Hypertoric varieties and hyperplane arrangements

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Hypertoric varieties and hyperplane arrangements Kyoto Univ. June 16, 2018

Motivation - Study of the geometry of symplectic variety Symplectic variety (Y 0, ω) Very special but interesting even dim algebraic variety Topic Classification and finding new examples The case of dim 4 is open. Universal (Poisson) deformation and its construction Its discriminant locus D C is hyperplane arr. with W -action. Good (= crepant) resolution π : (Y, ω) (Y 0, ω) construction and counting all crepant resolutions of Y 0. This is related to the structure of D C. We will study these problems for hypertoric variety.

Contents Intro - symplectic variety and Poisson deformation Hypertoric variety - definition and its universal Poisson deformation Application 1 - Application 2 - classification of affine hypertoric varieties by matroids counting good resolutions of affine hypertoric variety by hyperplane arr.

Intro - symplectic variety and Poisson deformation

Symplectic variety (Y 0, ω) : symplectic variety def normal alg. var. with holomo. symplectic form ω on (Y 0 ) reg. (Y 0, ω) : conical symplectic variety def affine symplectic variety with good C -action. Example V V /Γ : Symplectic quotient singularity (Γ Sp(V V )) eg. ADE-type surface singularity (A n : z n+1 xy = 0). O : Nilpotent orbit closure in g M 0 (Q, v, w) : (affine) quiver variety Y (A, 0) : (affine) hypertoric variety

Symplectic resolution = crepant resolution For symplectic variety (Y 0, ω), π : (Y, ω) (Y 0, ω) : symplectic resolution def π ω extends to a symplectic form ω on whole Y. Remark For resolution π : Y Y 0 of sympleccitc variety Y 0, π : symplectic resolution π : crepant, i.e., π K Y0 = K Y

Poisson variety and Poisson deformation (Y, ω) has natural Poisson str. (Y, {, } 0 ) (Poisson str. bracket on O Y satisfying Leipnitz rule & Jacobi id) Poisson deformation (Y, {, } 0 ) (Y, {, }) : Poisson deform. of (Y, {, } 0 ) 0 S flat Definition (UPD (Universal Poisson deformation)) (Y univ, {, }) S is the universal Poisson deformation of Y. def (infinitesimal) Poisson deformation (X, {, } ) Spec A, X = Y univ S Spec A Y univ Spec A!f S

UPD of conical symplectic variety Theorem (Namikawa) π : Y Y 0 : proj. sympl. resolution of conical sympl. variety Y 0 There exists UPDs Y univ, Y0 univ of Y and Y 0. Moreover, Y univ µ H 2 (Y, C) Y Y 0 Π π Y univ 0 0 0 ψ H 2 (Y, C)/W, where W is Namikawa-Weyl group W GL(H 2 (Y, C)). Discriminant locus D C :={h H 2 (Y, C) fiber of Y univ 0 H 2 /W H 2 H 2 at h is singular} Hyperplane Arrangement!! i.e., A C s.t. D C = H A H. µ W

Example of discriminant arrangement A C In general, W A C H 2 (Y, C) as reflection w.r.t. some H A C. Example of A C C 2 /G : ADE-type surface singularity (G SL 2 (C)) { W = WG (usual) Weyl group A C = Weyl arrangement Sym{ n+1 (C 2 /G) : symplectic quotient singularity W = Z/2Z WG A C = cone over extended Catalan arr. Cat [ n,n] Φ G (Cat [ n,n] Φ G := {H λ,k : λ, = k λ Φ G, n k n} h) Moreover, D C = H A C H has connection to birational geometry of Y 0 (see later). Goal Describe the diagram of UPD for hypertoric variety.

Hypertoric variety - definition and its universal Poisson deformation

Q. What is hypertoric variety? A. algebraic variety with combinatorial flavor (like toric variety). Combinatorics {hyperplane arrangement H α B } Geometry {hypertoric variety Y (A, α)},where matrices A and B satisfy 0 Z n d B Z n A Z d 0 Philosophy Read off the geometric properties of Y (A, α) from the combinatorics of associated hyperplane arrangement H α B.

What is hypertoric variety? Example Let A = (1 1 1), For α = (1, 1, 1), ) ( 0 1 = ( b b b ). BT = 1 1 2 3 0 1 1 α set HB := {Hi : bi, = α i }. 0 Y(A,α)= Translation Singular pt 0 Y(A, 0)

What is hypertoric variety? Example ( ) Let A = (1 1 1), B T = 1 0 1 0 1 1 = ( b 1 b 2 b 3 ). For α = (1, 1, 1), set HB α := {H i : b i, = α i }. 0 Translation Y(A,α)= 0 Singular pt Y(A, 0)

What is hypertoric variety? Example ( ) Let A = (1 1 1), B T = 1 0 1 0 1 1 = ( b 1 b 2 b 3 ). For α = (1, 1, 1), set HB α := {H i : b i, = α i }. 0 [1:0:0] [0:0:1] Y(A,α)= [0:1:0] Translation Singular pt 0 Y(A, 0)

What is hypertoric variety? Example ( ) Let A = (1 1 1), B T = 1 0 1 0 1 1 = ( b 1 b 2 b 3 ). For α = (1, 1, 1), set HB α := {H i : b i, = α i }. 0 Translation Y(A,α)= 0 Singular pt Y(A, 0)

What is hypertoric variety? Example ( ) Let A = (1 1 1), B T = 1 0 1 0 1 1 = ( b 1 b 2 b 3 ). For α = (1, 1, 1), set HB α := {H i : b i, = α i }. 0 Translation Y(A,α)= 0 Singular pt Y(A, 0)

What is hypertoric variety? Example ( ) Let A = (1 1 1), B T = 1 0 1 0 1 1 = ( b 1 b 2 b 3 ). For α = (1, 1, 1), set HB α := {H i : b i, = α i }. 0 Translation Y(A,α)= 0 Singular pt Y(A, 0)

What is hypertoric variety? Example ( ) Let A = (1 1 1), B T = 1 0 1 0 1 1 = ( b 1 b 2 b 3 ). For α = (1, 1, 1), set HB α := {H i : b i, = α i }. 0 Translation Y(A,α)= 0 Singular pt Y(A, 0)

Hypertoric variety A Mat m n (Z) s.t. unimodular matrix, i.e., d d-minors=0, ±1 Take B as 0 Z n d B Z n A N = Z d 0 exact. ( ) ( ) B T = b 1 b 2 b n A = a1 a 2 a n n T d C (C2n = C n C n, ω C := dz j dw j ), j=1 Hamiltonian action t (z i, w i ) := (t a i z i, t a i w i ) moment map µ : C 2n C d : (z, w) n j=1 z jw j a j Define hypertoric variety as the quotient µ 1 (0)/T d C.

Hypertoric variety Definition (Hypertoric variety Y (A, α)) For each parameter α := A α ( α Z n ), the quotient space Y (A, α) := µ 1 (0) α ss //T d C is hypertoric variety. (µ 1 (0) α ss µ 1 (0) is α-semistable set) Inclusion µ 1 (0) α ss µ 1 (0) 0 ss = µ 1 (0) projective morphism π : Y (A, α) Y (A, 0) Fact For generic α, 1 Y (A, α) is 2(n d) = 2 rank B T dim smooth symplectic variety. 2 π : Y (A, α) Y (A, 0) is (conical) symplectic resolution.

Example: hypertoric variety associated to graph Example (Toric quiver variety) G : directed graph, Z E A G N Z V : e ij e i e j, 0 1 2 3 G 4 1 1 1 0 0 0 1 0 0 1 0 0 A G = 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 1 Y (A G, α) : toric quiver variety dim Y (A G, α) = 2(circuit rank of G)

Example Example (A 2 -type surface singularity) ( ) 1 0 1 A = A G =, G = 0 1 1, B T = ( 1 1 1 ) Y (A, α) S A2 : minimal resolution of S A2 π Y (A, 0) S A2 : {u 3 xy = 0} : A 2 -type singularity Remark In general, for G = edge graph of l + 1-gon, Y (A G, 0) = S Al : A l -type surface singularity.

Description of UPD for hypertoric variety We want to determine the UPDs for π : Y (A, α) Y (A, 0). Theorem (Braden-Licata-Proudfoot-Webster, N-.) X (A, α) µ C d l 1 π Y (A, α) Y (A, 0) Π WB X (A, 0)/W B 0 0 ψ C d /W B µ WB. l {}}{{}} s { Assume B T = ( c 1 c 1 c s c s ), where c k1 ±c k2 if k 1 k 2. Description of W B -action and discriminant locus D C W B := S l1 S ls C d = Span C (a 1,......, a n ). A C ={H C d codim H = 1 and H is generated by some a i s}.

Application 1 - classification of affine hypertoric varieties Y (A, 0) by matroids

Classification of Y (A, 0) What operation to A preserves the isomorphism calss of Y (A, 0)? Definition A A def A is obtained from A by a seq. of the followings: elementary row operations over Z, interchanging column vectors, i.e., a i a j, multiplying column vector by 1, i.e., a i a i. A A Y (A, 0) = Y (A, 0) : T n d C -eq. iso as conical sympl. var. Theorem (Arbo-Proudfoot, N-.) Y (A, 0) = Y (A, 0) as conical symplectic variety A A. Remark M(A) : regular matroid associated to A whose dual is M(B T ). A A M(A) = M(A ) M(B T ) = M(B T ).

Corollary of Theorem By classification theorem, in particular, for toric quiver varieties, Y (A G, 0) = Y (A G, 0) M(A G ) = M(A G ). Theorem (Whitney s 2-isomorphism theorem) Let G and G be graphs without isolated vertices. Then, M(A G ) = M(A G ) G can be transformed into G by a sequence of the following operations; (i) Vertex identification and vertex cleaving (ii) Whitney twist (i). (ii)

4-dimensional classification We only have to classify M(B T ) of rank = n d = 1 2 dim Y (A, 0). Theorem (N-. Classification of 4-dimensional Y (A, 0)) Every 4-dimensional Y (A, 0) is isomorphic to one of the followings; (i) S Al1 1 S A l2 1. (ii) O min (l 1, l 2, l 3 ):= ( u1 x 1 ) 3 1 u 2 x 2 sl y 3 y 2 u 3 3 All 2 2-minors of ul 1 1 x 1 x 3 y 1 u l 2 2 x 2 = 0 y 3 y 2 u l. 3 3 l 1 l {}}{{}} 2 { (i) B T = ( 1 1 0 0 ) G = 0 0 1 1 l 1 l {}}{ 2 l {}}{{}} 3 { (ii) B T = ( 1 1 0 0 1 1 ) G = 0 0 1 1 1 1

6-dimensional classification (1) G = (2) G = (3) G = (4) G = (5) G =

Application 2 - counting good resolutions of affine hypertoric variety by hyperplane arr.

Counting crepant resolutions of Y (A, 0) It is important to study good (=crepant) resolutions of singularity. Fact For generic α Z d, π α : Y (A, α) Y (A, 0) gives a crepant resolution. Conversely, all crepant resolutions are obtained by this form. Remark A := {H R d codim H = 1 and H is generated by some a i s}. α Z d : generic def α / D := H A Question Which α gives you different resolutions? H.

Example Example 1 0 AG = 0 1 1 1 A = AG N R. a3 a2 a4 a1 ( ) 1 1 1 1 1 0 T 1 1, G =, B =, 0 0 11 0 0 α := {H : b, = α } Rn d. Recall for α, HB i i i 0 0 Question: Which chambers give different resolutions? Note WB := S2 -action on A NR Answer: Chambers in a fundamental domain of WB -action.

Example Example 1 0 1 1 ( ) A G = 1 1 1 0 0 1 1 1, G =, B T =, 0 0 1 1 1 1 0 0 A = A G N R. Recall for α, HB α := {H i : b i, = α i } R n d. 0 0 Question: Which chambers give different resolutions? Note W B := S 2 -action on A N R Answer: Chambers in a fundamental domain of W B -action.

Example Example 1 0 1 1 ( ) A G = 1 1 1 0 0 1 1 1, G =, B T =, 0 0 1 1 1 1 0 0 A = A G N R. Recall for α, HB α := {H i : b i, = α i } R n d. 0 0 Question: Which chambers give different resolutions? Note W B := S 2 -action on A N R Answer: Chambers in a fundamental domain of W B -action.

Example Example 1 0 1 1 ( ) A G = 1 1 1 0 0 1 1 1, G =, B T =, 0 0 1 1 1 1 0 0 A = A G N R. Recall for α, HB α := {H i : b i, = α i } R n d. 0 0 Question: Which chambers give different resolutions? Note W B := S 2 -action on A N R Answer: Chambers in a fundamental domain of W B -action.

Example Example 1 0 1 1 ( ) A G = 1 1 1 0 0 1 1 1, G =, B T =, 0 0 1 1 1 1 0 0 A = A G N R. Recall for α, HB α := {H i : b i, = α i } R n d. 0 0 Question: Which chambers give different resolutions? Note W B := S 2 -action on A N R Answer: Chambers in a fundamental domain of W B -action.

Example Example 1 0 1 1 ( ) A G = 1 1 1 0 0 1 1 1, G =, B T =, 0 0 1 1 1 1 0 0 A = A G N R. Recall for α, HB α := {H i : b i, = α i } R n d. 0 0 Question: Which chambers give different resolutions? Note W B := S 2 -action on A N R Answer: Chambers in a fundamental domain of W B -action.

Example Example 1 0 1 1 ( ) A G = 1 1 1 0 0 1 1 1, G =, B T =, 0 0 1 1 1 1 0 0 A = A G N R. Recall for α, HB α := {H i : b i, = α i } R n d. 0 0 Question: Which chambers give different resolutions? Note W B := S 2 -action on A N R Answer: Chambers in a fundamental domain of W B -action.

Example Example 1 0 1 1 ( ) A G = 1 1 1 0 0 1 1 1, G =, B T =, 0 0 1 1 1 1 0 0 A = A G N R. Recall for α, HB α := {H i : b i, = α i } R n d. 0 0 Question: Which chambers give different resolutions? Note W B := S 2 -action on A N R Answer: Chambers in a fundamental domain of W B -action.

Counting crepant resolutions of Y (A, 0) l 1 l {}}{{}} s { Assume B T ( ) = c1 c 1 c s c s. Recall W B :=S l1 S ls of a j s. R d =Span R ({a j }) as permutation Theorem (Namikawa, Braden-Licata-Proudfoot-Webster) Each chamber in a fundamental domain of W B -action gives all different crepant resolutions of Y (A, 0). Corollary The number of crepant resolutions of Y (A, 0) is #{chamber of A} #W B = χ A( 1) l 1! l s!, where χ A (t) is the characteristic polynomial of A.

Counting crepant resolutions of O min (l 1, l 2, l 3 ) Lemma For O min (l 1, l 2, l 3 ), the corresponding A is (the essentialization of) A l1,l 2,l 3 in R l 1+l 2 +l 3 : H ijk : x i + y j + z k = 0 (1 i l 1, 1 j l 2, 1 k l 3 ) Hi x 1,i 2 : x i1 x i2 = 0 (1 i 1 < i 2 l 1 ) H y j 1,j 2 : y j1 y j2 = 0 (1 j 1 < j 2 l 2 ) Hk z 1,k 2 : z k1 z k2 = 0 (1 k 1 < k 2 l 3 ) Proposition (Edelman-Reiner) χ Al1,l 2,l 3 (t) = t 2 (t 1)(t 2) (t (l 1 + l 2 1)) (l 3 = 1) l 1 +l 2 l 1 +l 2 1 t 2 (t 1) (t i) (t j) (l 3 = 2) i=l 1 +1 j=l 2 +1

Counting crepant resolutions of O min (l 1, l 2, l 3 ) Corollary The number of crepant resolutions of O min (l 1, l 2, l 3 ) is ( l1 +l 2 ) l 1 (l 3 = 1) ( l 1 +l 2 +1 l 1 )( l 1 +l 2 +1 l 2 ) l 1 +l 2 +1 (l 3 = 2) This gives a geometric meaning of A l1,l 2,l 3 considered by Edelman and Reiner. Remark For l 1, l 2, l 3 3, it is known A l1,l 2,l 3 is NOT free arrangement. χ A3,3,3 (t) = t 2 (t 1)(t 5)(t 7)(t 4 23t 3 +200t 2 784t +1188) Question Compute the number of chambers of A l1,l 2,l 3 for l 1, l 2, l 3 3.

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