Birational geometry of quiver varieties Gwyn Bellamy jt. with A. Craw, T. Schedler (S. Rayan and H. Weiss) Thursday, 25th June 2020 Gwyn Bellamy Birational geometry of quiver varieties
Plan Introduction Quiver varieties Anisotropic roots Isotropic roots Hyperpolygon spaces Gwyn Bellamy Birational geometry of quiver varieties
Introduction - Quiver varieties, as introduced by Nakajima, are ubiquitous in geometric representation theory. - Large class of examples of symplectic singularities, together with an associated symplectic resolution given by variation of geometric invariant theory (VGIT). Questions: (A) Can one obtain all symplectic resolutions via VGIT? (B) What is the birational transformation that occurs when we cross a GIT wall? Gwyn Bellamy Birational geometry of quiver varieties
Quiver varieties - Q = ( Q 0 , Q 1 ) a finite quiver with double Q . - v ∈ N Q 0 dimension vector. - Space of representations of dimension v : � Hom ( C t ( a ) , C h ( a ) ) . Rep ( Q , v ) = a ∈ Q 1 - Carries (Hamiltonian) action of G ( v ) = � i ∈ Q 0 GL ( C v i ). - Corresponding moment map µ : Rep ( Q , v ) → g ( v ) where g ( v ) = Lie G ( v ). Gwyn Bellamy Birational geometry of quiver varieties
Quiver varieties Definition The quiver variety associated to ( Q , v ) is M 0 := µ − 1 (0) / / G ( v ) . Proposition (B-Schedler) M 0 has symplectic singularities. Q. When does M 0 admit a symplectic resolution? Gwyn Bellamy Birational geometry of quiver varieties
Factorization - Z Q 0 has symmetric form ( − , − ). - Applying Crawley-Boevey’s factorization, M 0 ( v ) ∼ = M 0 ( v 1 ) × · · · × M 0 ( v k ) where each v i ≤ v is either (1) v i = n δ i for δ i minimal imaginary, ( δ i , δ i ) = 0; or (2) anisotropic root: ( v i , v i ) < 0. - M 0 ( v ) admits a symplectic resolution iff every factor M 0 ( v i ) admits a symplectic resolution. - Hilbert schemes give symplectic resolutions in case (1). Gwyn Bellamy Birational geometry of quiver varieties
Anisotropic roots In the case where v is an anisotropic root, ( v , v ) < 0, have: Theorem (B-Schedler) M 0 ( v ) admits a symplectic resolution iff v indivisible or ”(2 , 2) case”. The ”(2 , 2) case” is v = 2 u with u indivisible, ( u , u ) = − 2. This situation is exceptional. Gwyn Bellamy Birational geometry of quiver varieties
Anisotropic roots - birational geometry Assume v anisotropic and indivisible. Set � � θ ∈ Q Q 0 | θ ( v ) = 0 Λ = . For each θ ∈ Λ, consider space � � µ − 1 (0) θ = M ∈ µ − 1 (0) | θ (dim M ′ ) ≤ 0 , ∀ M ′ subrep M space of θ -semistable objects. Definition M θ := µ − 1 (0) θ / / G ( v ) . Always a Poisson morphism M θ → M 0 . Gwyn Bellamy Birational geometry of quiver varieties
Anisotropic roots - birational geometry Λ reg set of all θ ∈ Λ with M θ smooth. Proposition (B-Craw-Schedler = BCS) Λ reg complement to finitely many hyperplanes H α . - Hyperplanes H α are either ”interior” or ”boundary”, depending on α . - Fix C ⊂ Λ reg a chamber and θ ∈ C . - C lies in a unique (closed) chamber F of the boundary arrangement. Gwyn Bellamy Birational geometry of quiver varieties
Anisotropic roots - birational geometry Slice to arrangement in Λ = Q 3 , showing chambers in F (boundary,interior). C e ⊥ 1 e ⊥ 2 Gwyn Bellamy Birational geometry of quiver varieties
Anisotropic roots - birational geometry - Define L C : Λ → Pic ( M θ ) Q by � (det R i ) ⊗ ϑ i L C ( ϑ ) = i ∈ Q 0 - Here R i tautological bundle of rank v i . Theorem (BCS) 1 L C is an isomorphism with L C ( C ) = Amp ( M θ ). 2 L C = L C ′ if C , C ′ ⊂ F . 3 L C ( F ) = Mov ( M θ ). Surjectivity of L C requires McGerty-Nevins theorem on surjectivity of the Kirwan map. Gwyn Bellamy Birational geometry of quiver varieties
Application Corollary (BCS) Let v be arbitrary. Every (projective) symplectic resolution of M 0 ( v ) is given by a quiver variety. Need to exclude (2 , 2) case above. Corollary (BCS) Assume v a root. If M 0 ( v ) admits a symplectic resolution then #resolutions = | π 0 ( F ∩ Λ reg ) | . Gwyn Bellamy Birational geometry of quiver varieties
Q -factorial terminalizations Assume v is not indivisible. Proposition (B-Schedler) For generic θ ∈ Λ, M θ → M 0 is a Q -factorial terminalization. BCS: - Chamber structure still exists. - L C is always injective. - Know to be surjective in certain cases. - Expect it always to be an isomorphism. All results make sense in this generality provided L C is an isomorphism. Gwyn Bellamy Birational geometry of quiver varieties
Isotropic roots - Q affine Dynkin quiver. - v = n δ with δ minimal imaginary root. - ∆ = { e 1 , . . . , e r } simple roots in finite root system Φ. Hyperplanes - A I = { β + m δ | β ∈ Φ , − n < m < n , m � = 0 } . - A B = { δ } ∪ Φ + . Then - H α for α ∈ A I are ”interior” hyperplane. - H α for α ∈ A B are ”boundary” hyperplane. Gwyn Bellamy Birational geometry of quiver varieties
Isotropic roots Theorem (B-Craw) - Λ reg = Λ � � α H α , where α ∈ A I ∪ A B . - F = { θ ∈ Λ | θ ( δ ) ≥ 0 , θ ( e i ) ≥ 0 , i = 1 , . . . , r } . W Φ be the (finite) Weyl group of Φ. Theorem (B-Craw) - W = S 2 × W Φ acts on Λ with fundamental domain F . = M C ′ iff C ′ = w ( C ) some w ∈ W . - M C ∼ Gwyn Bellamy Birational geometry of quiver varieties
Application - Γ ⊂ SL (2 , C ) finite group associated to Q . - S n ≀ Γ = S n ⋊ Γ n acts on C 2 n . - Symplectic resolution of quotient given by Hilb n �� � → C 2 n / ( S n ≀ Γ) C 2 / Γ where � C 2 / Γ minimal resolution of C 2 / Γ. Corollary (B-Craw) Every (projective) symplectic resolution of C 2 n / ( S n ≀ Γ) is of the form M θ for some θ ∈ Λ reg . Gwyn Bellamy Birational geometry of quiver varieties
Hyperpolygon spaces 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 2 1 1 1 Let n ≥ 4 and ( Q , v ) star quiver with n outer vertices. - M θ ( n ) a ”hyperpolygon space”. - As a hyperh¨ ahler manifold, compactification of cotangent bundle of polygon moduli space. - dim M θ = 2( n − 3). Gwyn Bellamy Birational geometry of quiver varieties
A quotient singularity - Notice for n = 4, M 0 ∼ = C 2 / BD 8 . 1 1 1 1 2 2 1 1 1 1 Gwyn Bellamy Birational geometry of quiver varieties
A quotient singularity - Notice for n = 4, M 0 ∼ = C 2 / BD 8 . Theorem (B-Schedler) The group Q 8 × Z 2 D 8 acts on C 4 such that C 4 / ( Q 8 × Z 2 D 8 ) admits a symplectic resolution. Theorem (B,Donten–Bury-Wi´ sniewski) The quotient C 4 / ( Q 8 × Z 2 D 8 ) admits 81 (projective) symplectic resolutions. Gwyn Bellamy Birational geometry of quiver varieties
A quotient singularity Theorem (B-Craw-Rayan-Schedler-Weiss) As symplectic singularities, C 4 / ( Q 8 × Z 2 D 8 ) ∼ = M 0 (5) . Easy to recover count of 81 from hyperplane arrangement in Λ. Theorem (B-Craw-Rayan-Schedler-Weiss) = Q n with For n ≥ 4, we have Λ ∼ - Λ reg = { θ | θ 1 ± θ 2 ± · · · ± θ n � = 0 , θ 1 , . . . , θ n � = 0 } . - F = { θ | θ i ≥ 0 } . - W = S n 2 . Gwyn Bellamy Birational geometry of quiver varieties
The End! Thanks for listening. Gwyn Bellamy Birational geometry of quiver varieties
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