Deformations of Poisson structures on Hilbert schemes Brent Pym - PowerPoint PPT Presentation

Deformations of Poisson structures on Hilbert schemes Brent Pym Based on joint work in progress with Mykola Matviichuk (McGill) & Travis Schedler (Imperial)

Plan 2 Holonomicity: nondegeneracy condition for holomorphic Poisson structures (P.–Schedler) ( ∧ • T X , d π ) ⊗ D X Char ( X , π ) ⊂ T ∗ X ( X , π ) � � Holonomic ⇐ ⇒ Char ( X , π ) Lagrangian = ⇒ # char leaves < ∞ � �� � conj ⇐ ⇒ Symplectic leaf is characteristic if modular vector field ∆ π is tangent Motivation: ( ∧ • T X , d π ) is perverse, so deformation theory is “topological” This talk: an illustrative example Hilb n ( X , π ) its Hilbert scheme ( X , π ) smooth Poisson surface �

Poisson surface := C -surface X with π ∈ H 0 ( K − 1 X ) 3 X = P 2 deg ( K − 1 X ) = 3 Y = cubic ∂ X := Zeros ( π ) ⊂ X ∂ 2 X := singular locus of ∂ X Nondegenerate on X ◦ := X \ ∂ X : π ∼ = ∂ q ∧ ∂ p ∆ π = 0 On smooth locus of ∂ X : π ∼ = u ∂ u ∧ ∂ v ∆ π = ∂ v Characteristic leaves: X ◦ , ∂ 2 X ⇒ ω := π − 1 log symplectic holonomic ⇐ ⇒ ∂ X reduced ⇐ Theorem (Enriques, Kodaira; Bartocci–Macr´ ı, Ingalls) If ( X , π ) is a projective Poisson surface, then ( X , π ) is birational to: ( P 1 × C ( C 2 ( P 2 , cubic ) T ∗ ( curve ) Λ , u ∂ u ∧ ∂ v ) Λ , ∂ q ∧ ∂ p ) K 3 Consequently, ∂ X is locally quasi-homogeneous.

Poisson cohomology of log symplectic surface ( X , π ) 4 j i ∂ 2 X X ◦ Characteristic leaves: X Theorem (Goto, P.–Schedler) ⇒ ( ∧ • T X , d π ) ∼ = Rj ∗ C X 0 ⊕ i ∗ i ∗ K − 1 ∂ 2 X quasi-homogeneous = X [ − 2] , so  H j ( X ◦ ; C ) j � = 2   HP j ( X , π ) ∼ H 2 ( X ◦ ; C ) H 0 ( i ∗ K − 1 = ⊕ X ) j = 2  � �� � � �� �  defs . of ω smoothings of ∂ 2 X Sketch of proof. 1 Restriction to open leaf: j ∗ ( ∧ • T X , d π ) ∼ X ◦ , d ) ∼ = (Ω • = C X ◦ 2 Therefore (adjunction): ∧ • T X → Rj ∗ C X ◦ 3 Splitting: Rj ∗ C X ◦ ∼ = Ω • X (log ∂ X ) → ∧ • T X via quasihomogeneous “log comparision” of [Castro-Jim´ enez–Narv´ aez-Macarro–Mond] X (log ∂ X ) ∼ 4 Show that ∧ • T X / Ω • = i ∗ i ∗ K − 1 X [ − 2]

Hilbert schemes of a Poisson surface ( X , π ) 5 Sym n ( X ) := X n / S n Hilb n ( X ) := { length- n subschemes of X } ← − � �� � � �� � smooth Poisson [Beauville, Bottacin, Mukai] singular Poisson variety For instance: Hilb 2 ( X ) = Bl ∆ Sym 2 ( X ) = ( Sym 2 ( X ) \ ∆) ⊔ P ( T X ) � �� � � �� � reduced schemes 1-jets × × × Case X compact K¨ ahler, π nondegenerate: Same for Hilb n ( X ) [Beauville, Mukai], so hyperK¨ ahler [Calabi, Yau] Albanese fibres are “irreducible” [Beauville] – only other known IHSMs (up to deformation) are O’Grady’s M 6 , M 10 Unobstructed deformations parameterized by H 2 ( Hilb n ( X ); C ) [Beauville, Bogomolov]

Symplectic leaves of Hilb n ( X ) 6 W ∈ Hilb n ( X ) ∂ W := W ∩ ∂ X (scheme-theoretic) � W , W ′ in same symplectic leaf ⇐ ⇒ ∂ W = ∂ W ′ ∂ W Example Leaf × × Hilb n ( X ◦ ) × ∅ × × W { W } × × × × ( Bl ∂ W X ) ◦ n − 1 points × × ×

Characteristic leaves 7 Locally: modular vector field ∆ π on X lifts to ∆ π Hilb Proposition leaf of W is characteristic (i.e. ∆ π Hilb tangent) ⇐ ⇒ ∂ W fixed by ∆ π Conjecture (Matviichuk–P.–Schedler) For n ≥ 2 , we have: Hilb n ( X ) holonomic ⇐ ⇒ # char leaves < ∞ ⇐ ⇒ every point in X has type A k , k ≥ 0 i.e. local equation x 2 = y k +1 Cases proven so far: both = ⇒ , both both ⇐ = for n = 2 or ∂ X smooth second ⇐ = for k ≤ 2 Key point: type A k ⇐ ⇒ linearization of ∆ π nonzero

Deformations 8 Theorem (Matviichuk–P.–Schedler) For ( X , π ) connected log sympl. surf., ∂ X locally quasi-hgns, n ≥ 2 : HP 2 ( Hilb n ( X )) H 2 ( Hilb n ( X ◦ )) ⊕ H 0 ( i ∗ K − 1 = X ) HP 2 ( X ) ⊕ ∧ 2 H 1 ( X ◦ ; C ) = ⊕ C · c 1 ( E ) � �� � � �� � � �� � log Albanese Hilb ( Def ( X ,π )) Hilb ( Quant ( X ,π )) [Hitchin, Nevins–Stafford, Rains] Corollary (Ran) If ∂ X is smooth then deformations are unobstructed. Corollary Rains’ Hilbert schemes of noncommutative rational surfaces form irreducible components in the moduli space of Poisson varieties

Deformations – proof 9 Theorem (Matviichuk–P.–Schedler) For ( X , π ) connected log sympl. surf., ∂ X locally quasi-hgns, n ≥ 2 : HP 2 ( Hilb n ( X )) H 2 ( Hilb n ( X ◦ )) ⊕ H 0 ( i ∗ K − 1 = X ) HP 2 ( X ) ⊕ ∧ 2 H 1 ( X ◦ ; C ) = ⊕ C · c 1 ( E ) � �� � � �� � � �� � Hilb ( Def ( X ,π )) log Albanese Hilb ( Quant ( X ,π )) [Hitchin, Nevins–Stafford, Rains] Sketch. 1 throw out codim 4 (higher Hartogs); look at char. leaves ∗ C Hilb n − 1 X ◦ ⊗ H 0 ( i ∗ K − 1 2 codim 0: Rj ∗ C Hilb n X ◦ , codim 2: Rj ′ X ) 3 codim 1: no contributions, codim 3: could a priori only make HP 2 smaller, but doesn’t (local calculation, or interpret deformations) THANK YOU!