Lefschetz pencils and the symplectic topology of complex surfaces - PowerPoint PPT Presentation

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

Symplectic 4-manifolds A (compact) symplectic 4-manifold ( M 4 , ω ) is a smooth 4-manifold with a sym- plectic 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. SYMPLECTIC SMOOTH � � surgery SW invariants Thurston, Gompf... Taubes ⇒ Classification problems. Complex surfaces are fairly well understood, but their topology as smooth or symplectic manifolds remains mysterious. 1

Example: Horikawa surfaces X 1 X 2 2 : 1 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) ( b + 2 = 21, b − X 1 , X 2 are homeomorphic 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 K¨ ahler forms) symplectomorphic? Remark: projecting to CP 1 , Horikawa surfaces carry genus 2 fibrations. 2

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 ) ∼ z 2 1 + z 2 2 . ( ⇒ sing. fibers are nodal) vanishing cycle s s M f S 2 × × s 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) 3

Symplectic manifolds and Lefschetz pencils Algebraic geometry: → CP N . X complex surface + ample line bundle ⇒ projective embedding X ֒ 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 X → S 2 with distinguished − 1-sections. f : ˆ ˆ (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 K¨ ahler forms [ ω i ] = K X i ) are symplectomorphic iff generic pencils of curves in the pluricanonical linear systems | mK X i | define topologically equivalent Lefschetz fibrations with sections for some m (or for all m ≫ 0). 4

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

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 i τ 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) s s γ 1 γ r γ 1 γ r γ i +1 ∼ γ i γ i +1 × . . . × × . . . × × . . . × × . . . × γ − 1 i +1 γ i γ i +1 { genus g Lefschetz fibrations with n sections } / isomorphism ↑ ↓ 1-1 (if 2 − 2 g − n < 0) � factorizations in Map g,n � � Hurwitz equiv. δ = � (pos. Dehn twists) + global conj. 6

Classification in low genus • g = 0 , 1: only holomorphic fibrations ( ⇒ ruled surfaces, elliptic surfaces). • g = 2, assuming sing. fibers are irreducible: r 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 2 τ 4 τ 3 ( τ 1 · τ 2 · τ 3 · τ 4 · τ 5 ) 6 = 1 τ 1 τ 5 ( τ 1 · τ 2 · τ 3 · τ 4 ) 10 = 1 (up to a technical assumption; argument relies on pseudo-holomorphic curves) • g ≥ 3: intractable (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

Canonical pencils on Horikawa surfaces On X 1 and X 2 , generic pencils in the linear systems | K X i | 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 connected curve C (6 , 12) ⊂ CP 1 × CP 1 5Σ 0 ∪ Σ ∞ ⊂ F 6 A 8

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 X i | (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 s s ⇑ M ⇓ ? “alg. vanishing cycles” (ODP degenerations) f (span [ π ∗ H 2 ( P 1 × P 1 )] ⊥ � = [ π ∗ H 2 ( F 6 )] ⊥ ) S 2 × × (but... φ ∈ G 2 suggests where to start looking for exotic matching paths?) 9