Estimated transversality in symplectic geometry and projective maps. Denis AUROUX

Transcription

1 Estimated transversality in symplectic geometry and projective maps Denis AUROUX

2 Ample bundles over almost-complex manifolds GW invariants: holomorphic maps from complex manifolds to a symplectic manifold Dual point of view: (approx.) holomorphic maps from a symplectic manifold to complex manifolds (Donaldson) Tool: estimated transversality for approx. holomorphic sections of very ample bundles ( good linear systems, maps to CP m ) (X 2n, J) almost-complex, compact (L k, k ) line { bundles are asympt. very ample if if k (v, Jv) > c k v 2, c k + curvature = O(1) F (0,2) k ω k = if k is symplectic, J is ω k -tame. Example: c 1 (L k ) = k[ω] and J is ω-compatible. Asympt. holomorphic sections of L k : { s k C r,g k = O(1) (rescaling : g k = c k g) s k C r,g k = O(c 1/2 k ) curvature + look into X at small scale non-integrability 0. 1

3 Estimated transversality of jets Asympt. holomorphic sections s k,0,..., s k,m Γ(L k ) approx. holomorphic maps f k : X CP m Need estimated transversality for the jets of these maps. E k = C m+1 L k asympt. very ample vector bundles, holom. jet bundles J r E k = r ( T X (1,0)) j sym E k. j=0 S k = asympt. holomorphic stratifications of J r E k : finite Whitney stratifications, transverse to the fibers ; all strata are asympt. holomorphic submanifolds, with bounded curvature away from lower-dimensional strata. The jet j r s k is η-transverse to S k if dist(j r s k (x), S k,a ) < η the graph of j r s k is transverse at x to T S k,a, with minimum angle > η. Theorem 1 S k asympt. holomorphic stratifications of J r E k ; δ > 0 ; s k asympt. holomorphic sections of E k for large enough k, asympt. holomorphic sections σ k of E k s.t. (1) σ k s k C r+1,g k < δ ; (2) j r σ k is η (δ) -transverse to S k. 2

4 Estimated transversality of jets Ingredients of proof : transversality is an open property transv. to all strata by successive perturbations start with lowest dim. strata ; S k are Whitney only work away from lower-dim. strata very localized asympt. holomorphic sections of L k in coords.: s k,x,i (z) = z i z i n n exp( 1 4 c k z 2 ) local trivializations of J r E k local transversality result for functions C n C p (Donaldson) a localized small perturbation of s k yields estimated transversality to S k,a over a small ball globalization argument using openness, combine local perturbations to obtain transversality everywhere Theorem 1 also holds for families indexed by t [0, 1] objects are canonical up to isotopy (and even independent of the chosen J as long as L k remain asympt. very ample) 3

5 Boardman stratifications of holomorphic jet spaces Boardman stratification of jets of holomorphic maps C n C m : f : C n C m holomorphic singular loci Σ i (f) = {x, dim Ker df(x) = i} Σ i1,...,i r (f) = Σ ir (f Σi1,...,i r 1 (f)) stratification of J r (C n, C m ) by Σ I. Sections s k of E k = C m+1 L k f k = Ps k : X s 1 k (0) CPm j r s k = (s k, s k, s k,... ) ; use local approx. holom. coordinates to identify J r (X, CP m ) with J r (C n, C m ) Boardman stratification of J r E k : - S 0 = {j r s(x), s(x) = 0} - S I = {j r s(x), s(x) 0, j r Ps(x) Σ I } These stratifications are asympt. holomorphic by Theorem 1, for large k we get s k Γ(E k ) s.t. j r s k uniformly transverse to Boardman stratifications. 4

6 Generic projective maps s k asympt. holomorphic sections of C m+1 L k, j r s k uniformly transverse to Boardman stratifications : the base loci Z k = s 1 k (0) are smooth symplectic codim. 2m + 2 submanifolds. Local model: f k (z 1,..., z n ) = (z 1 :z 2 :...:z m+1 ) the holomorphic r-jets of f k = Ps k behave similarly to those of generic holomorphic maps between complex manifolds singular loci Σ I (f k ) = stratified symplectic submanifolds of X Z k, of the expected codimension Away from singular loci, estimated transversality + asympt. holomorphicity f k f k holomorphic local models for f k Near Σ I (f k ), need to ensure f k f k obtain some control over f k. Idea: the antiholomorphic part of the jet of f k should vanish in the normal directions to Σ I (f k ). 5

7 Generic projective maps Suitable perturbation to kill the antiholomorphic jet of f k along normal directions to singular loci obtain approx. holomorphic projective maps, topologically conjugate near every point of X to generic holomorphic maps between complex manifolds (in local approx. holomorphic coordinates). m = 1 : symplectic Lefschetz pencils (Donaldson) m = 2 : maps to CP 2 (D. A.) m 2n : projective immersions/embeddings (Muñoz-Presas-Sols) general case : in progress 6

8 Symplectic Lefschetz pencils (s 0, s 1 ) Γ(C 2 L k ) suitably chosen symplectic Lefschetz pencil : Σ α = {x X, s 0 + αs 1 = 0} (α CP 1 ) symplectic hypersurfaces, smooth except for finitely many singular points. Base locus Z = {s 0 = s 1 = 0} (codim. 4). Projective map f = (s 0 :s 1 ) : X Z CP 1 : local model f(z) = z z 2 n near critical points. Blow up Z Lefschetz fibration ˆX CP 1 γ i ˆX CP 1 monodromy = (generalized) Dehn twist Monodromy = θ : π 1 (C {pts}) Map ω (Σ 2n 2, Z) Map ω (Σ, Z) := π 0 ({φ Symp(Σ, ω), φ U(Z) = Id}) symplectic invariants. 7

9 Symplectic maps to CP 2 (s 0, s 1, s 2 ) Γ(C 3 L k ) suitably chosen f = (s 0 : s 1 : s 2 ) : X Z CP 2. Fibers = codimension 4 symplectic submanifolds, intersecting at the base locus Z (codim. 6), singular along a smooth symplectic curve R X. Local singular models near R : 1. (z 1,..., z n ) (z z 2 n 1, z n ) points where R is transverse to the fibers of f 2. (z 1,..., z n ) (z 3 1 z 1 z n + z z 2 n 1, z n ) cusp points of D = f(r) The critical curve D = f(r) is symplectic, with nodes (both orientations) and cusp singularities. Fiber above a smooth point of D = obtained by collapsing a vanishing cycle (Lagrangian S n 2 ) in the generic fiber Σ 2n 4. Monodromy around D : θ : π 1 (C 2 D) Map ω (Σ 2n 4, Z) θ(geometric generator) = generalized Dehn twist. Up to cancellation of nodes in D, for k 0 the topology of f k is a symplectic invariant. 8

10 Monodromy and braid groups Perturbation D = singular branched cover of CP 1. CP 2 { } D deg D = d CP 1 π : (x 0 : x 1 : x 2 ) (x 0 : x 1 ) Monodromy = ρ : π 1 (C {pts}) B d (braid group) Monodromy around each crit. point = (half-twist) δ, δ { 2, 1, 2, 3} : δ = 1 + δ = 2 δ = 3 δ = 2 (Moishezon-Teicher, Auroux-Katzarkov) 9

11 The higher dimensional case Monodromies of f : X Z CP 2 : ρ : π 1 (C {pts}) B d (describes D) θ : π 1 (C 2 D) Map ω (Σ 2n 4, Z) (describes f) Restricting to the hypersurface W 2n 2 = s 1 2 (0), f W = (s 0 : s 1 ) : W Z CP 1 is a symplectic Lefschetz pencil, monodromy θ = θ i : π 1 (C {q 1,..., q d }) Map ω (Σ 2n 4, Z) X Σ W 2n 2 D CP 2 CP 1 The monodromy invariants (ρ, θ) determine the manifold X up to symplectomorphism. 10

12 Dimensional induction (X 2n, ω) symplectic, s 0,..., s n Γ(L k ) well-chosen. Σ r = {s r+1 = = s n = 0} smooth symplectic submanifold, dim Σ r = 2r, Σ n = X. s r 1 and s r define a SLP on Σ r, generic fiber Σ r 1, base locus Σ r 2. Monodromy : θ r : π 1 (C {pts}) Map ω (Σ r 1, Σ r 2 ) (s r 2 : s r 1 : s r ) : Σ r Σ r 3 CP 2, singular locus D r CP 2, deg D r = d r 1. Monodromy : ρ r : π 1 (C {pts}) B dr 1 and θ r 1 ρ r and θ r 1 (description of Σ r by a map to CP 2 ) determine θ r (description of Σ r by a SLP) explicitly. Symplectic invariants characterizing X : (θ r, ρ r+1, ρ r+2,..., ρ n ), 1 r n. r = n : SLP r = 2 : n 2 braid factorizations + word in Map g,n r = 1 : n 1 braid factorizations + word in S N. 11