 y FJ(D)1908 1448 y FI(k)q FM(+)p FI(p)p 1839 1473 195 4 v 1894 1549 a FJ(v)1937 1525 y FE(00)1934 1569 y FI(p)2044 1492 y FJ(;)1234 1733 y FL(r)1303 1745 y FI(a)1366 1733 y FQ(=)1486 1676 y FJ(@)p 1464 1713 93 4 v 1464 1790 a(@)g(a)1585 1733 y FL(\000)1686 1676 y FQ(1)p 1678 1713 59 4 v 1678 1790 a FH({)1760 1654 y Fy(X)1805 1831 y FI(j)1915 1671 y FQ(\012)1975 1686 y FI(j;)p FM(\(2\))p 1904 1713 219 4 v 1904 1790 a FJ(a)18 b FL(\000)g FJ(z)2092 1761 y FE(0)2088 1813 y FI(j)2133 1733 y FJ(;)1245 1997 y FL(r)1314 2009 y FI(t)1366 1997 y FQ(=)1479 1941 y FJ(@)p 1464 1978 79 4 v 1464 2054 a(@)5 b(t)1571 1997 y FL(\000)1708 1941 y FQ(1)p 1664 1978 131 4 v 1664 2054 a(2)p FH({)s FJ(t)1818 1918 y Fy(X)1834 2093 y FI(p;q)1951 1997 y FQ(\012)2011 2009 y FI(k)q FM(+)p FI(p;k)q FM(+)p FI(q)2277 1997 y FJ(;)456 2220 y FQ(where)27 b(w)n(e)g(denoted)h(\012)1192 2235 y FI(i;)p FM(\(2\))1347 2220 y FQ(=)1435 2158 y Fy(P)1522 2245 y FI(p)1574 2220 y FQ(\012)1634 2232 y FI(i;k)q FM(+)p FI(p)1827 2220 y FQ(=)22 b(\012)1974 2234 y FI(W)2045 2214 y Fx(0)2036 2253 y FG(i)2068 2234 y FI(;W)2159 2214 y Fx(00)2150 2252 y FF(1)2200 2234 y FE(\012\001\001\001)o(\012)p FI(W)2434 2218 y Fx(0)q(0)2425 2251 y FG(m)2485 2220 y FQ(.)605 2322 y(Let)28 b(us)f(iden)n(tify)539 2484 y(\()p FJ(W)661 2450 y FE(0)649 2504 y(1)738 2484 y FL(\012)18 b FJ(W)911 2450 y FE(0)899 2504 y FM(1)955 2484 y FL(\012)g(\001)c(\001)g(\001)19 b(\012)f FJ(W)1327 2450 y FE(0)1315 2504 y FI(k)1374 2484 y FL(\012)g FJ(W)1547 2450 y FE(00)1535 2504 y FM(1)1608 2484 y FL(\012)g(\001)c(\001)g(\001) k(\012)g FJ(W)1979 2450 y FE(0)q(0)1967 2504 y FI(m)2031 2484 y FQ(\))2063 2450 y Fk(g)1058 2638 y FQ(=)1145 2559 y Fy(M)1188 2738 y FI(\025)1271 2638 y FQ(\()p FJ(W)1393 2604 y FE(0)1381 2659 y(1)1470 2638 y FL(\012)g FJ(W)1643 2604 y FE(0)1631 2659 y FM(1)1687 2638 y FL(\012)g(\001)c(\001)g(\001) 19 b(\012)f FJ(W)2059 2604 y FE(0)2047 2659 y FI(k)2106 2638 y FL(\012)g FJ(V)2237 2650 y FI(\025)2281 2638 y FQ(\))2313 2604 y Fk(g)2371 2638 y FL(\012)g FQ(\()p FJ(V)2554 2604 y FE(\003)2534 2659 y FI(\025)2610 2638 y FL(\012)g FJ(W)2783 2604 y FE(00)2771 2659 y FM(1)2844 2638 y FL(\012)g(\001)c(\001)g(\001)k(\012)g FJ(W)3215 2604 y FE(00)3203 2659 y FI(m)3267 2638 y FQ(\))3299 2604 y Fk(g)3338 2638 y FJ(:)456 2850 y FQ(Under)27 b(this)h(iden)n (ti\014cation,)g(one)f(has:)1705 2986 y(\012)1765 3001 y FI(j;)p FM(\(2\))1925 2986 y FQ(=)22 b(\012)2072 3000 y FI(W)2143 2981 y Fx(0)2134 3020 y FG(j)2166 3000 y FI(;V)2225 3009 y FG(\025)2268 2986 y FJ(;)1442 3069 y Fy(X)1459 3244 y FI(p;q)1576 3148 y FQ(\012)1636 3160 y FI(k)q FM(+)p FI(p;k)q FM(+)p FI(q)1925 3148 y FQ(=)g FJ(D)2081 3160 y FI(V)2134 3141 y Fx(\003)2120 3181 y FG(\025)2196 3148 y FQ(=)h FJ(D)2353 3160 y FI(V)2392 3169 y FG(\025)2435 3148 y FJ(:)456 3367 y FQ(Th)n(us,)j(w)n(e)h(see)f (that)i(the)f(sp)r(ecialization)f(exactly)g(coincides)h(with)g(the)g (pro)r(duct)g(of)g(KZ)f(con-)456 3466 y(nections)h(on)g FJ(X)970 3436 y FM(0)963 3490 y FI(k)q FM(+1)1106 3466 y FL(\002)18 b FJ(X)1265 3436 y FM(0)1258 3487 y FI(m)1321 3466 y FQ(.)605 3566 y(Finally)-7 b(,)30 b(in)g(order)e(to)i(justify)g (that)g(our)f(calculation)g(of)g(the)h(sp)r(ecialization)f(functor)h (is)456 3665 y(v)-5 b(alid,)34 b(w)n(e)f(ha)n(v)n(e)f(to)h(c)n(hec)n(k) f(that)i(our)e(connection)h(satis\014es)f(the)h(non-in)n(tegralit)n(y)f (prop)r(ert)n(y)456 3765 y(\(6.3.4\))o(.)i(This)21 b(follo)n(ws)f(from) g(the)h(fact)g(that)g(in)g(our)f(case,)h(the)g(op)r(erator)e FJ(A)p FQ(\()p FJ(t)24 b FQ(=)f(0\))d(is)h(giv)n(en)f(b)n(y)489 3832 y FM(1)p 466 3846 81 4 v 466 3893 a(2)p Fz({)556 3865 y FJ(D)625 3877 y FI(V)664 3885 y FG(i)695 3865 y FQ(.)36 b(Th)n(us,)25 b(its)h(eigen)n(v)-5 b(alues)24 b(are)g(of)i(the)f(form)g FL(h)p FJ(\025;)14 b(\025)h FQ(+)f(2)p FJ(\032)p FL(i)p FJ(=)p FQ(2)p FH({)s FQ(.)36 b(Since)25 b FL(h)p FJ(\025;)14 b(\025)h FQ(+)f(2)p FJ(\032)p FL(i)22 b(2)i FH(Q)5 b FQ(,)456 3964 y(and)28 b FH({)39 b FJ(=)-52 b FL(2)26 b FH(Q)6 b FQ(,)35 b(these)29 b(eigen)n(v)-5 b(alues)28 b(can)h(not)f(di\013er)i(b)n(y)e(a)h(non-zero)e(in)n(teger)h (\(this)i(is)f(the)g(only)456 4064 y(place)e(where)g(w)n(e)g(use)h(the) f(condition)h FH({)36 b FJ(=)-51 b FL(2)23 b FH(Q)6 b FQ(!\).)605 4164 y(This)28 b(completes)f(the)h(pro)r(of)f(of)g(the)h (gluing)f(axiom.)p 3384 4164 4 57 v 3388 4111 50 4 v 3388 4164 V 3437 4164 4 57 v 605 4316 a(Therefore,)e(w)n(e)h(see)f (that)h(the)g(KZ)g(connection)f(giv)n(en)g(ab)r(o)n(v)n(e)g(do)r(es)g (de\014ne)h(a)g(gen)n(us)f(zero)456 4416 y(complex)37 b(mo)r(dular)g(functor.)67 b(Th)n(us,)40 b(it)e(de\014nes)g(a)f (structure)h(of)f(a)h(w)n(eakly)e(rigid)h(tensor)456 4516 y(category)25 b(on)j FL(R)p FJ(ep)1057 4528 y FI(f)1099 4516 y FA(g)p FQ(.)605 4666 y FP(Pr)n(oposition)j FQ(6.5.4)p FP(.)40 b FO(The)34 b(we)l(akly)h(rigid)g(tensor)f(structur)l(e)e(on)h FL(R)p FJ(ep)2901 4678 y FI(f)2944 4666 y FQ(\()p FA(g)p FQ(\))h FO(de\014ne)l(d)g(as)456 4766 y(ab)l(ove)c(c)l(oincides)i(with) e(the)g(Drinfeld)h(c)l(ate)l(gory)g(structur)l(e)d(de\014ne)l(d)i(in)g (Se)l(ction)f FQ(1.4)p FO(.)605 4917 y FQ(The)g(pro)r(of)g(of)g(this)h (prop)r(osition)e(immediately)h(follo)n(ws)f(from)h(the)h(de\014nition) f(of)h(Drin-)456 5016 y(feld's)e(category)-7 b(.)35 b(This,)27 b(in)h(particular,)e(sho)n(ws)h(that)h(this)g(category)d(is)j(rigid.) 605 5116 y(The)39 b(reader)f(migh)n(t)h(notice)g(that)h(the)g(pro)r(of) e(of)h(the)h(gluing)f(axiom)f(giv)n(en)h(ab)r(o)n(v)n(e)f(is)456 5216 y(essen)n(tially)31 b(the)h(same)g(pro)r(of)f(w)n(e)h(used)g(in)h (Chapter)e(1)h(to)g(pro)n(v)n(e)e(the)j(asso)r(ciativit)n(y)d(axiom)p eop %%Page: 158 24 158 161 bop 456 226 a FM(158)445 b(6.)30 b(MODULI)f(SP)-5 b(A)n(CES)28 b(AND)h(COMPLEX)f(MODULAR)h(FUNCTOR)456 425 y FQ(for)35 b(the)h(Drinfeld's)f(category|only)f(no)n(w)g(w)n(e)i (ha)n(v)n(e)e(the)i(language)e(of)h(connections)g(with)456 525 y(regular)25 b(singularities)h(and)h(sp)r(ecialization)g(functors)g (in)h(our)e(disp)r(osition,)h(whic)n(h)h(allo)n(ws)e(us)456 624 y(to)h(mak)n(e)g(all)g(the)h(argumen)n(ts)e(absolutely)h(rigorous.) 1448 815 y FK(6.6.)46 b(Twisted)32 b FL(D)r FK(-mo)s(dules)605 965 y FQ(So)25 b(far,)g(w)n(e)g(ha)n(v)n(e)f(only)h(discussed)g(mo)r (dular)g(functors)f(in)i(whic)n(h)f(the)h(bundle)g(of)f(confor-)456 1065 y(mqal)j(blo)r(c)n(ks)f(carries)g(a)h(natural)f(\015at)i (connection)f(\(\\mo)r(dular)f(functors)h(with)h(zero)e(cen)n(tral)456 1164 y(c)n(harge"\).)65 b(Comparing)37 b(this)h(with)g(the)h (discussion)e(of)h(Section)g(5.7,)h(w)n(e)f(see)f(that)h(these)456 1264 y(mo)r(dular)29 b(functors)h(corresp)r(ond)e(to)i(categories)e (with)i FJ(p)2260 1234 y FM(+)2315 1264 y FJ(=p)2399 1234 y FE(\000)2481 1264 y FQ(=)d(1.)44 b(Ho)n(w)n(ev)n(er,)28 b(most)i(in)n(ter-)456 1363 y(esting)24 b(examples)g(|)i(for)e (example,)h(the)g(category)e(of)i(represen)n(tations)e(of)i(quan)n(tum) g(groups)456 1463 y(at)30 b(ro)r(ots)f(of)h(unit)n(y)h(|)f(do)g(not)g (satisfy)g(this)h(prop)r(ert)n(y)-7 b(.)44 b(As)30 b(w)n(as)g (discussed)f(in)i(Section)f(5.7,)456 1563 y(the)k(w)n(a)n(y)e(to)i (incorp)r(orate)e(mo)r(dular)h(categories)f(with)j FJ(p)2298 1533 y FM(+)2352 1563 y FJ(=p)2436 1533 y FE(\000)2525 1563 y FL(6)p FQ(=)e(1)h(is)f(to)h(de\014ne)g(mo)r(dular)456 1662 y(functor)26 b(with)i(cen)n(tral)e(c)n(harge.)34 b(In)27 b(top)r(ological)f(language,)f(it)i(w)n(as)f(de\014ned)h(as)f (a)h(pro)5 b(jectiv)n(e)456 1762 y(represen)n(tation)30 b(of)h(the)i(to)n(w)n(er)d(of)i(mapping)f(class)g(groups,)h(or,)g(more) f(precisely)-7 b(,)32 b(as)f(a)g(rep-)456 1862 y(resen)n(tation)h(of)i (a)g(suitable)g(cen)n(tral)f(extension)h(of)g(this)g(to)n(w)n(er)f (with)i(the)f(cen)n(tral)f(elemen)n(t)456 1961 y(acting)27 b(b)n(y)g(the)h(\014xed)g(constan)n(t)e FJ(K)34 b FQ(\(m)n (ultiplicativ)n(e)27 b(cen)n(tral)g(c)n(harge\).)605 2061 y(In)j(order)f(to)h(giv)n(e)f(an)h(analogous)e(description)i(of)g (the)g(mo)r(dular)g(functor)g(with)g(cen)n(tral)456 2160 y(c)n(harge,)h(w)n(e)g(need)i(to)e(in)n(tro)r(duce)h(the)g(appropriate) e(formalizm)i(|)g(namely)-7 b(,)33 b(the)f(notion)g(of)456 2260 y(t)n(wisted)h FL(D)r FQ(-mo)r(dules.)54 b(This)34 b(is)f(done)g(in)g(this)h(section;)i(in)d(the)h(next)g(section,)g(w)n (e)f(will)h(use)456 2360 y(this)27 b(formalizm)g(to)h(de\014ne)g(mo)r (dular)f(functor)g(with)h(cen)n(tral)f(c)n(harge.)605 2459 y(As)e(b)r(efore,)g(the)h(simplest)f(w)n(a)n(y)e(to)i(describ)r(e) g(suc)n(h)g(mo)r(dular)f(functors)g(is)h(to)g(replace)f(the)456 2559 y(requiremen)n(t)19 b(that)h(the)g(bundle)h(carry)d(a)i(\015at)g (connection)g(b)n(y)f(a)h FO(pr)l(oje)l(ctively)25 b(\015at)20 b FQ(connection,)456 2659 y(i.e.,)25 b(a)f(connection)g(suc)n(h)h(that) f([)p FL(r)1541 2671 y FI(X)1605 2659 y FJ(;)14 b FL(r)1711 2671 y FI(Y)1768 2659 y FQ(])e FL(\000)g(r)1949 2674 y FM([)p FI(X)q(;Y)j FM(])2145 2659 y FQ(is)24 b(an)h(op)r(erator)d(of) j(m)n(ultiplication)g(b)n(y)f(a)456 2759 y(function)30 b(\(dep)r(ending)g(on)f FJ(X)r(;)14 b(Y)k FQ(\).)43 b(Equiv)-5 b(alen)n(tly)e(,)29 b(w)n(e)g(can)g(sa)n(y)f(that)i(the)g(sheaf)f(of)g (sections)456 2858 y(carries)19 b(a)h(pro)5 b(jectiv)n(e)20 b(action)g(of)h(the)h(algebra)d(of)i(v)n(ector)e(\014elds,)k(or)d(that) h(the)h(sheaf)e(of)h(sections)456 2958 y(is)27 b(a)g(pro)5 b(jectiv)n(e)27 b FL(D)r FQ(-mo)r(dule.)605 3058 y(Ho)n(w)n(ev)n(er,)39 b(w)n(e)f(w)n(an)n(t)f(to)h(describ)r(e)g(the)h(failure)e(of)h(the)h (connection)f(to)g(b)r(e)g(\015at)g(more)456 3157 y(precisely)-7 b(,)34 b(b)n(y)g(describing)f(the)i(corresp)r(onding)d(cen)n(tral)h (extension)g(of)h(the)g(Lie)g(algebra)f(of)456 3257 y(v)n(ector)26 b(\014elds.)37 b(It)28 b(can)f(b)r(e)h(done)f(as)g(follo)n(ws.)605 3356 y(First,)i(let)g(us)g(sa)n(y)f(that)h(an)g FL(O)1570 3368 y FI(S)1618 3356 y FQ(-mo)r(dule)g(is)g(a)f(sheaf)h(of)g(Lie)g (algebras)d(if)k(w)n(e)e(ha)n(v)n(e)g(a)h(Lie)456 3456 y(algebra)d(structure)i(on)h(lo)r(cal)f(sections;)g(the)h(Lie)g(brac)n (k)n(et)e(do)r(es)h(not)g(ha)n(v)n(e)g(to)g(b)r(e)h FL(O)3142 3468 y FI(S)3190 3456 y FQ(-linear.)456 3556 y(F)-7 b(or)24 b(example,)g(the)h(sheaf)g(\002)1360 3568 y FI(S)1432 3556 y FQ(of)g(v)n(ector)e(\014elds)i(on)f FJ(S)29 b FQ(is)c(a)f(sheaf)g(of)h(Lie)f(algebras.)34 b(A)25 b(map)g(of)456 3655 y(Lie)j(algebra)f(shea)n(v)n(es)f(is)j(just)g(a)f(sheaf)g (morphism)g(whic)n(h)g(preserv)n(es)f(b)r(oth)i(the)g(Lie)f(brac)n(k)n (et)456 3755 y(and)f FL(O)683 3767 y FI(S)731 3755 y FQ(-mo)r(dule)h(structure.)605 3909 y FP(Definition)k FQ(6.6.1)p FP(.)40 b FQ(A)26 b(cen)n(tral)f(extension)g(of)h(\002)2198 3921 y FI(S)2272 3909 y FQ(is)g(a)f(sheaf)h(of)g(Lie)g(algebras)e FL(A)i FQ(on)g FJ(S)456 4008 y FQ(along)g(with)i(giv)n(en)f(maps)g(of)h (shea)n(v)n(es)d(of)j(Lie)g(algebras)d(giving)i(a)g(short)g(exact)g (sequence)1489 4175 y(0)c FL(!)g(O)1726 4187 y FI(S)1825 4124 y( )1797 4175 y FL(\000)-45 b(!)23 b(A)2043 4128 y FI(")2012 4175 y FL(\000)-56 b(!)24 b FQ(\002)2193 4187 y FI(S)2263 4175 y FL(!)f FQ(0)-1955 b(\(6.6.1\))456 4322 y(\(here)27 b FL(O)734 4334 y FI(S)810 4322 y FQ(is)g(considered)g (as)g(sheaf)g(of)h(Lie)f(algebras)f(with)i(zero)e(brac)n(k)n(et\))h (suc)n(h)g(that:)605 4443 y(1.)41 b FJ( )s FQ(\(1\))28 b(is)f(cen)n(tral)g(in)h FL(A)p FQ(.)605 4542 y(2.)41 b(F)-7 b(or)27 b FJ(a;)14 b(b)23 b FL(2)g(A)p FJ(;)14 b(f)32 b FL(2)23 b FJ(O)1395 4554 y FI(S)1444 4542 y FQ(,)k(w)n(e)h(ha)n(v)n(e)e([)p FJ(a;)14 b(f)9 b(b)p FQ(])22 b(=)h FJ(f)9 b FQ([)p FJ(a;)14 b(b)p FQ(])j(+)h(\()p FJ(")p FQ(\()p FJ(a)p FQ(\))p FJ(f)9 b FQ(\))p FJ(b)p FQ(.)605 4663 y(A)24 b(mo)r(dule)g FL(F)32 b FQ(o)n(v)n(er)22 b FL(A)j FQ(is)e(a)h(quasicoheren)n(t)e FL(O)r FQ(-mo)r(dule)i(with)h (the)f(action)f(of)h FL(A)h FQ(\(as)e(a)h(Lie)456 4763 y(algebra\))i(on)i FL(F)36 b FQ(whic)n(h)28 b(agrees)f(with)h(the)h FL(O)r FQ(-mo)r(dule)f(structure:)38 b FJ( )s FQ(\()p FJ(f)9 b FQ(\))p FJ(s)24 b FQ(=)f FJ(f)9 b(s;)42 b(f)33 b FL(2)24 b(O)r FJ(;)14 b(s)24 b FL(2)456 4863 y(F)8 b FQ(.)605 5016 y FP(Remark)32 b FQ(6.6.2)p FP(.)39 b FQ(This)20 b(is)f(a)g(sp)r(ecial)f(case)h(of)g(a)g(more)f(general)g (notion)h(of)g FO(A)n(tiyah)k(algebr)l(a)p FQ(,)456 5116 y(in)32 b(whic)n(h)h FL(O)866 5128 y FI(S)947 5116 y FQ(is)f(replaced)g(b)n(y)g(an)h(arbitrary)d(sheaf)j(of)f(asso)r(ciativ) n(e)f(algebras)g(o)n(v)n(er)f FL(O)r FQ(,)35 b(see)456 5216 y([)p FK(BS)p FQ(])27 b(for)g(details.)p eop %%Page: 159 25 159 162 bop 1518 226 a FM(6.6.)29 b(TWISTED)g FE(D)r FM(-MODULES)939 b(159)605 425 y FQ(One)31 b(can)f(easily)g(see)h(that)g (lo)r(cally)f(in)h FJ(S)5 b FQ(,)32 b(w)n(e)e(can)h(c)n(ho)r(ose)f(a)g (lifting,)j(i.e.)46 b(a)31 b(morphism)456 525 y(of)40 b FL(O)629 537 y FI(S)677 525 y FQ(-mo)r(dules)g FJ(a)9 b FQ(:)32 b(\002)44 b FL(!)g(A)p FQ(;)j(then)40 b(the)h(brac)n(k)n(et)e (can)g(b)r(e)i(written)f(as)g([)p FJ(a)p FQ(\()p FJ(X)7 b FQ(\))p FJ(;)14 b(a)p FQ(\()p FJ(Y)k FQ(\)])45 b(=)456 624 y FJ(a)p FQ(\([)p FJ(X)r(;)14 b(Y)k FQ(]\))h(+)f FJ(c)p FQ(\()p FJ(X)r(;)c(Y)k FQ(\),)28 b(where)f FJ(c)p FQ(\()p FJ(X)r(;)14 b(Y)19 b FQ(\))k FL(2)h(O)30 b FQ(is)d(a)g(2-co)r (cycle)g(on)g(\002)2612 636 y FI(S)2659 624 y FQ(.)605 782 y FP(Examples)k FQ(6.6.3)p FP(.)189 b FQ(1.)41 b(Let)22 b FL(A)1738 794 y FE(O)1820 782 y FQ(=)h FL(O)1974 794 y FI(S)2029 782 y FL(\010)7 b FQ(\002)2166 794 y FI(S)2236 782 y FQ(\(direct)22 b(sum)g(as)g FL(O)2831 794 y FI(S)2879 782 y FQ(-mo)r(dules\),)h(with)711 882 y(the)29 b(brac)n(k)n(et)d(giv)n (en)i(b)n(y)f([)p FJ(X)f FQ(+)18 b FJ(f)t(;)c(Y)37 b FQ(+)19 b FJ(g)s FQ(])k(=)h([)p FJ(X)r(;)14 b(Y)k FQ(])h(+)f FJ(X)7 b FQ(\()p FJ(g)s FQ(\))19 b FL(\000)f FJ(Y)h FQ(\()p FJ(f)9 b FQ(\),)28 b(where)g FJ(X)r(;)14 b(Y)42 b FL(2)711 981 y FQ(\002)p FJ(;)14 b(f)t(;)g(g)28 b FL(2)f(O)r FQ(,)k(and)e([)p FJ(X)r(;)14 b(Y)k FQ(])30 b(is)f(the)h(usual)f(brac)n(k)n(et)g(of)g(v)n (ector)f(\014elds.)43 b(This)30 b(pla)n(ys)e(the)711 1081 y(role)f(of)g(a)h(trivial)f(cen)n(tral)f(extension.)605 1181 y(2.)41 b(Let)31 b FJ(L)f FQ(b)r(e)h(a)f(line)g(bundle)h(on)g FJ(S)5 b FQ(,)31 b FL(L)p FQ(|)f(sheaf)h(of)f(sections)g(of)g FJ(L)p FQ(.)45 b(De\014ne)31 b FL(A)3143 1193 y FE(L)3224 1181 y FQ(as)f(the)711 1280 y(algebra)24 b(of)j(\014rst)f(order)e (di\013eren)n(tial)i(op)r(erators)e(in)j FJ(L)p FQ(.)36 b(If)26 b(w)n(e)g(c)n(ho)r(ose)f(a)h(lo)r(cal)f(trivial-)711 1380 y(ization)j(of)g FJ(L)p FQ(,)f(then)i(sections)e(of)h FL(A)g FQ(ha)n(v)n(e)f(the)i(form)e FJ(@)i FQ(=)23 b FJ(X)i FQ(+)18 b FJ(f)t(;)c(X)30 b FL(2)24 b FQ(\002)3070 1392 y FI(S)3118 1380 y FJ(;)14 b(f)32 b FL(2)24 b(O)3373 1392 y FI(S)3421 1380 y FQ(.)711 1482 y(In)k(other)g(w)n(ords,)f(c)n (hoice)g(of)i(trivialization)e FL(Lj)2210 1494 y FI(U)2318 1435 y FE(\030)2290 1482 y FL(\000)-40 b(!)24 b(O)r(j)2513 1494 y FI(U)2598 1482 y FQ(de\014nes)k(an)g(isomorphism)711 1589 y FL(A)777 1601 y FE(L)828 1589 y FL(j)851 1601 y FI(U)958 1542 y FE(\030)929 1589 y FL(\000)-39 b(!)23 b(A)1127 1601 y FE(O)1186 1589 y FL(j)1209 1601 y FI(U)1264 1589 y FQ(.)605 1688 y(3.)41 b(Let)29 b FL(A)f FQ(b)r(e)h(a)f(cen)n (tral)f(extension)h(of)g(\002,)g FJ(k)f FL(2)d FH(C)2196 1658 y FE(\002)2258 1688 y FQ(.)39 b(Then)29 b(w)n(e)e(can)h(de\014ne)h (the)f(cen)n(tral)711 1788 y(extension)23 b FL(A)1140 1758 y FI(k)1181 1788 y FQ(;)j(as)d(a)g(sheaf)g(of)h(Lie)f(algebras,)g (it)h(coincides)f(with)h FL(A)p FQ(,)h(but)f(the)g(em)n(b)r(ed-)711 1888 y(ding)c FL(O)26 b(!)d(A)1152 1857 y FI(k)1213 1888 y FQ(giv)n(en)d(b)n(y)f FJ( )s(=k)s FQ(,)j(where)d FJ( )24 b FQ(is)c(the)g(em)n(b)r(edding)h FL(O)k(!)e(A)p FQ(.)35 b(Equiv)-5 b(alen)n(tly)e(,)711 1987 y(if)32 b(w)n(e)f(lo)r(cally)f(c)n (ho)r(ose)g(a)h(lifting)h(\002)d FL(!)g(A)i FQ(so)g(that)g(the)h (extension)f FL(A)g FQ(is)h(giv)n(en)e(b)n(y)h(a)711 2087 y(2-co)r(cycle)h FJ(c)p FQ(\()p FJ(X)r(;)14 b(Y)19 b FQ(\),)35 b(then)f FL(A)1666 2057 y FI(k)1741 2087 y FQ(is)f(giv)n(en)g(b)n(y)g(the)h(2-co)r(cycle)e FJ(k)s(c)p FQ(\()p FJ(X)r(;)14 b(Y)k FQ(\),)35 b(whic)n(h)f(also)711 2186 y(sho)n(ws)25 b(that)g(it)h(is)g(w)n(ell-de\014ned)f(for)g FJ(k)h FQ(=)d(0.)36 b(One)25 b(can)g(easily)g(c)n(hec)n(k)f(that)i(for) f(in)n(teger)711 2286 y FJ(k)s FQ(,)h(one)e(has)g FL(A)1166 2303 y FE(L)1212 2286 y FG(k)33 b FQ(=)23 b(\()p FL(A)1462 2298 y FE(L)1513 2286 y FQ(\))1545 2256 y FI(k)1586 2286 y FQ(.)36 b(Using)25 b(this,)g(w)n(e)g(will)g(de\014ne)g(for)g(an)n(y)f FJ(k)i FL(2)d FH(C)46 b FQ(the)25 b(\\sheaf)711 2386 y(of)j(\014rst)f(order)f(di\013eren)n(tial)i(op)r(erators)d(in)j FL(L)2131 2356 y FI(k)2172 2386 y FQ(")f(b)n(y)1696 2534 y FL(A)1762 2551 y FE(L)1808 2535 y FG(k)1872 2534 y FQ(=)22 b(\()p FL(A)2057 2546 y FE(L)2108 2534 y FQ(\))2140 2500 y FI(k)2181 2534 y FJ(:)605 2692 y FQ(No)n(w)34 b(one)f(can)h(easily)f(see)h(that)g(ev)n(ery)f(pro)5 b(jectiv)n(ely)33 b(\015at)h(connection)f FL(r)i FQ(in)f(a)g(v)n(ector) 456 2792 y(bundle)26 b FJ(E)31 b FQ(de\014nes)25 b(a)h(cen)n(tral)e (extension)i FL(A)g FQ(of)f(\002)h(suc)n(h)f(that)h FL(r)g FQ(de\014nes)f(a)h(true)f(action)g(of)h FL(A)456 2892 y FQ(b)n(y)k(\014rst)h(order)f(di\013eren)n(tial)h(op)r(erators)e(in)i FJ(E)5 b FQ(.)47 b(In)32 b(other)e(w)n(ords,)h(failure)f(of)h(a)g(pro)5 b(jectiv)n(ely)456 2991 y(\015at)38 b(connection)g(to)g(b)r(e)h(\015at) f(can)g(b)r(e)h(describ)r(ed)f(b)n(y)g(a)g(cen)n(tral)g(extension)g FL(A)g FQ(of)h(the)g(Lie)456 3091 y(algebra)25 b(of)j(v)n(ector)e (\014elds.)605 3248 y FP(Exer)n(cise)32 b FQ(6.6.4)p FP(.)40 b FQ(Let)35 b FJ(L)g FQ(b)r(e)h(a)f(line)h(bundle)g(on)f FJ(S)5 b FQ(.)60 b(Sho)n(w)35 b(that)g FL(A)2874 3260 y FE(L)2960 3248 y FQ(is)h(isomorphic)456 3348 y(to)h(the)g(Lie)g (algebra)f(of)h(v)n(ector)f(\014elds)h(on)g(the)g(total)g(space)g(of)g FJ(L)g FQ(whic)n(h)g(comm)n(ute)g(with)456 3448 y(the)c(action)f(of)g FH(C)1011 3418 y FE(\002)1106 3448 y FQ(on)h FJ(L)f FQ(b)n(y)g (dilations;)j(lo)r(cally)-7 b(,)33 b(suc)n(h)g(v)n(ector)e(\014elds)i (ha)n(v)n(e)e(the)i(form)g FJ(@)j FQ(=)456 3547 y FJ(X)25 b FQ(+)19 b FJ(f)9 b FQ(\()p FJ(s)p FQ(\))p FJ(u@)879 3559 y FI(u)923 3547 y FJ(;)14 b(X)31 b FL(2)c FQ(\002)1207 3559 y FI(S)1254 3547 y FJ(;)14 b(f)34 b FL(2)26 b(O)1513 3559 y FI(S)1562 3547 y FQ(,)j(where)g FJ(u)g FQ(is)g(co)r(ordinate)f (along)g(the)h(\014b)r(ers)g(of)g FJ(L)p FQ(.)42 b(Using)456 3647 y(this,)f(sho)n(w)d(that)h(an)g(action)f(of)g FL(A)1620 3664 y FE(L)1666 3647 y FG(k)49 b FQ(on)38 b(a)h(v)n(ector)e(bundle)i FJ(E)44 b FQ(on)39 b FJ(S)k FQ(is)c(the)g(same)f(as)g(a)456 3747 y(mono)r(dromic)28 b(\015at)h(connection)f(on)h(the)g(pullbac)n(k) f FJ(\031)2144 3716 y FE(\003)2183 3747 y FJ(E)34 b FQ(of)29 b FJ(E)34 b FQ(to)29 b FJ(L)2629 3716 y FE(\002)2710 3747 y FQ(=)c FJ(L)18 b FL(n)h(f)p FQ(zero)26 b(section)p FL(g)456 3846 y FQ(suc)n(h)34 b(that)h(the)g(mono)r(drom)n(y)f(of)h (this)g(connection)f(around)g(the)h(zero)f(section)h(is)f(equal)h(to) 456 3946 y FJ(e)495 3916 y FE(\000)p FM(2)p FI(\031)r FM(i)p FI(k)680 3946 y FQ(.)605 4103 y(As)25 b(with)h(the)f(usual)f (\015at)h(connections,)g(w)n(e)g(can)f(also)g(use)h(the)g(language)e (of)i FL(D)r FQ(-mo)r(dules.)456 4203 y(The)k(appropriate)f (generalization)f(of)j(the)f(notion)h(of)f FL(D)r FQ(-mo)r(dule)h(is)f (the)h(notion)f(of)g(t)n(wisted)456 4303 y FL(D)r FQ(-mo)r(dule.)605 4460 y FP(Definition)j FQ(6.6.5)p FP(.)40 b FQ(A)33 b FO(twiste)l(d)h(she)l(af)h(of)h(di\013er)l(ential)f(op)l(er)l(ators)f FQ(on)e FJ(S)38 b FQ(is)32 b(a)g(sheaf)h(of)456 4560 y(asso)r(ciativ)n(e)25 b(algebras)g FL(U)36 b FQ(on)27 b FJ(S)32 b FQ(and)27 b(an)g(em)n(b)r(edding)g FL(O)2239 4572 y FI(S)2310 4560 y FJ(,)-14 b FL(!)24 b(U)35 b FQ(suc)n(h)27 b(that)h(lo)r(cally)-7 b(,)27 b(the)g(pair)456 4659 y(\()p FL(U)8 b FJ(;)14 b FL(O)25 b FJ(,)-14 b FL(!)24 b(U)8 b FQ(\))22 b(is)f(isomorphic)f(to)h(\()p FL(D)r FJ(;)14 b FL(O)26 b FJ(,)-14 b FL(!)23 b(D)r FQ(\).)36 b(A)21 b FO(twiste)l(d)j FL(D)r FO(-mo)l(dule)e FQ(is)f(a)g(sheaf)g(of)g(mo)r (dules)456 4759 y(o)n(v)n(er)k(a)j(t)n(wisted)f(sheaf)h(of)f (di\013eren)n(tial)h(op)r(erators.)605 4917 y(It)f(turns)g(out)f(that)h (the)g(notions)f(of)h(t)n(wisted)g(shea)n(v)n(es)d(of)j(di\013eren)n (tial)f(op)r(erators)f(and)i(of)456 5016 y(cen)n(tral)21 b(extensions)h(of)h(\002)g(are)e(equiv)-5 b(alen)n(t.)35 b(Namely)-7 b(,)24 b(for)e(a)h(t)n(wisted)g(sheaf)f FL(U)31 b FQ(of)23 b(d.o.,)g(w)n(e)g(can)456 5116 y(de\014ne)32 b(the)g(subsheaf)f FL(U)1238 5128 y FM(1)1308 5116 y FQ(of)g(di\013eren)n(tial)h(op)r(erators)e(of)i(\014rst)f(order.)48 b(A)32 b(reader)f(can)g(easily)456 5216 y(c)n(hec)n(k)d(that)h FL(U)914 5228 y FM(1)981 5216 y FQ(is)g(closed)g(under)g(the)g(Lie)g (brac)n(k)n(et)f([)p FJ(a;)14 b(b)p FQ(])25 b(=)h FJ(ab)19 b FL(\000)g FJ(ba)29 b FQ(and)g(the)g(action)g(of)g FL(U)3407 5228 y FM(1)p eop %%Page: 160 26 160 163 bop 456 226 a FM(160)445 b(6.)30 b(MODULI)f(SP)-5 b(A)n(CES)28 b(AND)h(COMPLEX)f(MODULAR)h(FUNCTOR)456 425 y FQ(on)c FL(O)j FQ(b)n(y)e FJ(@)5 b FQ(\()p FJ(f)k FQ(\))23 b(=)f([)p FJ(@)5 b(;)14 b(f)9 b FQ(])25 b(de\014nes)h(an)g (isomorphism)e FL(U)2177 437 y FM(1)2215 425 y FJ(=)p FL(O)2376 378 y FE(\030)2347 425 y FL(\000)-39 b(!)23 b FQ(\002,)j(and)g(th)n(us)f FL(U)2985 437 y FM(1)3048 425 y FQ(is)h(a)f(cen)n(tral)456 525 y(extension)i(of)g(\002.)37 b(Con)n(v)n(ersely)-7 b(,)25 b(if)j FL(A)g FQ(is)g(a)f(cen)n(tral)g (extension)g(of)g(\002,)h(then)g(de\014ne)1257 702 y FL(U)8 b FQ(\()p FL(A)p FQ(\))24 b(=)f FJ(U)9 b FQ(\()p FL(A)p FQ(\))p FJ(=)p FQ(\()p FJ(f)27 b FL(\000)18 b FJ( )s FQ(\()p FJ(f)9 b FQ(\)\))p FJ(;)181 b(f)31 b FL(2)24 b(O)r FJ(;)456 885 y FQ(where)g FJ(U)9 b FQ(\()p FL(A)p FQ(\))26 b(is)e(the)i(univ)n(ersal)d(en)n(v)n(eloping)h(algebra)f(of)i FL(A)g FQ(\(as)g(an)f FL(O)r FQ(-mo)r(dule,)i(it)f(is)g(isomor-)456 985 y(phic)i(to)h FL(O)21 b(\010)d(A)h(\010)f FJ(S)1131 954 y FM(2)1168 985 y FQ(\()p FL(A)p FQ(\))h FL(\010)f FJ(:)c(:)g(:)g FQ(\).)605 1159 y FP(Lemma)31 b FQ(6.6.6)p FP(.)40 b FO(The)29 b(functors)g FL(U)i(7!)23 b(U)1885 1171 y FM(1)1922 1159 y FO(,)29 b FL(A)24 b(7!)f(U)8 b FQ(\()p FL(A)p FQ(\))29 b FO(ar)l(e)g(inverse)g(to)f(e)l(ach)i(other) f(and)456 1258 y(thus)22 b(give)h(an)g(e)l(quivalenc)l(e)h(of)f(c)l (ate)l(gories)h(of)g(twiste)l(d)f(algebr)l(as)h(of)f(di\013er)l(ential) h(op)l(er)l(ators)g(and)456 1358 y(c)l(entr)l(al)30 b(extensions)f(of)i FQ(\002)p FO(.)40 b(In)30 b(p)l(articular,)h(twiste)l(d)f FL(D)r FO(-mo)l(dules)h(ar)l(e)g(the)f(same)g(as)h(mo)l(dules)456 1457 y(over)f(c)l(entr)l(al)g(extensions)f(of)i FQ(\002)p FO(.)605 1632 y FQ(W)-7 b(e)28 b(refer)f(the)h(reader)e(to)h([)p FK(BS)q FQ(])g(for)g(the)h(pro)r(of)f(\(easy\))h(and)f(more)g(details.) 605 1806 y FP(Example)k FQ(6.6.7)p FP(.)40 b FQ(F)-7 b(or)26 b(a)g(line)h(bundle)g FJ(L)p FQ(,)f(let)h FL(D)2168 1818 y FI(L)2245 1806 y FQ(b)r(e)g(the)g(sheaf)f(of)h(di\013eren)n (tial)f(op)r(era-)456 1905 y(tors)c(in)h FJ(L)p FQ(.)35 b(This)23 b(is)g(a)g(t)n(wisted)g(sheaf)g(of)g(d.o.,)h(whic)n(h)f (corresp)r(onds)e(to)i(the)h(cen)n(tral)e(extension)456 2005 y FL(A)522 2017 y FE(L)572 2005 y FQ(.)605 2179 y(In)41 b(a)f(similar)g(w)n(a)n(y)-7 b(,)43 b(w)n(e)e(can)f(de\014ne)h (the)h(\\sheaf)e FL(D)2331 2196 y FE(L)2377 2179 y FG(k)50 b FQ(of)41 b(di\013eren)n(tial)f(op)r(erators)f(in)456 2278 y FJ(L)513 2248 y FI(k)553 2278 y FQ(")32 b(as)g(the)h(t)n(wisted) f(sheaf)h(of)f(d.o.)52 b(corresp)r(onding)30 b(to)j(the)g(cen)n(tral)e (extension)h FL(A)3186 2295 y FE(L)3232 2279 y FG(k)3306 2278 y FQ(\(see)456 2378 y(Example)26 b(6.6.3\):)1669 2556 y FL(D)1733 2573 y FE(L)1779 2556 y FG(k)33 b FQ(=)22 b FL(U)8 b FQ(\()p FL(A)2088 2573 y FE(L)2134 2556 y FG(k)j FQ(\))p FJ(:)-1775 b FQ(\(6.6.2\))605 2750 y FP(Exer)n(cise)32 b FQ(6.6.8)p FP(.)40 b FQ(Let)34 b(us)h(c)n(ho)r(ose)f(lo)r(cal)g (trivializations)f(of)i FJ(L)p FQ(:)50 b FJ(')2801 2762 y FI(\013)2858 2750 y FQ(:)30 b FL(Lj)2991 2762 y FI(U)3036 2770 y FG(\013)3146 2703 y FE(\030)3118 2750 y FL(\000)-39 b(!)34 b(O)r(j)3352 2762 y FI(U)3397 2770 y FG(\013)456 2850 y FQ(\(where)19 b FL(f)p FJ(U)819 2862 y FI(\013)866 2850 y FL(g)g FQ(is)g(an)h(op)r(en)g(co)n(v)n(er)d(of)j FJ(S)5 b FQ(\),)21 b(and)f(let)g FJ(f)2035 2862 y FI(\013\014)2146 2850 y FQ(=)i FJ(')2287 2862 y FI(\014)2332 2850 y FJ(')2386 2820 y FE(\000)p FM(1)2386 2870 y FI(\013)2499 2850 y FL(2)h(O)r FQ(\()p FJ(U)2734 2862 y FI(\013)2785 2850 y FL(\\)s FJ(U)2900 2862 y FI(\014)2944 2850 y FQ(\))d(b)r(e)g(the)g (corre-)456 2959 y(sp)r(onding)h(transition)g(functions.)35 b(Sho)n(w)22 b(that)g(this)g(de\014nes)f(isomorphisms)g FJ(')2941 2971 y FI(\013)2997 2959 y FQ(:)28 b FL(D)3112 2976 y FI(L)3158 2960 y FG(k)3198 2959 y FL(j)3221 2971 y FI(U)3266 2979 y FG(\013)3364 2912 y FE(\030)3336 2959 y FL(\000)-40 b(!)456 3059 y(D)r(j)545 3071 y FI(U)590 3079 y FG(\013)636 3059 y FQ(,)34 b(and)e(the)g(transition)g(functions) g FJ(')1806 3071 y FI(\014)1851 3059 y FJ(')1905 3029 y FE(\000)p FM(1)1905 3080 y FI(\013)2004 3059 y FQ(:)d FL(D)r FQ(\()p FJ(U)2211 3071 y FI(\013)2280 3059 y FL(\\)22 b FJ(U)2414 3071 y FI(\014)2458 3059 y FQ(\))31 b FL(!)g(D)r FQ(\()p FJ(U)2790 3071 y FI(\013)2859 3059 y FL(\\)22 b FJ(U)2993 3071 y FI(\014)3037 3059 y FQ(\),)34 b(when)e(re-)456 3159 y(stricted)27 b(to)h(v)n(ector)e(\014elds,)i(are)e(giv)n(en)h(b)n (y)1478 3380 y FJ(')1532 3392 y FI(\014)1577 3380 y FJ(')1631 3345 y FE(\000)p FM(1)1631 3400 y FI(\013)1729 3380 y FQ(:)h FJ(v)e FL(7!)d FJ(v)f FQ(+)c FJ(k)2153 3323 y(v)s FQ(\()p FJ(f)2269 3335 y FI(\013\014)2357 3323 y FQ(\))p 2153 3360 237 4 v 2207 3436 a FJ(f)2248 3448 y FI(\013\014)2399 3380 y FJ(:)605 3604 y FQ(Note)29 b(that)g(the)g(righ)n(t-hand)e(side)h (is)h(not)f(a)h(v)n(ector)e(\014eld)i(but)g(a)f(\014rst)g(order)g (di\013eren)n(tial)456 3704 y(op)r(erator.)605 3878 y FP(Exer)n(cise)k FQ(6.6.9)p FP(.)40 b FQ(Sho)n(w)25 b(that)h(if)g FJ(L)f FQ(admits)h(a)g(\015at)f(connection,)h(then)g FL(D)2964 3890 y FI(L)3037 3878 y FL(')d(D)r FQ(,)j FL(A)3306 3890 y FE(L)3380 3878 y FL(')456 3977 y FJ(A)518 3989 y FE(O)576 3977 y FQ(.)605 4151 y(As)e(for)f(usual)h FL(D)r FQ(-mo)r(dules,)h(w)n(e)e(can)h(de\014ne)g(the)g(category)e FJ(R)q(S)2589 4163 y FI(L)2635 4147 y FG(c)2669 4151 y FQ(\()p 2701 4085 90 4 v FJ(M)9 b(;)14 b(M)9 b FQ(\))24 b(\(where)g FJ(L)f FQ(is)h(a)456 4254 y(v)n(ector)18 b(bundle)i(o)n(v)n(er)p 1131 4187 V 18 w FJ(M)8 b FQ(\))20 b(as)f(the)h(category)e(of)h(v)n(ector)f(bundles)i(o)n(v)n(er)p 2622 4187 V 18 w FJ(M)28 b FQ(\(up)20 b(to)g(meromorphic)456 4354 y(equiv)-5 b(alence\))30 b(with)i(an)e(action)h(of)g FL(D)1655 4366 y FI(L)1701 4349 y FG(c)1766 4354 y FQ(on)g(the)g(sheaf) g(of)g(sections)f(suc)n(h)h(that)g FJ(E)36 b FQ(admits)31 b(a)456 4453 y(trivialization)26 b(suc)n(h)g(that)i(the)f(action)g(of)g (v)n(ector)f(\014elds)h(has)f(\014rst)h(order)f(p)r(oles.)37 b(Indeed,)27 b(the)456 4553 y(regularit)n(y)g(condition)h(is)h(lo)r (cal,)g(and)g(lo)r(cally)f(a)h(t)n(wisted)g(sheaf)g(of)g(di\013eren)n (tial)g(op)r(erators)e(is)456 4652 y(isomorphic)f(to)h(the)h(usual)g (sheaf)f FL(D)j FQ(of)e(di\013eren)n(tial)f(op)r(erators.)605 4752 y(Similarly)-7 b(,)27 b(w)n(e)g(can)h(de\014ne)g(the)g(sp)r (ecialization)e(functor)1204 4930 y FJ(S)5 b(p)1302 4942 y FI(D)1371 4930 y FQ(:)27 b FJ(R)q(S)1536 4942 y FI(L)1582 4925 y FG(c)1617 4930 y FQ(\()p 1649 4863 V FJ(M)9 b(;)14 b(M)9 b FQ(\))23 b FL(!)g FJ(R)q(S)2142 4942 y FI(L)2188 4925 y FG(c)2222 4930 y FQ(\()p FJ(N)9 b(D)r(;)14 b(N)2514 4895 y FE(\002)2570 4930 y FJ(D)r FQ(\))p FJ(:)456 5116 y FQ(Note)22 b(that)h(restriction)f(of)g FJ(L)g FQ(to)h(the)g(\014b)r (er)g FJ(N)1876 5128 y FI(d)1914 5116 y FJ(D)i FL(')e FH(C)2150 5086 y FE(\002)2234 5116 y FQ(is)g(necessarily)e(trivial,)i (so)f(that)h(there)456 5216 y(is)k(no)g(need)h(to)g(c)n(hange)e(the)i (de\014nition)g(of)g(mono)r(dromic)e(connection.)p eop %%Page: 161 27 161 164 bop 1094 226 a FM(6.7.)29 b(MODULAR)g(FUNCTOR)h(WITH)g(CENTRAL) f(CHAR)n(GE)518 b(161)1087 425 y FK(6.7.)46 b(Mo)s(dular)31 b(functor)i(with)e(cen)m(tral)i(c)m(harge)605 575 y FQ(In)27 b(order)f(to)h(apply)g(the)g(tec)n(hnique)g(of)g(t)n(wisted)h FJ(D)r FQ(-mo)r(dules)e(to)h(mo)r(dular)g(functors,)g(w)n(e)456 674 y(m)n(ust)d(c)n(ho)r(ose)f(a)h(line)h(bundle)f FJ(L)g FQ(on)g(eac)n(h)g(of)g(the)h(mo)r(duli)g(spaces)e FJ(M)2622 686 y FI(g)r(;n)2745 674 y FQ(in)i(a)f(consisten)n(t)g(w)n(a)n(y)-7 b(.)456 774 y(W)g(e)27 b(will)g(use)f(the)h(so-called)f FO(determinant)j(line)g(bund)t(le)f FQ(in)n(tro)r(duced)e(b)n(y)h (Grothendiec)n(k)f(\(see)456 874 y([)p FK(KM)o FQ(])k(for)f(details\).) 42 b(As)30 b(b)r(efore,)f(in)h(order)e(to)h(de\014ne)h(a)f(line)g (bundle)h(on)g FJ(M)2923 886 y FI(g)r(;n)3051 874 y FQ(w)n(e)f(need)h (to)456 973 y(de\014ne)d(a)h(line)f(bundle)i(on)e FJ(S)32 b FQ(for)27 b(ev)n(ery)g(family)g(of)h(curv)n(es)e FJ(C)2400 985 y FI(S)2476 973 y FQ(o)n(v)n(er)g FJ(S)5 b FQ(.)605 1073 y(Before)27 b(doing)g(so,)h(let)g(us)g(in)n(tro)r(duce)f(some)g (notation.)38 b(Let)28 b FJ(L)f FQ(b)r(e)h(a)g(\014nite-dimensional)456 1172 y(v)n(ector)g(space;)i(w)n(e)g(de\014ne)g(one)f(dimensional)h(v)n (ector)e(space)h(det)14 b FJ(L)30 b FQ(as)f(the)h(highest)g(exterior) 456 1272 y(p)r(o)n(w)n(er)c(of)i FJ(L)p FQ(:)1648 1415 y(det)14 b FJ(L)22 b FQ(=)h(\003)2002 1380 y FM(dim)10 b FI(L)2173 1415 y FJ(L:)456 1558 y FQ(More)26 b(generally)-7 b(,)26 b(for)g(a)h(\014nite)h(complex)e(of)h(\014nite-dimensional)g(v)n (ector)f(spaces)g FJ(L)3093 1528 y FE(\017)3154 1558 y FQ(=)c FJ(:)14 b(:)g(:)23 b FL(!)456 1665 y FJ(L)513 1677 y FI(i)p FE(\000)p FM(1)655 1665 y FL(!)31 b FJ(L)826 1677 y FI(i)883 1665 y FL(!)g FJ(L)1054 1677 y FI(i)p FM(+1)1196 1665 y FL(!)f FJ(:)14 b(:)g(:)g FQ(,)33 b(w)n(e)f(denote)g (det)14 b FJ(L)2061 1635 y FE(\017)2129 1665 y FQ(=)2224 1603 y Fy(N)2317 1665 y FQ(\(det)g FJ(L)2535 1677 y FI(i)2562 1665 y FQ(\))2594 1635 y FM(\()p FE(\000)p FM(1\))2731 1610 y FG(i)2762 1665 y FQ(,)33 b(where,)g(for)f(a)f(one-)456 1765 y(dimensional)c(v)n(ector)f(space)h FJ(X)7 b FQ(,)27 b(w)n(e)g(let)h FJ(X)1828 1735 y FM(1)1888 1765 y FQ(=)22 b FJ(X)7 b FQ(,)27 b FJ(X)2177 1735 y FE(\000)p FM(1)2289 1765 y FQ(=)22 b FJ(X)2452 1735 y FE(\003)2490 1765 y FQ(.)605 1919 y FP(Exer)n(cise)32 b FQ(6.7.1)p FP(.)40 b FQ(Sho)n(w)27 b(that)g(there)h(is)f(a)h(canonical)e(isomorphism)1397 2074 y(det)14 b FJ(L)1583 2039 y FE(\017)1644 2074 y FQ(=)1731 1995 y Fy(O)1857 2074 y FQ(\(det)g FJ(H)2094 2039 y FI(i)2121 2074 y FQ(\()p FJ(L)2210 2039 y FE(\017)2248 2074 y FQ(\)\))2312 2039 y FM(\()p FE(\000)p FM(1\))2449 2014 y FG(i)2480 2074 y FJ(:)605 2240 y FQ(This)38 b(de\014nition)h (can)e(b)r(e)i(generalized)e(to)h(v)n(ector)f(bundles)h(o)n(v)n(er)e(a) i(smo)r(oth)g(base)g FJ(S)5 b FQ(:)456 2340 y(if)36 b FJ(E)k FQ(is)c(a)f(v)n(ector)g(bundle)h(of)f(dimension)h FJ(n)f FQ(o)n(v)n(er)f FJ(S)5 b FQ(,)38 b(then)e(w)n(e)f(de\014ne)h (line)g(bundle)g(det)14 b FJ(L)456 2439 y FQ(b)n(y)33 b(det)14 b FJ(L)33 b FQ(=)f(\003)951 2409 y FI(n)996 2439 y FJ(E)5 b FQ(.)56 b(Again,)34 b(this)g(can)g(b)r(e)g(trivially)f (generalized)f(to)h(complexes)g(of)h(v)n(ector)456 2539 y(bundles,)28 b(and)f(w)n(e)g(ha)n(v)n(e)g(the)h(follo)n(wing)e(prop)r (osition.)605 2693 y FP(Lemma)31 b FQ(6.7.2)p FP(.)40 b FO(L)l(et)f FJ(E)1372 2663 y FE(\017)1411 2693 y FJ(;)14 b(F)1513 2663 y FE(\017)1591 2693 y FO(b)l(e)39 b(\014nite)h(c)l (omplexes)g(of)h(ve)l(ctor)f(bund)t(les)h(over)f FJ(S)5 b FO(,)43 b(and)456 2793 y(let)35 b FJ(f)18 b FQ(:)30 b FJ(E)756 2762 y FE(\017)828 2793 y FL(!)k FJ(F)1010 2762 y FE(\017)1083 2793 y FO(b)l(e)i(a)g(morphism)h(of)g(c)l(omplexes) f(of)h(ve)l(ctor)f(bund)t(les)g(which)h(is)f(a)g(quasi-)456 2895 y(isomorphism,)25 b(i.e.,)g(it)d(induc)l(es)f(isomorphism)i(of)g (the)e(c)l(ohomolo)l(gy)j(she)l(aves)e FL(H)2972 2865 y FI(i)3000 2895 y FJ(f)17 b FQ(:)28 b FL(H)3180 2865 y FI(i)3208 2895 y FQ(\()p FL(E)3291 2865 y FE(\017)3329 2895 y FQ(\))3413 2848 y FE(\030)3385 2895 y FL(\000)-40 b(!)456 2994 y(H)527 2964 y FI(i)554 2994 y FQ(\()p FL(F)654 2964 y FE(\017)693 2994 y FQ(\))p FO(.)38 b(Then)31 b FJ(f)38 b FO(de\014nes)30 b(an)g(isomorphism)i(of)e(the)g(line)h(bund)t (les)f FQ(det)14 b FJ(E)2859 2964 y FE(\017)2920 2994 y FL(')23 b FQ(det)14 b FJ(F)3202 2964 y FE(\017)3240 2994 y FO(.)605 3148 y FQ(This)25 b(lemma)h(allo)n(ws)e(one)h(to)g (de\014ne)h(det)14 b FL(E)33 b FQ(for)25 b(arbitrary)f(coheren)n(t)g FL(O)r FQ(-mo)r(dule)i FL(E)7 b FQ(,)26 b(gen-)456 3248 y(eralizing)20 b(the)h(case)g(when)g FL(E)29 b FQ(is)21 b(the)h(sheaf)f(of)g(sections)g(of)g(a)g(v)n(ector)f(bundle)i FJ(E)5 b FQ(.)35 b(Indeed,)23 b(ev)n(ery)456 3348 y(coheren)n(t)e FL(O)r FQ(-mo)r(dule)i(admits)g(a)f(resolution)f(b)n(y)h(v)n(ector)f (bundles:)35 b(w)n(e)22 b(can)g(\014nd)h(a)f(complex)h(of)456 3447 y(v)n(ector)k(bundles)i FJ(F)1077 3417 y FE(\017)1144 3447 y FQ(suc)n(h)g(that)g FL(H)1585 3417 y FM(0)1622 3447 y FQ(\()p FL(F)1722 3417 y FE(\017)1760 3447 y FQ(\))d(=)f FL(E)7 b FJ(;)14 b FL(H)2067 3417 y FI(i)2094 3447 y FQ(\()p FL(F)2194 3417 y FE(\017)2233 3447 y FQ(\))25 b(=)g(0)j(for)g FJ(i)d FL(6)p FQ(=)g(0.)40 b(By)29 b(de\014nition,)g (let)456 3547 y(det)14 b FL(E)30 b FQ(=)23 b(det)14 b FJ(F)941 3517 y FE(\017)979 3547 y FQ(.)36 b(Lemma)23 b(6.7.2)g(sho)n(ws)f(that)j(this)f(is)f(indep)r(enden)n(t)i(of)f(the)g (c)n(hoice)f(of)h(resolu-)456 3647 y(tion.)35 b(The)23 b(same)f(argumen)n(t)f(sho)n(ws)h(that)h(w)n(e)f(can)g(de\014ne)h(det) 14 b FL(E)2480 3616 y FE(\017)2541 3647 y FQ(for)22 b(a)h(complex)f(of) g(coheren)n(t)456 3746 y FL(O)r FQ(-mo)r(dules)30 b FL(E)931 3716 y FE(\017)970 3746 y FQ(,)h(and)g(that)g(it)g(only)f(dep)r(ends)h (on)g(the)g(quasi-isomorphism)d(class)i(of)g FL(E)3281 3716 y FE(\017)3320 3746 y FQ(;)i(in)456 3846 y(other)23 b(w)n(ords,)h(det)g(is)g(w)n(ell)g(de\014ned)h(on)f(the)g(deriv)n(ed)g (category)e(of)i(coheren)n(t)f FL(O)r FQ(-mo)r(dules)i(\(see)456 3945 y([)p FK(KM)o FQ(])j(for)f(details\).)605 4100 y FP(Remark)32 b FQ(6.7.3)p FP(.)39 b FQ(In)i([)p FK(KM)o FQ(],)j(the)c(determinan)n(t)g(line)g(bundle)h(is)f(de\014ned)g(as)f(a) h(pair,)456 4199 y(consisting)e(of)h(a)f(line)i(bundle)f(a)g(and)g (\\parit)n(y",)h(i.e.)71 b(an)39 b(elemen)n(t)g(of)g FH(Z)p FJ(=)o FQ(2)p FH(Z)-7 b FQ(.)65 b(P)n(arit)n(y)38 b(is)456 4299 y(imp)r(ortan)n(t)18 b(for)h(trac)n(king)e(correct)h (signs)g(in)i(isomorphisms)d(lik)n(e)i(det\()p FJ(F)13 b FL(\010)q FJ(G)p FQ(\))24 b FL(')f FQ(det)14 b FJ(F)f FL(\012)q FQ(det)h FJ(G)p FQ(.)456 4399 y(Ho)n(w)n(ev)n(er,)25 b(for)i(us)h(these)g(signs)e(are)h(not)h(imp)r(ortan)n(t,)f(and)g (therefore)g(w)n(e)g(omit)h(parit)n(y)-7 b(.)605 4553 y FP(Definition)32 b FQ(6.7.4)p FP(.)40 b FQ(Let)25 b FJ(C)1519 4565 y FI(S)1592 4553 y FQ(b)r(e)h(a)e(family)h(of)g(p)r(oin) n(ted)h(curv)n(es)d(o)n(v)n(er)h FJ(S)5 b FQ(.)35 b(W)-7 b(e)26 b(de\014ne)f(the)456 4652 y(corresp)r(onding)g FO(determinant)30 b(line)g(bund)t(le)f FJ(Q)1948 4664 y FI(S)2023 4652 y FQ(b)n(y)1045 4807 y FJ(Q)23 b FQ(=)g(\(det)14 b FJ(R)q(\031)1494 4819 y FE(\003)1532 4807 y FL(O)1598 4819 y FI(C)1646 4827 y FG(S)1692 4807 y FQ(\))1724 4773 y FE(\000)p FM(1)1836 4807 y FQ(=)1924 4728 y Fy(O)2049 4807 y FQ(\(det)h FJ(R)2275 4773 y FI(i)2302 4807 y FJ(\031)2349 4819 y FE(\003)2388 4807 y FL(O)2454 4819 y FI(C)2502 4827 y FG(S)2547 4807 y FQ(\))2579 4773 y FM(\()p FE(\000)p FM(1\))2716 4748 y FG(i)p FF(+1)2818 4807 y FJ(;)-2385 b FQ(\(6.7.1\))456 4962 y(where)27 b FJ(\031)k FQ(is)c(the)h(pro)5 b(jection)27 b FJ(C)1454 4974 y FI(S)1525 4962 y FL(!)c FJ(S)5 b FQ(.)605 5116 y(Note)29 b(that)g(this)g(de\014nition)g(do)r (es)f(not)h(use)g(the)g(mark)n(ed)e(p)r(oin)n(ts.)40 b(Also,)29 b(this)g(de\014nition)456 5216 y(is)34 b(v)-5 b(alid)35 b(ev)n(en)g(if)g(the)h(family)f FJ(C)1505 5228 y FI(S)1588 5216 y FQ(is)g(singular:)51 b(in)35 b(this)g(case,)h FJ(R)2586 5185 y FI(i)2613 5216 y FJ(\031)2660 5228 y FE(\003)2699 5216 y FL(O)2765 5228 y FI(C)2813 5236 y FG(S)2894 5216 y FQ(needs)f(not)g(b)r(e)g(a)p eop %%Page: 162 28 162 165 bop 456 226 a FM(162)445 b(6.)30 b(MODULI)f(SP)-5 b(A)n(CES)28 b(AND)h(COMPLEX)f(MODULAR)h(FUNCTOR)456 425 y FQ(v)n(ector)23 b(bundle,)j(but)f(is)f(alw)n(a)n(ys)f(a)h (coheren)n(t)g FL(O)1951 437 y FI(S)1999 425 y FQ(-mo)r(dule,)h(and)g (th)n(us)f(det)14 b FJ(R)2871 395 y FI(i)2899 425 y FJ(\031)2946 437 y FE(\003)2984 425 y FL(O)3050 437 y FI(C)3098 445 y FG(S)3169 425 y FQ(is)24 b(a)g(line)456 525 y(bundle.)605 624 y(F)-7 b(or)36 b(readers)e(who)i(prefer)g(to)h(a)n(v)n(oid)d(using) i(the)h(notion)f(of)h(higher)e(direct)h(images,)i(it)456 724 y(su\016ces)27 b(to)g(sa)n(y)g(that)h(the)g(\014b)r(er)f(of)h(this) g(line)g(bundle)g(at)f(p)r(oin)n(t)h FJ(s)23 b FL(2)g FJ(S)33 b FQ(is)1329 879 y FJ(Q)1395 891 y FI(s)1453 879 y FQ(=)1541 800 y Fy(O)1666 879 y FQ(\(det)15 b FJ(H)1904 844 y FI(i)1931 879 y FQ(\()p FJ(C)2022 891 y FI(s)2058 879 y FJ(;)f FL(O)2161 891 y FI(C)2209 899 y FG(s)2245 879 y FQ(\)\))2309 844 y FM(\()p FE(\000)p FM(1\))2446 819 y FG(i)p FF(+1)2548 879 y FJ(:)605 1033 y FQ(F)-7 b(or)18 b(a)g(smo)r(oth)h(family)-7 b(,)20 b(this)f(de\014nition)g(can) g(b)r(e)g(simpli\014ed.)34 b(Let)19 b(us)f(assume)g(that)h FJ(C)3261 1045 y FI(S)3328 1033 y FQ(is)g(a)456 1133 y(non-singular)h(family)i(with)h(connected)f(\014b)r(er.)35 b(Then)22 b FJ(H)2249 1103 y FM(0)2286 1133 y FQ(\()p FJ(C)2377 1145 y FI(s)2413 1133 y FJ(;)14 b FL(O)r FQ(\))24 b(=)f FH(C)15 b FQ(,)29 b(and)22 b FJ(H)3000 1103 y FI(i)3027 1133 y FQ(\()p FJ(C)3118 1145 y FI(s)3155 1133 y FJ(;)14 b FL(O)r FQ(\))23 b(=)g(0)456 1233 y(for)28 b FJ(i)c(>)g FQ(1)29 b(\(since)f FL(O)j FQ(is)e(coheren)n(t\).)39 b(Th)n(us,)29 b(in)g(this)f(case)g FJ(Q)d FQ(=)f(det)14 b FJ(R)2662 1203 y FM(1)2699 1233 y FJ(\031)2746 1245 y FE(\003)2784 1233 y FQ(\()p FL(O)2882 1245 y FI(C)2930 1253 y FG(S)2976 1233 y FQ(\),)30 b(so)e(its)g(\014b)r(er)456 1332 y(at)f(p)r(oin)n(t)h FJ(s)f FQ(is)h(giv)n(en)f(b)n(y)1513 1475 y FJ(Q)1579 1487 y FI(s)1637 1475 y FQ(=)c(det\()p FJ(H)1948 1441 y FM(1)1985 1475 y FQ(\()p FJ(C)2076 1487 y FI(s)2113 1475 y FJ(;)14 b FL(O)2216 1487 y FI(C)2264 1495 y FG(s)2299 1475 y FQ(\)\))p FJ(:)-1930 b FQ(\(6.7.2\))456 1623 y(These)36 b(spaces)f(are)h(w)n(ell)g(kno)n(wn)g(\(see,)j(e.g.,)f ([)p FK(GH)p FQ(]\):)55 b(for)36 b(a)g(compact)g(complex)g(curv)n(e)g (of)456 1722 y(gen)n(us)c FJ(g)s FQ(,)j FJ(H)867 1692 y FM(1)904 1722 y FQ(\()p FJ(C)q(;)14 b FL(O)1099 1734 y FI(C)1156 1722 y FQ(\))34 b(is)g(a)f(v)n(ector)f(space)h(of)h (complex)f(dimension)h FJ(g)s FQ(.)54 b(In)34 b(particular,)g(for)456 1822 y FJ(g)25 b FQ(=)e(0)p FJ(;)14 b(H)764 1792 y FM(1)800 1822 y FQ(\()p FJ(C)q(;)g FL(O)995 1834 y FI(C)1052 1822 y FQ(\))23 b(=)g(0)k(and)h(th)n(us,)f(the)h(determinan)n(t)g(line)g (bundle)g(is)f(trivial.)605 1922 y(One)e(easily)f(sees)g(that)h (de\014nition)g(of)g(the)g(determinan)n(t)g(bundle)h FJ(Q)2745 1934 y FI(S)2817 1922 y FQ(is)f(functorial)f(in)i FJ(S)5 b FQ(,)456 2021 y(and)27 b(th)n(us,)g(w)n(e)g(ha)n(v)n(e)e(a)i (w)n(ell-de\014ned)g(line)g(bundle)h FJ(Q)2151 2033 y FI(M)2251 2021 y FQ(o)n(v)n(er)e(the)h(mo)r(duli)h(stac)n(k)e FJ(M)3143 2033 y FI(g)r(;n)3242 2021 y FQ(.)36 b(The)456 2121 y(same)23 b(de\014nition)h(also)f(w)n(orks)f(for)h(singular)g (curv)n(es,)h(and)f(th)n(us)h FJ(Q)2531 2133 y FI(M)2628 2121 y FQ(is)g(w)n(ell)g(de\014ned)g(o)n(v)n(er)e(the)456 2220 y(completion)p 880 2154 90 4 v 27 w FJ(M)970 2232 y FI(g)r(;n)1069 2220 y FQ(.)605 2320 y(Finally)-7 b(,)42 b(let)e(us)f(discuss)f(the)i(b)r(eha)n(vior)e(of)h(the)g(determinan)n (t)g(bundle)h(with)g(resp)r(ect)456 2420 y(to)34 b(gluing.)56 b(Let)35 b FJ(A)g FQ(b)r(e)f(a)g(\014nite)h(set,)h FJ(\013;)14 b(\014)39 b FL(2)c FJ(A)p FQ(|an)f(unordered)g(pair,)h FJ(A)2890 2390 y FE(0)2948 2420 y FQ(=)f FJ(A)23 b FL(n)g(f)p FJ(\013;)14 b(\014)t FL(g)p FQ(.)456 2519 y(Suc)n(h)39 b(a)f(pair)h(de\014nes)g(a)f(\\clutc)n(hing")g(map)h FJ(S)1994 2531 y FI(\013;\014)2111 2519 y FQ(:)32 b FJ(M)2247 2531 y FE(\003)p FI(;A)2396 2519 y FL(!)42 b FJ(N)9 b FQ(\()p FJ(D)2700 2489 y FM(0)2738 2519 y FQ(\),)42 b(where)d FJ(D)3158 2489 y FM(0)3234 2519 y FQ(is)g(the)456 2625 y(corresp)r(onding)25 b(comp)r(onen)n(t)j(of)f(the)h(b)r(oundary)f(in)p 2122 2558 V 28 w FJ(M)2211 2637 y FE(\003)p FI(;A)2315 2621 y Fx(0)2369 2625 y FQ(\(see)h(\(6.2.7\))o(\).)605 2779 y FP(Pr)n(oposition)j FQ(6.7.5)p FP(.)40 b FO(L)l(et)34 b FJ(S)1577 2791 y FI(\013;\014)1719 2779 y FO(b)l(e)h(as)h(ab)l(ove,)h FJ(Q)p FO({the)f(determinant)f(line)g(bund)t(le)g(on)456 2878 y FJ(M)537 2890 y FE(\003)p FI(;A)641 2874 y Fx(0)666 2878 y FO(.)60 b(L)l(et)36 b(us)g(also)i(denote)f(by)g FJ(Q)g FO(the)f(c)l(orr)l(esp)l(onding)i(line)g(bund)t(le)f(on)g FJ(N)9 b FQ(\()p FJ(D)3104 2848 y FM(0)3141 2878 y FQ(\))p FO(.)60 b(Then)456 2978 y FJ(S)512 2948 y FE(\003)507 3002 y FI(\013;\014)614 2978 y FQ(\()p FJ(Q)p FQ(\))30 b FO(is)g(c)l(anonic)l(al)t(ly)i(isomorphic)g(to)e(the)g(determinant)g (line)g(bund)t(le)g(over)h FJ(M)3116 2990 y FE(\003)p FI(;A)3223 2978 y FO(.)605 3138 y FQ(W)-7 b(e)28 b(omit)g(the)g(pro)r (of)f(of)g(this)h(prop)r(osition,)f(referring)f(the)i(reader)e(to)h([)p FK(BFM)p FQ(].)605 3238 y(No)n(w,)41 b(w)n(e)d(are)f(ready)h(to)g(form) n(ulate)g(the)h(de\014nition)f(of)h(the)g(mo)r(dular)f(functor)g(with) 456 3337 y(cen)n(tral)25 b(c)n(harge,)g(whic)n(h)h(is)g(parallel)f(to)h (De\014nition)h(6.4.1,)e(but)i(with)f(replacemen)n(t)g(of)g(v)n(ector) 456 3437 y(bundles)32 b(with)h(\015at)f(connection)g(b)n(y)g(v)n(ector) f(bundles)i(with)f(the)h(action)f(of)g(the)h(cen)n(tral)e(ex-)456 3537 y(tension)e FL(A)810 3549 y FI(Q)862 3532 y FG(c)927 3537 y FQ(of)g(the)g(Lie)g(algebra)f(of)h(v)n(ector)f(\014elds,)i(or,)e (equiv)-5 b(alen)n(tly)e(,)30 b(with)f(the)h(action)f(of)456 3636 y FL(D)520 3648 y FI(Q)572 3632 y FG(c)607 3636 y FQ(.)36 b(As)25 b(b)r(efore,)g(let)g FL(C)30 b FQ(b)r(e)25 b(an)g(ab)r(elian)f(category)f(o)n(v)n(er)g(the)i(\014eld)g FH(C)15 b FQ(,)32 b(and)24 b FJ(R)i FQ({)e(a)h(symmetric)456 3742 y(ob)5 b(ject)27 b(in)h(ind)p FL(\000C)1033 3711 y Fv(\002)p FM(2)1121 3742 y FQ(.)605 3896 y FP(Definition)k FQ(6.7.6)p FP(.)40 b FQ(A)25 b FO(c)l(omplex)j FL(C)5 b FO(-extende)l(d)26 b(mo)l(dular)i(functor)f(with)34 b FQ(\()p FO(additive)6 b FQ(\))30 b FO(c)l(en-)456 3995 y(tr)l(al)f(char)l(ge)i FJ(a)23 b FL(2)h FH(C)48 b FQ(is)28 b(the)g(follo)n(wing)e(collection)h(of)h(data:)605 4095 y(\(i\))d(F)-7 b(or)23 b(ev)n(ery)f(\014nite)j(set)f FJ(A)g FQ(and)g FJ(W)35 b FL(2)23 b(C)1896 4065 y Fv(\002)p FI(A)2002 4095 y FQ(,)h(a)g(\014nite-dimensional)f(v)n(ector)g(bundle)h (o)n(v)n(er)p 456 4131 V 456 4197 a FJ(M)545 4209 y FE(\003)p FI(;A)680 4197 y FQ(with)k(an)f(action)g(of)g FL(D)1391 4209 y FI(Q)1443 4193 y FG(a)1483 4197 y FQ(.)37 b(This)27 b(bundle)h(is)f(called)g(the)h FO(bund)t(le)h(of)i(c)l(onformal)f(blo)l (cks)p FQ(;)456 4303 y(its)d(\014b)r(er)h(at)f(a)h(p)r(oin)n(t)f FJ(C)j FL(2)p 1317 4236 V 23 w FJ(M)1407 4315 y FE(\003)p FI(;A)1542 4303 y FQ(will)e(b)r(e)g(denoted)g(b)n(y)f FL(h)p FJ(W)12 b FL(i)2395 4315 y FI(C)2451 4303 y FQ(.)605 4402 y(\(ii\))28 b(Isomorphisms)f FL(h)p FJ(X)7 b FL(i)1409 4414 y FI(C)1461 4398 y Fx(0)1505 4402 y FL(\012)18 b(h)p FJ(Y)h FL(i)1719 4414 y FI(C)1771 4398 y Fx(00)1839 4402 y FL(')k(h)p FJ(X)i Fw(\002)18 b FJ(Y)g FL(i)2234 4414 y FI(C)2286 4398 y Fx(0)2309 4414 y FE(t)p FI(C)2406 4398 y Fx(00)2450 4402 y FQ(.)605 4502 y(\(iii\))34 b FK(Gluing)i(isomorphisms.)47 b FQ(Let)33 b FJ(A)g FQ(b)r(e)g(a)f (\014nite)i(set,)g FJ(\013;)14 b(\014)36 b FL(2)c FJ(A)p FQ(|an)h(unordered)456 4609 y(pair,)c FJ(A)714 4579 y FE(0)763 4609 y FQ(=)d FJ(A)20 b FL(n)f(f)p FJ(\013;)14 b(\014)t FL(g)p FJ(;)g(W)38 b FL(2)26 b(C)1505 4579 y Fv(\002)p FI(A)1607 4554 y Fx(0)1633 4609 y FQ(.)42 b(F)-7 b(or)29 b(ev)n(ery)f(suc)n(h)h(collection,)h(w)n(e)f(require)f(an)h (isomor-)456 4709 y(phism)f(of)f FL(D)859 4721 y FI(Q)911 4705 y FG(a)952 4709 y FQ(-mo)r(dules)g(on)g FJ(M)1501 4721 y FE(\003)p FI(;A)1609 4709 y FQ(:)1343 4862 y FJ(G)1408 4874 y FI(\013;\014)1525 4862 y FQ(:)g FL(h)p FJ(W)k Fw(\002)18 b FJ(R)q FL(i)1947 4815 y FE(\030)1918 4862 y FL(\000)-39 b(!)23 b FJ(S)2106 4827 y FE(\003)2101 4882 y FI(\013;\014)2208 4862 y FJ(S)5 b(p)2306 4874 y FI(D)2366 4862 y FL(h)p FJ(W)12 b FL(i)p FJ(;)-2087 b FQ(\(6.7.3\))456 5015 y(where)39 b FJ(S)759 5027 y FI(\013;\014)875 5015 y FQ(:)32 b FJ(M)1011 5027 y FE(\003)p FI(;A)1161 5015 y FL(!)43 b FJ(N)9 b FQ(\()p FJ(D)1466 4985 y FM(0)1504 5015 y FQ(\))p FJ(;)14 b(D)1644 4985 y FM(0)1724 5015 y FL(\032)43 b FJ(M)1913 5027 y FE(\003)p FI(;A)2017 5011 y Fx(0)2043 5015 y FQ(,)f(is)e(the)g(\\clutc)n(hing")e (\(6.2.7\))o(,)43 b(and)c FJ(R)i FQ(is)456 5114 y(placed)28 b(at)g(p)r(ositions)g(with)g(indices)h FJ(\013;)14 b(\014)t FQ(.)40 b(\(Since)28 b FJ(S)2134 5126 y FI(\013;\014)2270 5114 y FQ(is)g(a)g FH(C)2478 5084 y FE(\002)2540 5114 y FQ(-bundle,)h(the)g(de\014nition)g(of)456 5214 y FJ(S)512 5184 y FE(\003)507 5237 y FI(\013;\014)642 5214 y FQ(causes)d(no)i (problems.\))p eop %%Page: 163 29 163 166 bop 1094 226 a FM(6.7.)29 b(MODULAR)g(FUNCTOR)h(WITH)g(CENTRAL) f(CHAR)n(GE)518 b(163)605 425 y FQ(\(iv\))30 b FK(V)-8 b(acuum)34 b(propagation.)42 b FQ(W)-7 b(e)30 b(ha)n(v)n(e)e(a)h (distinguished)h(elemen)n(t)f FK(1)d FL(2)g FQ(Ob)14 b FL(C)5 b FQ(,)30 b(and)456 525 y(for)d(ev)n(ery)f FJ(\013)d FL(2)h FJ(A)p FQ(,)k(w)n(e)f(require)g(an)g(isomorphism)f(of)i FL(D)2231 537 y FI(Q)2283 521 y FG(a)2324 525 y FQ(-mo)r(dules)f(on)g FJ(M)2873 537 y FE(\003)p FI(;A)2981 525 y FQ(:)1490 670 y FJ(G)1555 682 y FI(\013)1612 670 y FQ(:)g FL(h)p FJ(W)k Fw(\002)18 b FK(1)p FL(i)2017 623 y FE(\030)1989 670 y FL(\000)-39 b(!)23 b FJ(S)2177 636 y FE(\003)2172 690 y FI(\013)2219 670 y FL(h)p FJ(W)12 b FL(i)p FJ(;)-1940 b FQ(\(6.7.4\))456 805 y(where)27 b FJ(S)747 817 y FI(\013)803 805 y FQ(:)h FJ(M)935 817 y FE(\003)p FI(;A)1065 805 y FL(!)23 b FJ(M)1252 820 y FE(\003)p FI(;A)p FE(n)p FI(\013)1465 805 y FQ(is)k(the)h(op)r(erator)e(of)h(erasing)f(the)i(p)r (oin)n(t)g FJ(\013)p FQ(.)605 905 y(These)f(data)g(ha)n(v)n(e)g(to)g (satisfy)h(the)g(follo)n(wing)e(prop)r(erties:)612 1022 y FK(F)-8 b(unctorialit)m(y:)41 b FJ(W)35 b FL(7!)23 b(h)p FJ(W)12 b FL(i)1588 1034 y FI(C)1670 1022 y FQ(is)25 b(functorial)g(in)h FJ(W)12 b FQ(,)26 b(and)f(the)h(isomorphisms)e (\(ii\)-\(iv\))711 1122 y(are)j(functorial)g(isomorphisms.)612 1221 y FK(Compatibilit)m(y:)38 b FQ(the)21 b(isomorphisms)d (\(ii\)-\(iv\))j(are)f(compatible)f(with)i(eac)n(h)e(other)h(and)711 1321 y(with)25 b(the)f(comm)n(utativit)n(y)-7 b(,)25 b(asso)r(ciativit)n(y)-7 b(,)23 b(and)h(unit)h(morphisms)f(in)g FL(V)7 b FJ(ec)3071 1333 y FI(f)3138 1321 y FQ(\(cf.)36 b(Def-)711 1420 y(inition)28 b(4.2.1\).)612 1520 y FK(Normalization:)39 b FL(h)p FK(1)p FJ(;)14 b FK(1)p FL(i)1471 1535 y Fz(P)1512 1518 y FF(1)1566 1520 y FQ(=)22 b FH(C)15 b FQ(.)605 1670 y(Note)27 b(that)h(the)g(requiremen)n(t)e(that)i FL(h)p FJ(W)12 b FL(i)28 b FQ(b)r(e)g(a)f FL(D)2197 1682 y FI(Q)2249 1666 y FG(a)2317 1670 y FQ(is)g(equiv)-5 b(alen)n(t)27 b(to)h(sa)n(ying)d(that)j(the)456 1769 y(pull-bac)n(k)34 b(of)h FL(h)p FJ(W)12 b FL(i)36 b FQ(to)f(the)g (total)g(space)g(of)g(the)g(line)g(bundle)h FJ(Q)f FQ(has)g(a)f(mono)r (dromic)g(\015at)456 1869 y(connection)22 b(with)i(the)g(mono)r(drom)n (y)e FJ(e)1684 1839 y FE(\000)p FM(2)p FI(\031)r(a)1873 1869 y FQ(around)g(the)i(zero)e(section)h(\(see)h(Exercise)d(6.6.4\).) 605 1969 y(Let)k(us)g(no)n(w)f(relate)h(it)g(to)g(the)g(top)r(ological) f(form)n(ulation)f(of)i(the)h(mo)r(dular)e(functor)h(with)456 2068 y(cen)n(tral)32 b(c)n(harge.)53 b(T)-7 b(o)33 b(do)g(it,)j(let)e (us)f(recall)g(the)h(de\014nition)f(of)h(the)g(cen)n(tral)e(c)n(harge)g (for)h(the)456 2168 y(mo)r(dular)27 b(functor)g(in)h(the)g(top)r (ological)e(setup)i(\(see)f(Section)h(5.7\).)605 2268 y(It)h(w)n(as)f(de\014ned)i(using)e(the)i(cen)n(tral)e(extension)g(of)h (the)h(usual)e(to)n(w)n(er)g(of)h(mapping)f(class)456 2367 y(groups)33 b(b)n(y)i(the)g(group)r(oid)f FJ(T)1409 2379 y FM(\006)1495 2367 y FQ(=)h FJ(T)1644 2382 y FI(H)1702 2366 y FF(1)1735 2382 y FM(\(\006)p FI(;)p Fz(R)p FM(\))1900 2367 y FQ(,)i(where)d(\006)h(is)g(a)g(closed)f(orien)n(ted)g(top)r (ological)456 2467 y(surface)f(of)h(gen)n(us)g FJ(g)s FQ(,)h(and)f(for)g(a)g(symplectic)g(real)f(v)n(ector)g(space)h FJ(V)18 b FQ(,)37 b FJ(T)2801 2479 y FI(V)2892 2467 y FQ(is)d(the)h(P)n(oincare)456 2566 y(\(fundamen)n(tal\))26 b(group)r(oid)f FJ(T)1394 2578 y FI(V)1475 2566 y FQ(=)d FJ(\031)1609 2578 y FM(1)1647 2566 y FQ(\(\003)1737 2578 y FI(V)1794 2566 y FQ(\))27 b(of)f(the)g(set)g(\003)2273 2578 y FI(V)2357 2566 y FQ(of)g(all)g(Lagrangian)d(subspaces)i(in)456 2666 y FJ(V)18 b FQ(.)37 b(Recall)28 b(also)e(that)i(for)f(ev)n(ery)f FJ(L)d FL(2)g FJ(T)1732 2678 y FI(V)1790 2666 y FJ(;)14 b FQ(Hom)1999 2678 y FI(T)2038 2686 y FG(V)2093 2666 y FQ(\()p FJ(L;)g(L)p FQ(\))22 b FL(')h FH(Z)o FQ(.)605 2766 y(It)28 b(is)f(w)n(ell)h(kno)n(wn)e(\(see,)i(e.g.,)f([)p FK(GH)p FQ(]\))h(that)g(for)f(a)g(connected)g(compact)g(complex)g(curv) n(e)456 2865 y(the)37 b(natural)f(map)h(of)h(shea)n(v)n(es)d FH(R)45 b FL(!)39 b(O)g FQ(induces)e(an)g(isomorphism)f(of)h(the)h (cohomology)456 2965 y(spaces)22 b FJ(H)782 2935 y FM(1)819 2965 y FQ(\()p FJ(C)q(;)14 b FH(R)q FQ(\))29 b FL(')23 b FJ(H)1228 2935 y FM(1)1265 2965 y FQ(\()p FJ(C)q(;)14 b FL(O)r FQ(\),)25 b(where)e(b)r(oth)h(sides)f(are)f(considered)g(as)h (v)n(ector)f(spaces)g(o)n(v)n(er)456 3065 y FH(R)p FQ(.)40 b(In)22 b(other)e(w)n(ords,)h(a)g(c)n(hoice)f(of)h(a)f(complex)h (structure)f(on)h(a)g(top)r(ological)e(surface)h(\006)h(de\014nes)456 3164 y(a)27 b(complex)g(structure)g(on)g(the)h(2)p FJ(g)s FQ(-dimensional)e(real)h(v)n(ector)f(space)h FJ(V)42 b FQ(=)23 b FJ(H)2920 3134 y FM(1)2957 3164 y FQ(\(\006)p FJ(;)14 b FH(R)p FQ(\).)605 3314 y FP(Theorem)32 b FQ(6.7.7)p FP(.)40 b FO(L)l(et)29 b FJ(C)36 b FO(b)l(e)30 b(a)g(c)l(omplex)g (curve.)39 b(Then)30 b(one)g(has)h(a)f(c)l(anonic)l(al)h(e)l(quiva-)456 3413 y(lenc)l(e)f(of)g(gr)l(oup)l(oids)1591 3548 y FJ(T)1640 3560 y FM(\006)1714 3548 y FQ(=)23 b FJ(\031)1849 3560 y FM(1)1887 3548 y FQ(\()p FJ(Q)1985 3513 y FE(\012)p FM(2)1985 3573 y FI(C)2092 3548 y FL(n)18 b(f)p FQ(0)p FL(g)p FQ(\))456 3694 y FO(wher)l(e,)31 b(as)f(b)l(efor)l(e,)h FJ(Q)1152 3706 y FI(C)1230 3694 y FQ(=)23 b(det)14 b FJ(H)1523 3663 y FM(1)1560 3694 y FQ(\()p FJ(C)q(;)g FL(O)1755 3706 y FI(C)1812 3694 y FQ(\))p FO(.)605 3843 y FP(Pr)n(oof.)41 b FQ(Let)30 b(us)h(note)f(that)h(the)g(complex)f (structure)g(on)g FJ(V)47 b FQ(=)27 b FJ(H)2778 3813 y FM(1)2815 3843 y FQ(\()p FJ(C)q(;)14 b FH(R)q FQ(\))37 b(de\014ned)31 b(b)n(y)456 3943 y(the)39 b(iden)n(ti\014cation)h FJ(V)62 b FL(')42 b FJ(H)1416 3913 y FM(1)1453 3943 y FQ(\()p FJ(C)q(;)14 b FL(O)r FQ(\),)44 b(agrees)38 b(with)i(the)g (symplectic)f(structure)g(in)h FJ(V)58 b FQ(as)456 4042 y(follo)n(ws:)1289 4177 y FL(h\001)p FJ(;)14 b(i)1410 4189 y FI(C)1466 4177 y FL(\001i)28 b FQ(is)f(symmetric)g(p)r(ositiv)n (e)h(de\014nite)-2155 b(\(6.7.5\))456 4317 y(where)30 b FL(h\001)p FJ(;)14 b FL(\001i)31 b FQ(is)g(the)g(symplectic)g(form,)h (and)e FJ(i)1939 4329 y FI(C)2023 4317 y FL(2)f FQ(End)2256 4329 y Fz(R)2302 4317 y FQ(\()p FJ(V)19 b FQ(\))31 b(is)g(the)g(op)r (erator)e(of)i(m)n(ultipli-)456 4418 y(cation)c(b)n(y)g(i)c(=)954 4353 y FL(p)p 1023 4353 107 4 v 65 x(\000)p FQ(1)k(in)h(the)g(complex)f (structure)g(de\014ned)h(b)n(y)f FJ(C)6 b FQ(.)605 4518 y(Let)23 b FJ(V)42 b FQ(b)r(e)23 b(an)f(arbitrary)f(symplectic)i(real)f (v)n(ector)f(space.)35 b(Denote)23 b(b)n(y)f FJ(H)2923 4530 y FI(V)3004 4518 y FQ(the)h(set)g(of)g(all)456 4617 y(complex)28 b(structures)h(on)g FJ(V)48 b FQ(satisfying)29 b(the)h(condition)f(ab)r(o)n(v)n(e.)40 b(It)30 b(is)f(usually)g(called) g FO(Sie)l(gel)456 4717 y(upp)l(er)d(half)i(plane)f(of)e FJ(V)19 b FQ(.)35 b(W)-7 b(e)25 b(quote)e(without)h(pro)r(of)f(the)i (follo)n(wing)d(standard)h(result,)i(whic)n(h)456 4817 y(can)i(b)r(e)h(found,)g(for)f(example,)g(in)h([)p FK(GH)p FQ(].)605 4966 y FP(Theorem)k FQ(6.7.8)p FP(.)40 b FJ(H)1318 4978 y FI(V)1405 4966 y FO(is)30 b(a)g(c)l(ontr)l(actible)h(sp)l(ac)l (e.)605 5116 y FP(Exer)n(cise)h FQ(6.7.9)p FP(.)40 b FQ(Sho)n(w)35 b(that)i(for)e FJ(V)56 b FQ(=)37 b FH(R)2052 5086 y FM(2)2131 5116 y FQ(with)g(the)f(standard)g(symplectic)g(form,) 456 5216 y FJ(H)525 5228 y FI(V)610 5216 y FQ(can)27 b(b)r(e)h(iden)n(ti\014ed)g(with)g(the)g(upp)r(er)g(half-plane)f(of)h FH(C)15 b FQ(.)p eop %%Page: 164 30 164 167 bop 456 226 a FM(164)445 b(6.)30 b(MODULI)f(SP)-5 b(A)n(CES)28 b(AND)h(COMPLEX)f(MODULAR)h(FUNCTOR)605 425 y FQ(Let)36 b(us)g(consider)e(the)i(line)g(bundle)h FJ(\025)f FQ(o)n(v)n(er)e FJ(H)2144 437 y FI(V)2201 425 y FQ(,)k(whose)d(\014b)r(er)h(at)g(p)r(oin)n(t)g FJ(h)g FL(2)h FJ(H)3295 437 y FI(V)3389 425 y FQ(is)456 525 y FJ(\025)504 537 y FI(h)576 525 y FQ(=)29 b(det)785 537 y Fz(C)845 525 y FJ(V)19 b FQ(;)34 b(here)c FJ(V)51 b FQ(is)31 b(considered)f(as)h FJ(g)s FQ(-dimensional)f(v)n(ector)g (space)g(o)n(v)n(er)g FH(C)52 b FQ(with)32 b(the)456 624 y(complex)27 b(structure)g(giv)n(en)g(b)n(y)g FJ(h)p FQ(.)37 b(Consider)26 b(the)i(total)g(space)f(of)g FJ(\025)2629 594 y FE(\012)p FM(2)2737 624 y FL(n)18 b(f)p FQ(zero)26 b(section)p FL(g)o FQ(.)605 778 y FP(Pr)n(oposition)31 b FQ(6.7.10)p FP(.)39 b FO(F)-6 b(or)44 b(every)g FJ(h)k FL(2)h FJ(H)2090 790 y FI(V)2148 778 y FO(,)e(one)d(has)g(c)l(anonic)l (al)h(e)l(quivalenc)l(e)g(of)456 878 y(gr)l(oup)l(oids)1094 1019 y FJ(\031)1141 1031 y FM(1)1179 1019 y FQ(\()p FJ(\025)1259 985 y FE(\012)p FM(2)1367 1019 y FL(n)18 b(f)p FO(zer)l(o)30 b(se)l(ction)p FL(g)o FQ(\))24 b FL(')e FJ(\031)2123 1031 y FM(1)2161 1019 y FQ(\()p FJ(\025)2241 984 y FE(\012)p FM(2)2241 1044 y FI(h)2349 1019 y FL(n)c(f)p FQ(0)p FL(g)p FQ(\))k FL(')h FJ(T)2726 1031 y FI(V)2783 1019 y FJ(:)-2350 b FQ(\(6.7.6\))605 1173 y(Ob)n(viously)-7 b(,)26 b(this)h(prop)r (osition)f(immediately)i(implies)f(the)g(statemen)n(t)g(of)g(the)h (theorem.)605 1273 y(Let)21 b(us)g(\014rst)g(construct)g(equiv)-5 b(alence)21 b FJ(\031)1850 1285 y FM(1)1887 1273 y FQ(\()p FJ(\025)1967 1243 y FE(\012)p FM(2)2063 1273 y FL(n)6 b(f)p FQ(zero)25 b(section)o FL(g)p FQ(\))e FL(')g FJ(\031)2808 1285 y FM(1)2845 1273 y FQ(\()p FJ(\025)2925 1237 y FE(\012)p FM(2)2925 1298 y FI(h)3021 1273 y FL(n)6 b(f)p FQ(0)p FL(g)p FQ(\).)33 b(This)456 1373 y(equiv)-5 b(alence)22 b(follo)n(ws)h(from)g(the)g(fact)h(that)f FJ(H)1892 1385 y FI(V)1974 1373 y FQ(is)g(con)n(tractible,)g(and)g(therefore,)g(em)n (b)r(edding)456 1472 y FJ(\025)504 1437 y FE(\012)p FM(2)504 1497 y FI(h)612 1472 y FL(n)18 b(f)p FQ(0)p FL(g)j FJ(,)-14 b FL(!)23 b FJ(\025)982 1442 y FE(\012)p FM(2)1090 1472 y FL(n)18 b(f)p FQ(zero)26 b(section)p FL(g)h FQ(is)h(a)f(homotop)n(y)f (equiv)-5 b(alence.)605 1579 y(Next,)27 b(let)f(us)f(construct)g(equiv) -5 b(alence)26 b FJ(\031)1899 1591 y FM(1)1936 1579 y FQ(\()p FJ(\025)2016 1543 y FE(\012)p FM(2)2016 1604 y FI(h)2121 1579 y FL(n)14 b(f)p FQ(0)p FL(g)p FQ(\))22 b FL(')h FJ(T)2494 1591 y FI(V)2551 1579 y FQ(.)36 b(T)-7 b(o)26 b(do)f(so,)h(let)g(us)f(rewrite)456 1678 y(the)d(de\014nition)g (of)f FJ(T)1093 1690 y FI(V)1172 1678 y FQ(as)g(follo)n(ws.)34 b(Let)21 b FJ(L)h FQ(b)r(e)g(the)g(tautological)e(v)n(ector)g(bundle)i (of)g(dimension)456 1778 y FJ(g)35 b FQ(o)n(v)n(er)30 b(\003)771 1790 y FI(V)829 1778 y FQ(:)46 b(it)33 b(is)f(a)g (sub-bundle)h(of)f(the)h(trivial)f(bundle)h FJ(V)41 b FL(\002)21 b FQ(\003)2587 1790 y FI(V)2677 1778 y FQ(suc)n(h)32 b(that)h(its)f(\014b)r(er)h(at)456 1877 y(p)r(oin)n(t)39 b FJ(L)j FL(2)h FJ(T)12 b FQ(\()p FJ(V)18 b FQ(\))40 b(is)f(the)h(subspace)e FJ(L)43 b FL(\032)f FJ(V)19 b FQ(.)72 b(Consider)38 b(the)i(one-dimensional)e(v)n(ector)456 1977 y(bundle)24 b FJ(\025)772 1989 y Fz(R)841 1977 y FQ(=)f(det)1044 1989 y Fz(R)1104 1977 y FJ(L)g FQ(=)f(\003)1329 1937 y FI(g)1329 2002 y Fz(R)1375 1977 y FJ(L)h FQ(\(to)h(a)n(v)n(oid)e (confusion,)i(w)n(e)f(use)h(subscript)f FH(R)30 b FQ(for)23 b(determinan)n(t)456 2082 y(of)30 b(v)n(ector)f(spaces)h(o)n(v)n(er)f FH(R)p FQ(\).)52 b(Then)31 b FJ(\025)1673 2046 y FE(\002)1673 2106 y Fz(R)1729 2082 y FJ(=)20 b FL(\006)g FQ(1)31 b(is)f(a)g(bundle)h (with)h(\014b)r(er)e FH(R)2823 2094 y FM(+)2915 2082 y FQ(o)n(v)n(er)f(\003)3154 2094 y FI(V)3211 2082 y FQ(;)j(th)n(us,)456 2186 y FJ(\025)504 2151 y FE(\002)504 2211 y Fz(R)560 2186 y FJ(=)18 b FL(\006)g FQ(1)23 b FL(!)g FQ(\003)932 2198 y FI(V)1017 2186 y FQ(is)k(a)g(homotop)n(y)g(equiv)-5 b(alence,)27 b(and)g FJ(T)2228 2198 y FI(V)2309 2186 y FQ(=)22 b FJ(\031)2443 2198 y FM(1)2481 2186 y FQ(\()p FJ(\025)2561 2151 y FE(\002)2561 2211 y Fz(R)2618 2186 y FJ(=)c FL(\006)g FQ(1\).)605 2286 y(No)n(w)28 b(let)h FJ(h)24 b FL(2)g FJ(H)1136 2298 y FI(V)1223 2286 y FQ(b)r(e)k(a)g (complex)g(structure)g(on)g FJ(V)19 b FQ(,)29 b FJ(\025)2372 2298 y FI(h)2439 2286 y FQ(=)24 b(det)14 b FJ(V)44 b FQ(=)24 b(\003)2896 2256 y FI(g)2934 2286 y FJ(V)19 b FQ(,)28 b(where)g FJ(V)48 b FQ(is)456 2385 y(considered)27 b(as)g(a)g(v)n(ector)g(space)g(o)n(v)n(er)f FH(C)49 b FQ(using)28 b(the)g(complex)g(structure)f FJ(h)p FQ(.)38 b(Th)n(us,)28 b(w)n(e)g(ha)n(v)n(e)456 2485 y(a)h(w)n(ell-de\014ned)g (map)h FJ(\025)1218 2497 y Fz(R)1291 2485 y FQ(=)d(\003)1441 2445 y FI(g)1441 2510 y Fz(R)1486 2485 y FJ(L)g FL(!)f FJ(\025)1727 2497 y FI(h)1798 2485 y FQ(=)g(\003)1947 2455 y FI(g)1985 2485 y FJ(V)19 b FQ(;)31 b(the)f(left-hand)g(side)g (is)g(one)f(dimensional)456 2585 y(real)20 b(v)n(ector)g(space,)i(the)g (righ)n(t)f(hand)g(side)h(is)f(one-dimensional)f(complex)h(v)n(ector)f (space.)35 b(This)456 2684 y(giv)n(es)26 b(rise)h(to)g(a)h(map)f FJ(\025)1215 2649 y FE(\002)1215 2709 y Fz(R)1272 2684 y FJ(=)17 b FL(\006)i FQ(1)j FL(!)h FJ(\025)1633 2654 y FE(\012)p FM(2)1741 2684 y FL(n)18 b(f)p FQ(0)p FL(g)p FQ(.)605 2784 y(This)i(completes)g(the)h(pro)r(of)f(of)g(the)h(prop)r (osition.)33 b(On)20 b(the)h(other)f(hand,)i(the)f(prop)r(osition)456 2883 y(immediately)27 b(implies)h(the)g(statemen)n(t)g(of)f(the)h (theorem.)p 3384 2883 4 57 v 3388 2831 50 4 v 3388 2883 V 3437 2883 4 57 v 605 3043 a FP(Example)j FQ(6.7.11)p FP(.)39 b FQ(Let)31 b FJ(g)f FQ(=)e(1,)j(so)f(\006)g(is)h(a)f(torus)g (\(or,)g(equiv)-5 b(alen)n(tly)e(,)31 b FJ(C)37 b FQ(is)31 b(an)f(elliptic)456 3143 y(curv)n(e\).)47 b(Then)32 b FJ(V)48 b FQ(=)30 b FH(R)1219 3113 y FM(2)1293 3143 y FQ(with)i(the)g(canonical)e(symplectic)i(form,)g(so)f(\003)2802 3155 y FI(V)2891 3143 y FQ(is)g(the)h(set)f(of)h(all)456 3243 y(real)e(one-dimensional)h(subspaces)f(in)i FH(R)1773 3212 y FM(2)1816 3243 y FQ(,)h(th)n(us)f(\003)2117 3255 y FI(V)2204 3243 y FL(')d FJ(S)2354 3212 y FM(1)2391 3243 y FQ(.)49 b(On)32 b(the)g(other)f(hand,)h(in)g(this)456 3342 y(case)26 b FJ(\025)679 3354 y FI(h)746 3342 y FQ(=)c(det)14 b FJ(V)42 b FQ(=)23 b FJ(V)46 b FQ(\(as)27 b(a)g(complex)g(v)n(ector)f (space\).)37 b(Th)n(us,)27 b(in)h(this)f(case)g(the)h(canonical)456 3444 y(map)f FJ(\031)687 3456 y FM(1)725 3444 y FQ(\(\003)815 3456 y FI(V)872 3444 y FQ(\))d FL(!)f FJ(\031)1081 3456 y FM(1)1118 3444 y FQ(\()p FJ(\025)1198 3409 y FE(\012)p FM(2)1198 3469 y FI(h)1307 3444 y FL(n)18 b(f)p FQ(0)p FL(g)p FQ(\))26 b(is)h(ob)n(vious.)605 3598 y(As)33 b(an)g(immediate)h (corollary)d(of)i(Theorem)f(6.7.7,)i(w)n(e)f(see)g(get)g(the)g(follo)n (wing)g(result.)456 3697 y(As)27 b(b)r(efore,)h(let)g FL(C)k FQ(b)r(e)c(an)f(ab)r(elian)g(category)f(o)n(v)n(er)g FL(C)5 b FQ(.)605 3851 y FP(Theorem)32 b FQ(6.7.12)p FP(.)39 b FO(The)24 b(notions)g(of)g FL(C)5 b FO(-extende)l(d)22 b(top)l(olo)l(gic)l(al)k(MF)e(with)g(multiplic)l(ative)456 3951 y(c)l(entr)l(al)29 b(char)l(ge)i FJ(K)k FO(and)30 b(of)g FL(C)5 b FO(-extende)l(d)29 b(c)l(omplex)h(MF)g(with)g(additive) i(c)l(entr)l(al)d(char)l(ge)i FJ(a)e FO(ar)l(e)456 4050 y(e)l(quivalent)h(if)g FJ(K)f FQ(=)23 b FJ(e)1149 4020 y FI(\031)r FM(i)p FI(a)1248 4050 y FO(.)605 4204 y FP(Cor)n(ollar)-6 b(y)33 b FQ(6.7.13)p FP(.)39 b FO(A)n(ny)24 b(MTC)i(c)l(ate)l(gory)g (over)g FH(C)46 b FO(gives)26 b(rise)f(to)g(a)g FL(C)5 b FO(-extende)l(d)24 b(c)l(om-)456 4303 y(plex)30 b(MF)g(with)g (additive)i(c)l(entr)l(al)e(char)l(ge)h FJ(a)e FO(such)h(that)1693 4445 y FJ(e)1732 4411 y FI(\031)r(ia)1860 4445 y FQ(=)22 b FJ(p)1989 4411 y FM(+)2044 4445 y FJ(=p)2128 4411 y FE(\000)2184 4445 y FJ(:)-1751 b FQ(\(6.7.7\))605 4587 y FO(Conversely,)27 b(every)d FL(C)5 b FO(-extende)l(d)23 b(c)l(omplex)h(MF)h(with)f(additive)i(c)l(entr)l(al)d(char)l(ge)i FJ(a)e FO(de\014nes)456 4687 y(a)37 b(we)l(akly)i(ribb)l(on)f(structur) l(e)e(on)h FL(C)5 b FO(;)42 b(if)c(this)g(c)l(ate)l(gory)g(is)g(rigid,) j(then)c(it)h(is)f(mo)l(dular,)k(and)456 4787 y FJ(p)498 4756 y FM(+)552 4787 y FJ(=p)636 4756 y FE(\000)722 4787 y FO(is)30 b(given)g(by)g(the)g(formula)h(ab)l(ove.)605 4940 y FQ(Note)k(that)g(for)g(mo)r(dular)f(functors)h(coming)g(from)f (rational)g(conformal)g(\014eld)h(theory)-7 b(,)456 5040 y(the)28 b(additiv)n(e)f(cen)n(tral)g(c)n(harge)e(is)j(giv)n(en)f(b)n (y)1795 5182 y FJ(a)c FQ(=)f FJ(c=)p FQ(2)p FJ(;)p eop %%Page: 165 31 165 168 bop 1094 226 a FM(6.7.)29 b(MODULAR)g(FUNCTOR)h(WITH)g(CENTRAL) f(CHAR)n(GE)518 b(165)456 425 y FQ(where)23 b FJ(c)h FQ(is)g(the)h(Virasoro)d(cen)n(tral)h(c)n(harge)f(of)j(the)f(theory)f (\(w)n(e)i(will)f(illustrate)g(it)g(in)h(the)f(next)456 525 y(c)n(hapter,)d(where)f(the)g(W)-7 b(ess-Zumino-Witten)21 b(mo)r(del)f(is)h(considered\).)33 b(Com)n(bining)20 b(this)h(with)456 624 y(the)28 b(corollary)d(ab)r(o)n(v)n(e,)h(w)n(e)h (see)g(that)h(in)g(this)g(case)f(one)g(has)1643 806 y FJ(K)h FQ(=)1840 750 y FJ(p)1882 720 y FM(+)p 1840 787 98 4 v 1840 863 a FJ(p)1882 839 y FE(\000)1971 806 y FQ(=)22 b FJ(e)2097 772 y FI(\031)r FM(i)p FI(c=)p FM(2)456 985 y FQ(\(cf.)37 b(Remark)27 b(3.1.20\).)p eop %%Page: 166 32 166 169 bop 456 226 a FM(166)445 b(6.)30 b(MODULI)f(SP)-5 b(A)n(CES)28 b(AND)h(COMPLEX)f(MODULAR)h(FUNCTOR)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF