Hypertoric varieties and hyperplane arrangements

Takahiro Nagaoka

Kyoto Univ.

June 16, 2018

Motivation - Study of the geometry of symplectic variety

Symplectic variety (Y0, ω) · · ·Very special but interestingeven 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 DC is hyperplane arr. with W -action.

“Good”(= crepant) resolution π : (Y , ω) → (Y0, ω)⇝ construction and counting all crepant resolutions of Y0.⇝ This is related to the structure of DC.

We will study these problems for hypertoric variety.

Takahiro Nagaoka

Contents

Intro - symplectic variety and Poisson deformation

Hypertoric variety - definition and its universalPoisson deformation

Application 1 - classification of affine hypertoricvarieties by matroids

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

Takahiro Nagaoka

Intro - symplectic variety and Poisson deformation

Takahiro Nagaoka

Symplectic variety

(Y0, ω) : symplectic varietydef⇔ normal alg. var. with holomo. symplectic form ω on (Y0)reg .

(Y0, ω) : conical symplectic varietydef⇔ affine symplectic variety with “good” C∗-action.

Example

V⊕V ∗/Γ : Symplectic quotient singularity (Γ ⊂ Sp(V ⊕V ∗))

⇝ eg. ADE -type surface singularity (An : zn+1 − xy = 0).

O : Nilpotent orbit closure in g

M0(Q, v,w) : (affine) quiver variety

Y (A, 0) : (affine) hypertoric variety

Takahiro Nagaoka

Symplectic resolution = crepant resolution

For symplectic variety (Y0, ω),

π : (Y , ω) → (Y0, ω) : symplectic resolutiondef⇔ π∗ω extends to a symplectic form ω on whole Y .

Remark

For resolution π : Y → Y0 of sympleccitc variety Y0,

π : symplectic resolution ⇔ π : crepant, i.e., π∗KY0 = KY

Takahiro Nagaoka

Poisson variety and Poisson deformation

(Y , ω) has natural Poisson str. (Y , {−,−}0)(Poisson str. · · · bracket on OY satisfying Leipnitz rule & Jacobi id)

Poisson deformation

(Y , {−,−}0) (Y, {−,−}) : Poisson deform. of (Y , {−,−}0)

0 S

⊂

flat

∈

Definition (UPD (Universal Poisson deformation))

(Yuniv , {−,−}) → S is the universal Poisson deformation of Y .def⇔ ∀ (infinitesimal) Poisson deformation (X , {−,−}′) → SpecA,

X ∼= Yuniv ×S SpecA Yuniv

SpecA S∃!f

Takahiro Nagaoka

UPD of conical symplectic variety

Theorem (Namikawa)

π : Y → Y0 : proj. sympl. resolution of conical sympl. variety Y0⇒ There exists UPDs Yuniv , Yuniv0 of Y and Y0. Moreover,

Y Y0

Yuniv Yuniv0

0 0

H2(Y ,C) H2(Y ,C)/W

⊃

π

⊃Π

µ

∋ ∋ψ

µW

, where W is Namikawa-Weyl group W ⊂ GL(H2(Y ,C)).

Discriminant locusDC:={h∈ H2(Y ,C) |fiber of Yuniv0 ×H2/WH2 →H2 at h is singular}

⇝ Hyperplane Arrangement !! i.e., ∃AC s.t. DC =∪

H∈AH.

Takahiro Nagaoka

Example of discriminant arrangement ACIn general, W ↷AC⊂H2(Y ,C) as reflection w.r.t. some H ∈ AC.

Example of ACC2/G : ADE -type surface singularity (G ⊂ SL2(C))

⇝{

W = WG (usual) Weyl groupAC = Weyl arrangement

Symn+1(C2/G ) : symplectic quotient singularity

⇝{

W = Z/2Z×WGAC = cone over extended Catalan arr. Cat

[−n,n]ΦG

(Cat[−n,n]ΦG

:= {Hλ,k : ⟨λ,−⟩ = k | λ ∈ ΦG ,−n ≤ k ≤ n} ⊂ h)

Moreover, DC =∪

H∈AC H has connection to birational geometryof Y0 (see later).

Goal

Describe the diagram of UPD for hypertoric variety.Takahiro Nagaoka

Hypertoric variety - definition and its universal Poissondeformation

Takahiro Nagaoka

Q. What is hypertoric variety?

A. algebraic variety with “combinatorial” flavor (like toric variety).

Combinatorics Geometry

{hyperplane arrangement HαB} {hypertoric variety Y (A, α)}

,where matrices A and B satisfy 0 Zn−d Zn Zd 0B A

Philosophy

Read off the geometric properties of Y (A, α) from thecombinatorics of associated hyperplane arrangement HαB .

Takahiro Nagaoka

What is hypertoric variety?

Example

Let A = (1 1 1), BT =(1 0 −10 1 −1

)= ( b1 b2 b3 ).

For α̃ = (1, 1, 1), set HαB := {Hi : ⟨bi,−⟩ = α̃i}.

Takahiro Nagaoka

0

Singular pt

0

Translation

Y(A,α)=

Y(A, 0)

What is hypertoric variety?

Example

Let A = (1 1 1), BT =(1 0 −10 1 −1

)= ( b1 b2 b3 ).

For α̃ = (1, 1, 1), set HαB := {Hi : ⟨bi,−⟩ = α̃i}.

Takahiro Nagaoka

0

Singular pt

0

Translation

Y(A,α)=

Y(A, 0)

What is hypertoric variety?

Example

Let A = (1 1 1), BT =(1 0 −10 1 −1

)= ( b1 b2 b3 ).

For α̃ = (1, 1, 1), set HαB := {Hi : ⟨bi,−⟩ = α̃i}.

Takahiro Nagaoka

0

Singular pt

0

Translation

Y(A,α)=

Y(A, 0)

[1:0:0]

[0:1:0] [0:0:1]

What is hypertoric variety?

Example

Let A = (1 1 1), BT =(1 0 −10 1 −1

)= ( b1 b2 b3 ).

For α̃ = (1, 1, 1), set HαB := {Hi : ⟨bi,−⟩ = α̃i}.

Takahiro Nagaoka

0

Singular pt

0

Translation

Y(A,α)=

Y(A, 0)

What is hypertoric variety?

Example

Let A = (1 1 1), BT =(1 0 −10 1 −1

)= ( b1 b2 b3 ).

For α̃ = (1, 1, 1), set HαB := {Hi : ⟨bi,−⟩ = α̃i}.

Takahiro Nagaoka

0

Singular pt

0

Translation

Y(A,α)=

Y(A, 0)

What is hypertoric variety?

Example

Let A = (1 1 1), BT =(1 0 −10 1 −1

)= ( b1 b2 b3 ).

For α̃ = (1, 1, 1), set HαB := {Hi : ⟨bi,−⟩ = α̃i}.

Takahiro Nagaoka

0

Singular pt

0

Translation

Y(A,α)=

Y(A, 0)

What is hypertoric variety?

Example

Let A = (1 1 1), BT =(1 0 −10 1 −1

)= ( b1 b2 b3 ).

For α̃ = (1, 1, 1), set HαB := {Hi : ⟨bi,−⟩ = α̃i}.

Takahiro Nagaoka

0

Singular pt

0

Translation

Y(A,α)=

Y(A, 0)

Hypertoric variety

A ∈ Matm×n(Z) s.t. unimodular matrix, i.e., ∀d × d-minors=0,±1

Take B as 0 Zn−d Zn N ∼= Zd 0B A exact.

BT =

(b1 b2 · · · bn

)A =

(a1 a2 · · · an

)

TdC ↷ (C2n = Cn⊕Cn, ωC :=n∑

j=1

dzj ∧ dwj), “Hamiltonian action”

t · (zi ,wi ) := (taizi , t−aiwi )

⇝ ∃ moment map µ : C2n → Cd : (z,w) 7→∑n

j=1 zjwjaj

Define hypertoric variety as the “quotient µ−1(0)/TdC”.

Takahiro Nagaoka

Hypertoric variety

Definition (Hypertoric variety Y (A, α))

For each parameter α := Aα̃ (α̃ ∈ Zn), the “quotient” space

Y (A, α) := µ−1(0)α−ss//TdC

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 rankBT dim smooth symplecticvariety.

2 π : Y (A, α) → Y (A, 0) is (conical) symplectic resolution.

Takahiro Nagaoka

Example: hypertoric variety associated to graph

Example (Toric quiver variety)

G : directed graph, Z|E | AG−−→→ N ⊂ Z|V | : eij 7→ ei − ej,

1•

0• 2• 4•

3•

G

⇝ AG =

1 1 1 0 0 0−1 0 0 1 0 00 −1 0 0 1 00 0 −1 0 0 10 0 0 −1 −1 −1

⇝ Y (AG , α) : toric quiver variety⇝ dimY (AG , α) = 2(circuit rank of G )

Takahiro Nagaoka

Example

Example (A2-type surface singularity)

A = AG =

(1 0 −10 1 −1

), G =

•• •

, BT =(1 1 1

)

Y (A, α) S̃A2 : minimal resolution of SA2

Y (A, 0) SA2 : {u3 − xy = 0} : A2-type singularity

∼

π

∼

Remark

In general, for G = edge graph of ℓ+ 1-gon,Y (AG , 0) = SAℓ : Aℓ-type surface singularity.

Takahiro Nagaoka

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-.)

Y (A, α) Y (A, 0)

X (A, α) X (A, 0)/WB

0 0

Cd Cd/WB

⊃

π

⊃ΠWB

µ

ψ∋ ∋

µWB.

Assume BT=

ℓ1︷ ︸︸ ︷ ℓs︷ ︸︸ ︷(c1 · · · c1 · · · cs · · · cs ), where ck1 ̸= ±ck2 if k1 ̸= k2.

Description of WB -action and discriminant locus DCWB := Sℓ1 × · · · ×Sℓs ↷ Cd = SpanC(a1, . . . . . . , an).AC={H⊂Cd | codimH = 1 and H is generated by some ai’s}.

Takahiro Nagaoka

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

Takahiro Nagaoka

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., ai ↔ aj,· multiplying column vector by −1, i.e., ai 7→ −ai.

A ∼ A′ ⇒Y (A, 0)∼=Y (A′, 0) : Tn−dC -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(BT ).

A ∼ A′ ⇔ M(A) ∼= M(A′) ⇔ M(BT ) ∼= M(B ′T ).

Takahiro Nagaoka

Corollary of Theorem

By classification theorem, in particular, for toric quiver varieties,

Y (AG , 0) ∼= Y (AG ′ , 0) ⇔ M(AG ) ∼= M(AG ′).

Theorem (Whitney’s 2-isomorphism theorem)

Let G and G ′ be graphs without isolated vertices. Then,M(AG ) ∼= M(AG ′)⇔ G ′ can be transformed into G by a sequence of the followingoperations;

(i) Vertex identification and vertex cleaving

(ii) Whitney twist

• ••

• •

(i)∼• •

• •• •

.• • •

• • •(ii)∼

• • •

• • •

Takahiro Nagaoka

4-dimensional classification

We only have to classify M(BT ) of rank = n − d = 12 dimY (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) SAℓ1−1 × SAℓ2−1 .

(ii) Omin(ℓ1, ℓ2, ℓ3):=(u1 x1 x3y1 u2 x2y3 y2 u3) ∈ sl3∣∣∣∣∣∣ All 2×2-minors of

uℓ11 x1 x3y1 uℓ22 x2y3 y2 u

ℓ33

= 0.

(i) BT =

ℓ1︷ ︸︸ ︷ ℓ2︷ ︸︸ ︷(1 · · · 1 0 · · · 0 )0 · · · 0 1 · · · 1

⇝ G =

(ii) BT =

ℓ1︷ ︸︸ ︷ ℓ2︷ ︸︸ ︷ ℓ3︷ ︸︸ ︷(1 · · · 1 0 · · · 0 1 · · · 1 )0 · · · 0 1 · · · 1 1 · · · 1

⇝ G =

Takahiro Nagaoka

6-dimensional classification

(1) G =

(2) G =

(3) G =

(4) G =

(5) G =

Takahiro Nagaoka

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

Takahiro Nagaoka

Counting crepant resolutions of Y (A, 0)

It is important to study “good”(=crepant) resolutions of singularity.

Fact

For “generic”α ∈ Zd , πα : Y (A, α) → Y (A, 0) gives a crepantresolution.

Conversely, all crepant resolutions are obtained by this form.

Remark

A := {H ⊂ Rd | codimH = 1 and H is generated by some ai’s}.

α ∈ Zd : generic def⇔ α /∈ D :=∪H∈A

H.

Question Which α gives you different resolutions?

Takahiro Nagaoka

Example

Example

AG =

1 0 −1 10 1 1 −1−1 −1 0 0

, G = • • • , BT = (1 −1 1 00 0 1 1

),

A = AG ⊂ NR. Recall for α, HαB := {Hi : ⟨bi,−⟩ = α̃i} ⊂ Rn−d .

Question: Which chambers give different resolutions ?⇝ Note WB := S2-action on A ⊂ NRAnswer: Chambers in a fundamental domain of WB -action.

Takahiro Nagaoka

0

0

a2 a3

a4

a1

Example

Example

AG =

1 0 −1 10 1 1 −1−1 −1 0 0

, G = • • • , BT = (1 −1 1 00 0 1 1

),

A = AG ⊂ NR. Recall for α, HαB := {Hi : ⟨bi,−⟩ = α̃i} ⊂ Rn−d .

Question: Which chambers give different resolutions ?⇝ Note WB := S2-action on A ⊂ NRAnswer: Chambers in a fundamental domain of WB -action.

Takahiro Nagaoka

0

0

Example

Example

AG =

1 0 −1 10 1 1 −1−1 −1 0 0

, G = • • • , BT = (1 −1 1 00 0 1 1

),

A = AG ⊂ NR. Recall for α, HαB := {Hi : ⟨bi,−⟩ = α̃i} ⊂ Rn−d .

Question: Which chambers give different resolutions ?⇝ Note WB := S2-action on A ⊂ NRAnswer: Chambers in a fundamental domain of WB -action.

Takahiro Nagaoka

0

0

Example

Example

AG =

1 0 −1 10 1 1 −1−1 −1 0 0

, G = • • • , BT = (1 −1 1 00 0 1 1

),

A = AG ⊂ NR. Recall for α, HαB := {Hi : ⟨bi,−⟩ = α̃i} ⊂ Rn−d .

Question: Which chambers give different resolutions ?⇝ Note WB := S2-action on A ⊂ NRAnswer: Chambers in a fundamental domain of WB -action.

Takahiro Nagaoka

0

0

Example

Example

AG =

1 0 −1 10 1 1 −1−1 −1 0 0

, G = • • • , BT = (1 −1 1 00 0 1 1

),

A = AG ⊂ NR. Recall for α, HαB := {Hi : ⟨bi,−⟩ = α̃i} ⊂ Rn−d .

Question: Which chambers give different resolutions ?⇝ Note WB := S2-action on A ⊂ NRAnswer: Chambers in a fundamental domain of WB -action.

Takahiro Nagaoka

0

0

Example

Example

AG =

1 0 −1 10 1 1 −1−1 −1 0 0

, G = • • • , BT = (1 −1 1 00 0 1 1

),

A = AG ⊂ NR. Recall for α, HαB := {Hi : ⟨bi,−⟩ = α̃i} ⊂ Rn−d .

Question: Which chambers give different resolutions ?⇝ Note WB := S2-action on A ⊂ NRAnswer: Chambers in a fundamental domain of WB -action.

Takahiro Nagaoka

0

0

Example

Example

AG =

1 0 −1 10 1 1 −1−1 −1 0 0

, G = • • • , BT = (1 −1 1 00 0 1 1

),

A = AG ⊂ NR. Recall for α, HαB := {Hi : ⟨bi,−⟩ = α̃i} ⊂ Rn−d .

Question: Which chambers give different resolutions ?⇝ Note WB := S2-action on A ⊂ NRAnswer: Chambers in a fundamental domain of WB -action.

Takahiro Nagaoka

0

0

Example

Example

AG =

1 0 −1 10 1 1 −1−1 −1 0 0

, G = • • • , BT = (1 −1 1 00 0 1 1

),

A = AG ⊂ NR. Recall for α, HαB := {Hi : ⟨bi,−⟩ = α̃i} ⊂ Rn−d .

Question: Which chambers give different resolutions ?⇝ Note WB := S2-action on A ⊂ NRAnswer: Chambers in a fundamental domain of WB -action.

Takahiro Nagaoka

0

0

Counting crepant resolutions of Y (A, 0)

Assume BT=

ℓ1︷ ︸︸ ︷ ℓs︷ ︸︸ ︷(c1 · · · c1 · · · cs · · · cs

).

Recall WB :=Sℓ1×· · ·×Sℓs ↷ Rd=SpanR({aj}) as permutationof aj’s.

Theorem (Namikawa, Braden-Licata-Proudfoot-Webster)

Each chamber in a fundamental domain of WB -action gives alldifferent crepant resolutions of Y (A, 0).

Corollary

The number of crepant resolutions of Y (A, 0) is

#{chamber of A}#WB

=|χA(−1)|ℓ1! · · · ℓs !

,

where χA(t) is the characteristic polynomial of A.

Takahiro Nagaoka

Counting crepant resolutions of Omin(ℓ1, ℓ2, ℓ3)

Lemma

For Omin(ℓ1, ℓ2, ℓ3), the corresponding A is (the essentializationof) Aℓ1,ℓ2,ℓ3 in Rℓ1+ℓ2+ℓ3 :

Hijk : xi + yj + zk = 0 (1 ≤ i ≤ ℓ1, 1 ≤ j ≤ ℓ2, 1 ≤ k ≤ ℓ3)Hxi1,i2 : xi1 − xi2 = 0 (1 ≤ i1 < i2 ≤ ℓ1)Hyj1,j2 : yj1 − yj2 = 0 (1 ≤ j1 < j2 ≤ ℓ2)Hzk1,k2 : zk1 − zk2 = 0 (1 ≤ k1 < k2 ≤ ℓ3)

Proposition (Edelman-Reiner)

χAℓ1,ℓ2,ℓ3 (t) =t2(t − 1)(t − 2) · · · (t − (ℓ1 + ℓ2 − 1)) (ℓ3 = 1)

t2(t − 1)ℓ1+ℓ2∏i=ℓ1+1

(t − i)ℓ1+ℓ2−1∏j=ℓ2+1

(t − j) (ℓ3 = 2)

Takahiro Nagaoka

Counting crepant resolutions of Omin(ℓ1, ℓ2, ℓ3)

Corollary

The number of crepant resolutions of Omin(ℓ1, ℓ2, ℓ3) is(ℓ1+ℓ2ℓ1

)(ℓ3 = 1)

(ℓ1+ℓ2+1ℓ1 )(ℓ1+ℓ2+1

ℓ2)

ℓ1+ℓ2+1(ℓ3 = 2)

This gives a geometric meaning of Aℓ1,ℓ2,ℓ3 considered by Edelmanand Reiner.

Remark

For ℓ1, ℓ2, ℓ3 ≥ 3, it is known Aℓ1,ℓ2,ℓ3 is NOT free arrangement.χA3,3,3(t) = t

2(t−1)(t−5)(t−7)(t4−23t3+200t2−784t+1188)

Question

Compute the number of chambers of Aℓ1,ℓ2,ℓ3 for ℓ1, ℓ2, ℓ3 ≥ 3.Takahiro Nagaoka

Thank you for listening!!

Takahiro Nagaoka