The universal covers of hypertoric varieties andBogomolov’s decomposition
Takahiro Nagaoka
Kyoto Univ.

Nov. 22. 2019
Toric Topology 2019 in Okayama@Okayama University of ScienceContents

Background and Motivation
Hypertoric varieties
π1 (YA (0)reg ) & universal coverings
Bogomolov’s decomposition

Takahiro Nagaoka

universal covers of hypertoric varieties

Background and Motivation

Takahiro Nagaoka

universal covers of hypertoric varieties

Background - conical symplectic varieties
Definition(Y, ω) is a conical symplectic variety if• ω is a holomorphic symplectic form on Yreg (+ conditions)• Y is aﬃne, and ∃C∗ ↷ Y with positive weight (like cone)
Example(C2n , ωC ) with scalar action
ADE-type surface singularity (ex. C2 /Zℓ ={xy − z ℓ =0})
dAﬃne hypertoric variety YA (0) := “µ−1C (0)//TC ”

Study the “universal covering” of Y !
Takahiro Nagaoka

universal covers of hypertoric varieties

Motivation 1 - the “singular” universal cover
Conjecture|π1 (Yreg )| < ∞.
Proposition (The existence of the universal cover)
Assume |π1 (Yreg )| < ∞.
Then,∃!a conical symplectic variety (Ye , ωe ) satisfying(Ye , ωe)⊃φ−1 (Yreg )∃!φ : finite(Y, ω)⊃φ| : universal cover.

Yreg

Problem 1
What are π1 (Yreg ) and the universal cover Ye ?
Takahiro Nagaoka

universal covers of hypertoric varieties

Motivation 2 - Bogomolov’s decomposition

Problem 2 ([Namikawa] Analogue of Bogomolov’s decomposition)
For any conical sympl. variety (Y, ω) with |π1 (Yreg )| < ∞,(Ye , ωe) ∼= ∃?

r∏(Ym , ωm ),m=1

where (Ym , ωm ) is irreducible, i.e., ωm is the unique sympl. form.
GoalAnswer these problems for hypertoric varieties.

Takahiro Nagaoka

universal covers of hypertoric varieties

Hypertoric varieties YA (α)
Combinatorics

Geometryα}{polytope PB{toric variety XA (α)}α}{hyperplane arrangement HB{hypertoric variety YA (α)}

Takahiro Nagaoka

universal covers of hypertoric varieties

What is hypertoric variety?
Example()Let B T = 10 01 −1−1 = ( b1 b2 b3 ) and α̃ = (1, 1, 1).

Takahiro Nagaoka

universal covers of hypertoric varieties

What is hypertoric variety?
Example()Let B T = 10 01 −1−1 = ( b1 b2 b3 ) and α̃ = (1, 1, 1).
α := {H : ⟨b , −⟩ = −α̃ }.
Set HBiii

Takahiro Nagaoka

universal covers of hypertoric varieties

What is hypertoric variety?
Example()Let B T = 10 01 −1−1 = ( b1 b2 b3 ) and α̃ = (1, 1, 1).
α := {H : ⟨b , −⟩ = −α̃ }.
Set HBiii0

Y(A,α)=
Translation

Singular pt0Y(A, 0)
Takahiro Nagaoka

universal covers of hypertoric varieties

What is hypertoric variety?
Example()Let B T = 10 01 −1−1 = ( b1 b2 b3 ) and α̃ = (1, 1, 1).
α := {H : ⟨b , −⟩ = −α̃ }.
Set HBiii

0

Y(A,α)=
Translation

Singular pt0

Y(A, 0)Takahiro Nagaoka

universal covers of hypertoric varieties

What is hypertoric variety?
Example()Let B T = 10 01 −1−1 = ( b1 b2 b3 ) and α̃ = (1, 1, 1).
α := {H : ⟨b , −⟩ = −α̃ }.
Set HBiii[1:0:0]0[0:0:1][0:1:0]Y(A,α)=
Translation

Singular pt0

Y(A, 0)Takahiro Nagaoka

universal covers of hypertoric varieties

What is hypertoric variety?
Example()Let B T = 10 01 −1−1 = ( b1 b2 b3 ) and α̃ = (1, 1, 1).
α := {H : ⟨b , −⟩ = −α̃ }.
Set HBiii0

Y(A,α)=
Translation

Singular pt0

Y(A, 0)Takahiro Nagaoka

universal covers of hypertoric varieties

What is hypertoric variety?
Example()Let B T = 10 01 −1−1 = ( b1 b2 b3 ) and α̃ = (1, 1, 1).
α := {H : ⟨b , −⟩ = −α̃ }.
Set HBiii0

Y(A,α)=
Translation

Singular pt0Y(A, 0)
Takahiro Nagaoka

universal covers of hypertoric varieties

What is hypertoric variety?
Example()Let B T = 10 01 −1−1 = ( b1 b2 b3 ) and α̃ = (1, 1, 1).
α := {H : ⟨b , −⟩ = −α̃ }.
Set HBiii0

Y(A,α)=
Translation

Singular pt0Y(A, 0)
Takahiro Nagaoka

universal covers of hypertoric varieties

Definition of hypertoric variety as symplectic reduction
0Zn−dBZnA

NA := Zd0ATTA := TdC ,→ TnC ↷ (C2n = Cn ⊕(Cn )∗ , ωC )µC : Tn
C -inv. moment map

CdDefinition (Hypertoric variety )YA (α) :=µ−1C (0) − {α-unstable pts}//TA : hypertoric variety
α :={H : ⟨b , −⟩= −α̃ }⊂Rn−d : the associated arrangement)(HBiiiFact
1Tn−d↷ (YA (0), ω) is a 2(n − d) dim conical sympl. var.
C2∃πα : (YA (α), ω) → (YA (0), ω) is a resolution for generic α.
Takahiro Nagaoka

universal covers of hypertoric varieties

Example
Example (A2 -type surface singularity C2 /Z3 ))(()
1 0 −1A=, BT = 1 1 10 1 −1
YA (α)∼^
2 /ZC3∼C2 /Z3: the minimal resolution of SA2παYA (0)∼{u3 − xy = 0} : A2 -type singularity

Translation

Takahiro Nagaoka

universal covers of hypertoric varieties4-dimensional classification

Fact (Classification of 4-dimensional YA (0))(i) C2 /Zℓ1 × C2 /Zℓ2 .
min(ii) O (ℓ1 , ℓ2 , ℓ3 ):= ℓu11 x1 x3
 (u x x )y11 u12 x32 ∈ sl3 All 2×2-minors of  y1 uℓ2 x2  = 0 .
2
 y3 y2 u3
y3 y2 uℓ33
1ℓℓ1ℓ22ℓℓ3
Takahiro Nagaokaℓ3
universal covers of hypertoric varietiesπ1 (YA (0)reg ) & universal coverings

Takahiro Nagaoka

universal covers of hypertoric varieties

Example (YA (0) := C2 /Zℓ ∼= {xy − uℓ = 0})
C2 → C2 /Zℓ =: YA (0) is universal cover & π1 (YA (0)reg ) = Zℓ .
ℓ
Simplificationmin )Example (YA (0) := OA2min ) ) = 1.
4-dim isolated singularity & π1 ((OAreg
2

Singular pt

0.
Sing(YA (0)) ↔ “multiplicated” or “non-general” locus of HBℓ1The slice ∼= C2 /Zℓ1 .
ℓ2⇝ π1 (YA (0)reg ) ̸= 1 ? (if ℓ1 ≥ 2).

ℓ3
Takahiro Nagaoka

universal covers of hypertoric varieties

Simplificationℓℓz }|1 {z }|s {()We can assume B T= b(1) · · · b(1) · · · b(s) · · · b(s) .
Lemmaℓ1 =· · ·=ℓs = 1 , i.e., YA (0) is simple ⇒ π1 (YA (0)reg )=1.
T⇝ Consider the simplification B := (b(1) · · · b(s) )Simplification

Natural expectation∃? finite quotient φ : YA (0) → YA (0) : universal cover,where 0Zn−dBZs

ATakahiro Nagaoka

NA := Zd−(n−s)

0 : exact.

universal covers of hypertoric varieties

The fundamental group π1 (YA (0)reg )Theorem ([N.])∃ finite φ : YA (0) ↠ YA (0) : universal cover, andπ1 (YA (0)reg ) ∼= Γ/Γ ∩ TA ,where Γ :=∏s

k=1 Z/ℓk ZTsCA

TExample (Omin (ℓ1 , ℓ2 , ℓ3 ))π1 (Omin (ℓ1 , ℓ2 , ℓ3 )reg ) ∼=Zℓ1 × Zℓ2 × Zℓ3() ,Z⟨ ℓg1 , ℓg2 , ℓg3 ⟩where g := gcd(ℓ1 , ℓ2 , ℓ3 ).

Takahiro Nagaoka

d−(n−s)TA := TC.

ℓ1
ℓ2
ℓ3

universal covers of hypertoric varieties

Bogomolov’s decomposition

Takahiro Nagaoka

universal covers of hypertoric varieties

Bogomolov’s decomposition
Problem ([Namikawa] Analogue of Bogomolov’s decomposition)
For any conical sympl. var. (Y, ω) with |π1 (Yreg )| < ∞,r∏∼e(Y , ωe ) = ∃?
(Ym , ωm ), where (Ym , ωm ) is irreducible.
m=1

A trivial example∏n(C2n , ωC ) ∼(C2 , dzm ∧ dwm ).
=m=1⊕
If A admits a block decomposition A = rm=1 Am , then clearlyr∏
YA (0) ∼YAm (0).
=m=1Naive Question
A is block indecomposable?
⇔ YA (0) is an irreducible conical symplectic variety.
Takahiro Nagaoka

universal covers of hypertoric varieties

Bogomolov’s decomposition
Problem ([Namikawa] Analogue of Bogomolov’s decomposition)
For any conical sympl. var. (Y, ω) with |π1 (Yreg )| < ∞,r∏∼e(Y , ωe ) = ∃?
(Ym , ωm ), where (Ym , ωm ) is irreducible.
m=1A trivial example∏n(C2n , ωC ) ∼(C2 , dzm ∧ dwm ).
=m=1⊕
If A admits a block decomposition A = rm=1 Am , then clearlyr∏
YA (0) ∼YAm (0).
=m=1Theorem ([N.])A is block indecomposable⇔ YA (0) is an irreducible conical symplectic variety.
Takahiro Nagaoka

universal covers of hypertoric varieties

Byproduct - Combinatorial classification

Theorem (Berchtold)∀ toric varieties X and X ′ of dim n − d,X∼
= X′ ⇔ X ∼= X ′ : Tn−d
C -equivariant.
Corollary ([N.])∀ two smooth (or aﬃne) hypertoric YA (α) and YA′ (α′ ),TFAE:∼ YA′ (α′ ) : C∗ -equivariant.
(i) YA (α) =(ii) (YA (α), ω) ∼= (YA′ (α′ ), ω ′ ) : C∗×Tn−d
C -equivariant.
α ∼ Hα , i.e., induces the same “fan”.
(iii) HB= B′′Takahiro Nagaoka

universal covers of hypertoric varietiesThank you for listening !

Takahiro Nagaoka

universal covers of hypertoric varieties