Lefschetz pencils and the symplectic topology of complex surfaces

Transcription

1 Lefschetz pencils and the symplectic topology of complex surfaces Denis AUROUX Massachusetts Institute of Technology

2 Symplectic 4-manifolds A (compact) symplectic 4-manifold (M 4, ω) is a smooth 4-manifold with a symplectic form ω Ω 2 (M), closed (dω = 0) and non-degenerate (ω ω > 0). Local model (Darboux): R 4, ω 0 = dx 1 dy 1 + dx 2 dy 2. E.g.: (CP n, ω 0 = i log z 2 ) complex projective surfaces. The symplectic category is strictly larger (Thurston 1976, Gompf 1994,...). Hierarchy of compact oriented 4-manifolds: COMPLEX PROJ. Classification problems. surgery Thurston, Gompf... SYMPLECTIC SW invariants Taubes SMOOTH Complex surfaces are fairly well understood, but their topology as smooth or symplectic manifolds remains mysterious. 1

3 Example: Horikawa surfaces X 1 2 : 1 X 2 2 : 1 Σ C (6,12) 5 Σ 0 CP 1 CP 1 F 6 = P(O P 1 O P 1(6)) X 1, X 2 projective surfaces of general type, minimal, π 1 = 1 X 1, X 2 are not deformation equivalent (Horikawa) X 1, X 2 are homeomorphic (b + 2 = 21, b 2 = 93, non-spin) Open problems: X 1, X 2 diffeomorphic? (expect: no, even though SW (X 1 ) = SW (X 2 )) (X 1, ω 1 ), (X 2, ω 2 ) (canonical Kähler forms) symplectomorphic? Remark: projecting to CP 1, Horikawa surfaces carry genus 2 fibrations. 2

4 Lefschetz fibrations A Lefschetz fibration is a C map f : M 4 S 2 with isolated non-degenerate crit. pts, where (in oriented coords.) f(z 1, z 2 ) z1 2 + z2. 2 ( sing. fibers are nodal) vanishing cycle M S 2 Monodromy around sing. fiber = Dehn twist Also consider: Lefschetz fibrations with distinguished sections. Gompf: Assuming [fiber] non-torsion in H 2 (M), M carries a symplectic form s.t. ω fiber > 0, unique up to deformation. (extends Thurston s result on symplectic fibrations) f 3

5 Symplectic manifolds and Lefschetz pencils Algebraic geometry: X complex surface + ample line bundle projective embedding X CP N. Intersect with a generic pencil of hyperplanes Lefschetz pencil (= family of curves, at most nodal, through a finite set of base points). Blow up base points Lefschetz fibration with distinguished sections. Donaldson: Any compact sympl. (X 4, ω) admits a symplectic Lefschetz pencil f : X \ {base} CP 1 ; blowing up base points, get a sympl. Lefschetz fibration ˆf : ˆX S 2 with distinguished 1-sections. (uses approx. hol. geometry : f = s 0 /s 1, s i C (X, L k ), L ample, sup s i sup s i ) In large enough degrees (fibers m[ω], m 0), Donaldson s construction is canonical up to isotopy; combine with Gompf s results Corollary: the Horikawa surfaces X 1 and X 2 (with Kähler forms [ω i ] = K Xi ) are symplectomorphic iff generic pencils of curves in the pluricanonical linear systems mk Xi define topologically equivalent Lefschetz fibrations with sections for some m (or for all m 0). 4

6 Monodromy vanishing cycle M Monodromy around sing. fiber = Dehn twist γ r S 2 γ 1 f Monodromy: ψ : π 1 (S 2 \ {p 1,..., p r }) Map g = π 0 Diff + (Σ g ) Mapping class group: e.g. for T 2 = R 2 /Z 2, Map 1 = SL(2, Z); τ a = Choosing an ordered basis γ 1,..., γ r for π 1 (S 2 \ {p i }), get (τ 1,..., τ r ) Map g, τ i = ψ(γ i ), τi = 1. ( ) , τ b = ( ) factorization of Id as product of positive Dehn twists. With n distinguished sections: ˆψ : π1 (R 2 \ {p i }) Map g,n Map g,n = π 0 Diff + (Σ, Σ) genus g with n boundaries. τ 1... τ r = δ 5 (monodromy at = boundary twist).

7 Factorizations Two natural equivalence relations on factorizations: 1. Global conjugation (change of trivialization of reference fiber) (τ 1,..., τ r ) (φτ 1 φ 1,..., φτ r φ 1 ) φ Map g 2. Hurwitz equivalence (change of ordered basis γ 1,..., γ r ) (τ 1,..., τ i, τ i+1,... τ r ) (τ 1,..., τ i+1, τ 1 i+1 τ iτ i+1,..., τ r ) (τ 1,..., τ i τ i+1 τ 1 i, τ i,..., τ r ) (generates braid group action on r-tuples) γ 1 γ r γ i γ i γ 1 γ i+1 γ r γ 1 i+1 γ iγ i+1 { genus g Lefschetz fibrations with n sections } / isomorphism { factorizations in Mapg,n δ = (pos. Dehn twists) 1-1 (if 2 2g n < 0) } / Hurwitz equiv. + global conj. 6

8 Classification in low genus g = 0, 1: only holomorphic fibrations ( ruled surfaces, elliptic surfaces). g = 2, assuming sing. fibers are irreducible: Siebert-Tian (2003): always isotopic to holomorphic fibrations, i.e. built from: (τ 1 τ 2 τ 3 τ 4 τ 5 τ 5 τ 4 τ 3 τ 2 τ 1 ) 2 = 1 (τ 1 τ 2 τ 3 τ 4 τ 5 ) 6 = 1 (τ 1 τ 2 τ 3 τ 4 ) 10 = 1 g 3: intractable τ 2 τ3 τ 4 τ 1 τ 5 (up to a technical assumption; argument relies on pseudo-holomorphic curves) (families of non-holom. examples by Ozbagci-Stipsicz, Smith, Fintushel-Stern, Korkmaz,...) The genus 2 fibrations on X 1, X 2 are different (e.g., different monodromy groups): X 1 : (τ 1 τ 2 τ 3 τ 4 τ 5 τ 5 τ 4 τ 3 τ 2 τ 1 ) 12 = 1 X 2 : (τ 1 τ 2 τ 3 τ 4 ) 30 = 1... but can t conclude from them! 7

9 Canonical pencils on Horikawa surfaces On X 1 and X 2, generic pencils in the linear systems K Xi have fiber genus 17 (with 16 base points), and 196 nodal fibers compare 2 sets of 196 Dehn twists in Map 17,16? Theorem: The canonical pencils on X 1 and X 2 are related by partial conjugation: (φt 1 φ 1,..., φt 64 φ 1, t 65,..., t 196 ) vs. (t 1,..., t 196 ) The monodromy groups G 1, G 2 Map 17,16 are isomorphic; unexpectedly, the conjugating element φ belongs to the monodromy group. Key point: CP 1 CP 1 and F 6 are symplectomorphic; the branch curves of π 1 : X 1 CP 1 CP 1 and π 2 : X 2 F 6 differ by twisting along a Lagrangian annulus. disconnected curve 5Σ 0 Σ F 6 A connected curve C (6,12) CP 1 CP 1 8

10 Perspectives Theorem: The canonical pencils on X 1 and X 2 are related by partial conjugation; G 1, G 2 Map 17,16 are isomorphic; φ belongs to the monodromy group. The same properties hold for pluricanonical pencils mk Xi (in larger Map g,n ) These pairs of pencils are twisted fiber sums of the same pieces. If φ were monodromy along an embedded loop (+ more) (X 1, ω 1 ) (X 2, ω 2 ) (but only seems to arise from an immersed loop) Question: compare these (very similar) mapping class group factorizations?? E.g.: matching paths (= Lagrangian spheres fibering above an arc). Expect: H 2 -classes represented by Lagrangian spheres? alg. vanishing cycles (ODP degenerations) (span [π H 2 (P 1 P 1 )] [π H 2 (F 6 )] ) M f S 2 (but... φ G 2 suggests where to start looking for exotic matching paths?) 9