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Seminar on Lefschetz fibrations

Sommersemester 2002


K. Cieliebak and D. Kotschick

  • Time and place: Tuesdays, 2 - 4 pm, HS 134
  • Contents: Lefschetz pencils have recently become of great interest because of Donaldson's theorem that every symplectic manifold admits a Lefschetz pencil. In this seminar we aim at understanding the proof of Donaldson's theorem, as well as applications in symplectic geometry. A holomorphic Lefschetz pencil on a complex surface X is a nontrivial holomorphic map from a blow-up of X to the Riemann sphere. The fibres are (possibly singular) complex curves. It is a classical result that every smooth projective surface admits a holomorphic Lefschetz pencil. There are corresponding notions of Lefschetz pencils in the topological and symplectic categories in which the fibres are smooth, respectively symplectic, submanifolds. Donaldson's theorem states that every symplectic manifold whose symplectic form is integral admits a symplectic Lefschetz pencil. Conversely, every topological Lefschetz pencil with fibres of genus at least 2 admits a compatible symplectic structure.
  • References:
    [GS] R.E. Gompf, A.I. Stipsicz, 4-manifolds and Kirby calculus, GSM 20, American Mathematical Society, Providence 1999
    [G] R.E. Gompf, A new construction of symplectic 4-manifolds, Ann. of Math. 142 (1995), 527-595
    [GH] P. Griffiths, J. Harris, Principles of algebraic geometry, Wiley, New York 1978
    [D1] S.K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996), no. 4, 666-705
    [D2] S.K. Donaldson, Lefschetz Fibrations in Symplectic Geometry, Doc.Math.J.DMV Extra Volume ICM II (1998), 309-314
    [D3] S.K. Donaldson, Lefschetz pencils on symplectic manifolds J. Differential Geom. 53 (1999), no. 2, 205-236
  • Intended audience: Diplom and Master degree students with an interest and some background in topology and geometry.
  • Prerequisites: Basic notions of geometry and topology. Some knowledge of symplectic and/or algebraic geometry is helpful but not necessary.
  • Seminar plan
  • Lecture notes (can only be accessed from within the department)