Motivation
Basic Construction
Our Results

K-theory of toric hyperKähler manifolds
V. UmaI.I.T Chennai

Conference on Toric Topology
2019
Okayama

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Outline1

Motivation

2

Basic Construction
Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

3

Our Results
K-ring of toric hyperKähler manifolds

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Outline1

Motivation

2

Basic Construction
Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

3

Our Results
K-ring of toric hyperKähler manifolds

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Outline1

Motivation

2

Basic Construction
Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

3

Our Results
K-ring of toric hyperKähler manifolds

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Earlier work
Toric hyperKähler manifolds [BD00] were defined by
Bielawski and Dancer who study their topology andgeometry.
The integral cohomology ring of toric hyperKähler
manifolds was studied by Konno [K00] who gives apresentation for the cohomology ring.
Recently Kuroki [KU11] studied equivariant cohomology
ring of toric hyperKähler manifolds in relation tocohomological rigidity problem.
Algebraic geometric analogue of toric hyperKähler varieties
was developed by Hausel and Sturmfels [HS02] andstudied in relation to the geometry of toric quiver varieties.
V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Earlier work
Toric hyperKähler manifolds [BD00] were defined by
Bielawski and Dancer who study their topology andgeometry.
The integral cohomology ring of toric hyperKähler
manifolds was studied by Konno [K00] who gives apresentation for the cohomology ring.
Recently Kuroki [KU11] studied equivariant cohomology
ring of toric hyperKähler manifolds in relation tocohomological rigidity problem.
Algebraic geometric analogue of toric hyperKähler varieties
was developed by Hausel and Sturmfels [HS02] andstudied in relation to the geometry of toric quiver varieties.
V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Earlier work
Toric hyperKähler manifolds [BD00] were defined by
Bielawski and Dancer who study their topology andgeometry.
The integral cohomology ring of toric hyperKähler
manifolds was studied by Konno [K00] who gives apresentation for the cohomology ring.
Recently Kuroki [KU11] studied equivariant cohomology
ring of toric hyperKähler manifolds in relation tocohomological rigidity problem.
Algebraic geometric analogue of toric hyperKähler varieties
was developed by Hausel and Sturmfels [HS02] andstudied in relation to the geometry of toric quiver varieties.
V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Earlier work
Toric hyperKähler manifolds [BD00] were defined by
Bielawski and Dancer who study their topology andgeometry.
The integral cohomology ring of toric hyperKähler
manifolds was studied by Konno [K00] who gives apresentation for the cohomology ring.
Recently Kuroki [KU11] studied equivariant cohomology
ring of toric hyperKähler manifolds in relation tocohomological rigidity problem.
Algebraic geometric analogue of toric hyperKähler varieties
was developed by Hausel and Sturmfels [HS02] andstudied in relation to the geometry of toric quiver varieties.
V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Our aimTo study K-theory of toric hyperKähler manifolds and toric
hyperKähler varieties
We would like to give a presentation of the K-ring using the
combinatorics of the associated hyperplane arrangement
Our earlier results on the K-ring of smooth projective toricvarieties, quasitoric manifolds and torus manifolds usedthe combinatorics of fan or polytope.
We wished to explore if methods used extend to the settingof toric hyperKähler manifolds.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Our aimTo study K-theory of toric hyperKähler manifolds and toric
hyperKähler varieties
We would like to give a presentation of the K-ring using the
combinatorics of the associated hyperplane arrangement
Our earlier results on the K-ring of smooth projective toricvarieties, quasitoric manifolds and torus manifolds usedthe combinatorics of fan or polytope.
We wished to explore if methods used extend to the settingof toric hyperKähler manifolds.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Our aimTo study K-theory of toric hyperKähler manifolds and toric
hyperKähler varieties
We would like to give a presentation of the K-ring using the
combinatorics of the associated hyperplane arrangement
Our earlier results on the K-ring of smooth projective toricvarieties, quasitoric manifolds and torus manifolds usedthe combinatorics of fan or polytope.
We wished to explore if methods used extend to the settingof toric hyperKähler manifolds.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Our aimTo study K-theory of toric hyperKähler manifolds and toric
hyperKähler varieties
We would like to give a presentation of the K-ring using the
combinatorics of the associated hyperplane arrangement
Our earlier results on the K-ring of smooth projective toricvarieties, quasitoric manifolds and torus manifolds usedthe combinatorics of fan or polytope.
We wished to explore if methods used extend to the settingof toric hyperKähler manifolds.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Notations
N := Zn ; M ' Hom(N, Z)
N 0 := Zm ; M 0 := Hom(N 0 , Z).
Let {e1 , . . . , em } be a basis of N 0∗ } be the dual basis of M 0 .
and {e1∗ , . . . , emα̂ := (α1 , . . . , αm ) ∈ MR0 := M 0 ⊗Z R.
Let v1 , . . . , vm be nonzero primitive vectors in N.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Notations
N := Zn ; M ' Hom(N, Z)
N 0 := Zm ; M 0 := Hom(N 0 , Z).
Let {e1 , . . . , em } be a basis of N 0∗ } be the dual basis of M 0 .
and {e1∗ , . . . , emα̂ := (α1 , . . . , αm ) ∈ MR0 := M 0 ⊗Z R.
Let v1 , . . . , vm be nonzero primitive vectors in N.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Notations
N := Zn ; M ' Hom(N, Z)
N 0 := Zm ; M 0 := Hom(N 0 , Z).
Let {e1 , . . . , em } be a basis of N 0∗ } be the dual basis of M 0 .
and {e1∗ , . . . , emα̂ := (α1 , . . . , αm ) ∈ MR0 := M 0 ⊗Z R.
Let v1 , . . . , vm be nonzero primitive vectors in N.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Notations
N := Zn ; M ' Hom(N, Z)
N 0 := Zm ; M 0 := Hom(N 0 , Z).
Let {e1 , . . . , em } be a basis of N 0∗ } be the dual basis of M 0 .
and {e1∗ , . . . , emα̂ := (α1 , . . . , αm ) ∈ MR0 := M 0 ⊗Z R.
Let v1 , . . . , vm be nonzero primitive vectors in N.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Notations
N := Zn ; M ' Hom(N, Z)
N 0 := Zm ; M 0 := Hom(N 0 , Z).
Let {e1 , . . . , em } be a basis of N 0∗ } be the dual basis of M 0 .
and {e1∗ , . . . , emα̂ := (α1 , . . . , αm ) ∈ MR0 := M 0 ⊗Z R.
Let v1 , . . . , vm be nonzero primitive vectors in N.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Smooth hyperplane arrangements
Hi := {x ∈ MR | hx, vi i + αi = 0} is a codimension 1 affine
subspace in MR with a normal oriented vector vi .
H := {H1 , . . . , Hm } is a hyperplane arrangement in MR .
H is simple if each nonempty intersection of k hyperplanes
has codimension k and if there are n hyperplanes with
nonempty intersection
H is smooth if H is simple and every n linearly
independent vectors from {v1 , . . . , vm } span N.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Smooth hyperplane arrangements
Hi := {x ∈ MR | hx, vi i + αi = 0} is a codimension 1 affine
subspace in MR with a normal oriented vector vi .
H := {H1 , . . . , Hm } is a hyperplane arrangement in MR .
H is simple if each nonempty intersection of k hyperplanes
has codimension k and if there are n hyperplanes with
nonempty intersection
H is smooth if H is simple and every n linearly
independent vectors from {v1 , . . . , vm } span N.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Smooth hyperplane arrangements
Hi := {x ∈ MR | hx, vi i + αi = 0} is a codimension 1 affine
subspace in MR with a normal oriented vector vi .
H := {H1 , . . . , Hm } is a hyperplane arrangement in MR .
H is simple if each nonempty intersection of k hyperplanes
has codimension k and if there are n hyperplanes with
nonempty intersection
H is smooth if H is simple and every n linearly
independent vectors from {v1 , . . . , vm } span N.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Smooth hyperplane arrangements
Hi := {x ∈ MR | hx, vi i + αi = 0} is a codimension 1 affine
subspace in MR with a normal oriented vector vi .
H := {H1 , . . . , Hm } is a hyperplane arrangement in MR .
H is simple if each nonempty intersection of k hyperplanes
has codimension k and if there are n hyperplanes with
nonempty intersection
H is smooth if H is simple and every n linearly
independent vectors from {v1 , . . . , vm } span N.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Smooth hyperplane arrangements
Hi := {x ∈ MR | hx, vi i + αi = 0} is a codimension 1 affine
subspace in MR with a normal oriented vector vi .
H := {H1 , . . . , Hm } is a hyperplane arrangement in MR .
H is simple if each nonempty intersection of k hyperplanes
has codimension k and if there are n hyperplanes with
nonempty intersection
H is smooth if H is simple and every n linearly
independent vectors from {v1 , . . . , vm } span N.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Smooth hyperplane arrangements
Hi := {x ∈ MR | hx, vi i + αi = 0} is a codimension 1 affine
subspace in MR with a normal oriented vector vi .
H := {H1 , . . . , Hm } is a hyperplane arrangement in MR .
H is simple if each nonempty intersection of k hyperplanes
has codimension k and if there are n hyperplanes with
nonempty intersection
H is smooth if H is simple and every n linearly
independent vectors from {v1 , . . . , vm } span N.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Smooth hyperplane arrangements
Hi := {x ∈ MR | hx, vi i + αi = 0} is a codimension 1 affine
subspace in MR with a normal oriented vector vi .
H := {H1 , . . . , Hm } is a hyperplane arrangement in MR .
H is simple if each nonempty intersection of k hyperplanes
has codimension k and if there are n hyperplanes with
nonempty intersection
H is smooth if H is simple and every n linearly
independent vectors from {v1 , . . . , vm } span N.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Exact sequences
Since H is smooth we have a surjective homomorphism
ρ : N 0 −→ N where ρ(ei ) := vi for 1 ≤ i ≤ m.
N 00 := ker(ρ) ' Zm−n and M 00 = Hom(N 00 , Z).
We get exact sequences of lattices:ιρ

0 −→ N 00 −→ N 0 −→ N −→ 0ρ∗ι∗

0 −→ M −→ M 0 −→ M 00 −→ 0
Since H is smooth (1) implies that α := ι∗ (α̂) 6= 0.

V. UmaK-theory of toric hyperKähler manifolds(1)Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Exact sequences
Since H is smooth we have a surjective homomorphism
ρ : N 0 −→ N where ρ(ei ) := vi for 1 ≤ i ≤ m.
N 00 := ker(ρ) ' Zm−n and M 00 = Hom(N 00 , Z).
We get exact sequences of lattices:ιρ

0 −→ N 00 −→ N 0 −→ N −→ 0ρ∗ι∗

0 −→ M −→ M 0 −→ M 00 −→ 0
Since H is smooth (1) implies that α := ι∗ (α̂) 6= 0.

V. UmaK-theory of toric hyperKähler manifolds(1)Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Exact sequences
Since H is smooth we have a surjective homomorphism
ρ : N 0 −→ N where ρ(ei ) := vi for 1 ≤ i ≤ m.
N 00 := ker(ρ) ' Zm−n and M 00 = Hom(N 00 , Z).
We get exact sequences of lattices:ιρ

0 −→ N 00 −→ N 0 −→ N −→ 0ρ∗ι∗

0 −→ M −→ M 0 −→ M 00 −→ 0
Since H is smooth (1) implies that α := ι∗ (α̂) 6= 0.

V. UmaK-theory of toric hyperKähler manifolds(1)Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Exact sequences
Since H is smooth we have a surjective homomorphism
ρ : N 0 −→ N where ρ(ei ) := vi for 1 ≤ i ≤ m.
N 00 := ker(ρ) ' Zm−n and M 00 = Hom(N 00 , Z).
We get exact sequences of lattices:ιρ

0 −→ N 00 −→ N 0 −→ N −→ 0ρ∗ι∗

0 −→ M −→ M 0 −→ M 00 −→ 0
Since H is smooth (1) implies that α := ι∗ (α̂) 6= 0.

V. UmaK-theory of toric hyperKähler manifolds(1)Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Exact sequences
Since H is smooth we have a surjective homomorphism
ρ : N 0 −→ N where ρ(ei ) := vi for 1 ≤ i ≤ m.
N 00 := ker(ρ) ' Zm−n and M 00 = Hom(N 00 , Z).
We get exact sequences of lattices:ιρ

0 −→ N 00 −→ N 0 −→ N −→ 0ρ∗ι∗

0 −→ M −→ M 0 −→ M 00 −→ 0
Since H is smooth (1) implies that α := ι∗ (α̂) 6= 0.

V. UmaK-theory of toric hyperKähler manifolds(1)Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Exact sequences
Since H is smooth we have a surjective homomorphism
ρ : N 0 −→ N where ρ(ei ) := vi for 1 ≤ i ≤ m.
N 00 := ker(ρ) ' Zm−n and M 00 = Hom(N 00 , Z).
We get exact sequences of lattices:ιρ

0 −→ N 00 −→ N 0 −→ N −→ 0ρ∗ι∗

0 −→ M −→ M 0 −→ M 00 −→ 0
Since H is smooth (1) implies that α := ι∗ (α̂) 6= 0.

V. UmaK-theory of toric hyperKähler manifolds(1)Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Exact sequences
We also get the corresponding exact sequences of vector
spaces:
ι

ρ

ρ∗

ι∗

R
R
NR −→ 00 −→ NR00 −→NR0 −→

RR
0 −→ MR −→MR0 −→
MR00 −→ 0

Induced exact sequence of tori:
1 −→ G := (S 1 )m−n ,→ T 0 := (S 1 )m −→ T := (S 1 )n −→ 1

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Exact sequences
We also get the corresponding exact sequences of vector
spaces:
ι

ρ

ρ∗

ι∗

R
R
NR −→ 00 −→ NR00 −→NR0 −→

RR
0 −→ MR −→MR0 −→
MR00 −→ 0

Induced exact sequence of tori:
1 −→ G := (S 1 )m−n ,→ T 0 := (S 1 )m −→ T := (S 1 )n −→ 1

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Exact sequences
We also get the corresponding exact sequences of vector
spaces:
ι

ρ

ρ∗

ι∗

R
R
NR −→ 00 −→ NR00 −→NR0 −→

RR
0 −→ MR −→MR0 −→
MR00 −→ 0

Induced exact sequence of tori:
1 −→ G := (S 1 )m−n ,→ T 0 := (S 1 )m −→ T := (S 1 )n −→ 1

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Exact sequences
We also get the corresponding exact sequences of vector
spaces:
ι

ρ

ρ∗

ι∗

R
R
NR −→ 00 −→ NR00 −→NR0 −→

RR
0 −→ MR −→MR0 −→
MR00 −→ 0

Induced exact sequence of tori:
1 −→ G := (S 1 )m−n ,→ T 0 := (S 1 )m −→ T := (S 1 )n −→ 1

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

HyperKalher structure on Hm
Consider Hm with 3 complex structures I, J, K induced bymultiplication by i, j and k respectively satisfying thequaternionic relations.
The diagonal torus T 0 = (S 1 )m ⊆ Sp(m) ⊆ SO(4m) actson Hm ' R4m preserving the Riemannian metric and theKahler forms ωI , ωJ , ωK corresponding to the complex
structures I, J and K respectively.
µ = (µI , µJ , µK ) : Hm −→ (MR0 )3denotes the hyperKähler moment map for the T 0 -action.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

HyperKalher structure on Hm
Consider Hm with 3 complex structures I, J, K induced bymultiplication by i, j and k respectively satisfying thequaternionic relations.
The diagonal torus T 0 = (S 1 )m ⊆ Sp(m) ⊆ SO(4m) actson Hm ' R4m preserving the Riemannian metric and theKahler forms ωI , ωJ , ωK corresponding to the complex
structures I, J and K respectively.
µ = (µI , µJ , µK ) : Hm −→ (MR0 )3denotes the hyperKähler moment map for the T 0 -action.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

HyperKalher structure on Hm
Consider Hm with 3 complex structures I, J, K induced bymultiplication by i, j and k respectively satisfying thequaternionic relations.
The diagonal torus T 0 = (S 1 )m ⊆ Sp(m) ⊆ SO(4m) actson Hm ' R4m preserving the Riemannian metric and theKahler forms ωI , ωJ , ωK corresponding to the complex
structures I, J and K respectively.
µ = (µI , µJ , µK ) : Hm −→ (MR0 )3denotes the hyperKähler moment map for the T 0 -action.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Definition of toric hyperKähler manifold
This further induces an action of G ,→ T 0 on Hm andµG := ι∗R ◦ µ : Hm −→ (MR00 )3 is the moment map for the
G-action on Hm .
Since α 6= 0, (α, 0, 0) is a regular value of µG .
Since H is smooth, G acts freely on µ−1
G (α, 0, 0) and−1µG(α, 0, 0)/G is a smooth manifold of dimension 4n.
X := µ−1
G (α, 0, 0)/G is called toric hyperKähler manifold
equipped with an action of the n-dimensional torus
T = T 0 /G which preserves the hyperKähler structure i.e
the induced Riemannian metric and complex structures
Iα , Jα , Kα .
V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Definition of toric hyperKähler manifold
This further induces an action of G ,→ T 0 on Hm andµG := ι∗R ◦ µ : Hm −→ (MR00 )3 is the moment map for the
G-action on Hm .
Since α 6= 0, (α, 0, 0) is a regular value of µG .
Since H is smooth, G acts freely on µ−1
G (α, 0, 0) and−1µG(α, 0, 0)/G is a smooth manifold of dimension 4n.
X := µ−1
G (α, 0, 0)/G is called toric hyperKähler manifold
equipped with an action of the n-dimensional torus
T = T 0 /G which preserves the hyperKähler structure i.e
the induced Riemannian metric and complex structures
Iα , Jα , Kα .
V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Definition of toric hyperKähler manifold
This further induces an action of G ,→ T 0 on Hm andµG := ι∗R ◦ µ : Hm −→ (MR00 )3 is the moment map for the
G-action on Hm .
Since α 6= 0, (α, 0, 0) is a regular value of µG .
Since H is smooth, G acts freely on µ−1
G (α, 0, 0) and−1µG(α, 0, 0)/G is a smooth manifold of dimension 4n.
X := µ−1
G (α, 0, 0)/G is called toric hyperKähler manifold
equipped with an action of the n-dimensional torus
T = T 0 /G which preserves the hyperKähler structure i.e
the induced Riemannian metric and complex structures
Iα , Jα , Kα .
V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Definition of toric hyperKähler manifold
This further induces an action of G ,→ T 0 on Hm andµG := ι∗R ◦ µ : Hm −→ (MR00 )3 is the moment map for the
G-action on Hm .
Since α 6= 0, (α, 0, 0) is a regular value of µG .
Since H is smooth, G acts freely on µ−1
G (α, 0, 0) and−1µG(α, 0, 0)/G is a smooth manifold of dimension 4n.
X := µ−1
G (α, 0, 0)/G is called toric hyperKähler manifold
equipped with an action of the n-dimensional torus
T = T 0 /G which preserves the hyperKähler structure i.e
the induced Riemannian metric and complex structures
Iα , Jα , Kα .
V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Complex line bundles on X

Let Cs be the 1-dimensional complex vector space withpsG-action induced by G ,→ T 0 −→ S 1 .
Ls := µ−1
G (α, 0, 0) ×G Cs is a complex line bundle on X
which is holomorphic with respect to the complex structure
Iα on X .

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Complex line bundles on X

Let Cs be the 1-dimensional complex vector space withpsG-action induced by G ,→ T 0 −→ S 1 .
Ls := µ−1
G (α, 0, 0) ×G Cs is a complex line bundle on X
which is holomorphic with respect to the complex structure
Iα on X .

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Cohomology ring presentation
Let X be a toric hyperKähler manifold
H = {H1 , . . . , Hm } be the associated smooth hyperplanearrangement
Ideal J in Z[x1 , . . . , xm ] generated byY

xs whenever

s∈I
m
X

\

Hs = ∅, I ⊆ [1, m]s∈Ihu, vs ixs , u ∈ M.

s=1

Theorem(Konno) There is an isomorphism of Z-algebras
φ : Z[x1 , . . . , xm ]/J −→ H ∗ (X ; Z) that sends xs to c1 (Ls ) for1 ≤ s ≤ m.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Cohomology ring presentation
Let X be a toric hyperKähler manifold
H = {H1 , . . . , Hm } be the associated smooth hyperplanearrangement
Ideal J in Z[x1 , . . . , xm ] generated byY

xs whenever

s∈I
m
X

\

Hs = ∅, I ⊆ [1, m]s∈Ihu, vs ixs , u ∈ M.

s=1

Theorem(Konno) There is an isomorphism of Z-algebras
φ : Z[x1 , . . . , xm ]/J −→ H ∗ (X ; Z) that sends xs to c1 (Ls ) for1 ≤ s ≤ m.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Cohomology ring presentation
Let X be a toric hyperKähler manifold
H = {H1 , . . . , Hm } be the associated smooth hyperplanearrangement
Ideal J in Z[x1 , . . . , xm ] generated byY

xs whenever

s∈I
m
X

\

Hs = ∅, I ⊆ [1, m]s∈Ihu, vs ixs , u ∈ M.

s=1

Theorem(Konno) There is an isomorphism of Z-algebras
φ : Z[x1 , . . . , xm ]/J −→ H ∗ (X ; Z) that sends xs to c1 (Ls ) for1 ≤ s ≤ m.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Cohomology ring presentation
Let X be a toric hyperKähler manifold
H = {H1 , . . . , Hm } be the associated smooth hyperplanearrangement
Ideal J in Z[x1 , . . . , xm ] generated byY

xs whenever

s∈I
m
X

\

Hs = ∅, I ⊆ [1, m]s∈Ihu, vs ixs , u ∈ M.

s=1

Theorem(Konno) There is an isomorphism of Z-algebras
φ : Z[x1 , . . . , xm ]/J −→ H ∗ (X ; Z) that sends xs to c1 (Ls ) for1 ≤ s ≤ m.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

Construction of toric hyperKähler manifolds
Cohomology ring of toric hyperKähler manifolds

Cohomology ring presentation
Let X be a toric hyperKähler manifold
H = {H1 , . . . , Hm } be the associated smooth hyperplanearrangement
Ideal J in Z[x1 , . . . , xm ] generated byY

xs whenever

s∈I
m
X

\

Hs = ∅, I ⊆ [1, m]s∈Ihu, vs ixs , u ∈ M.

s=1

Theorem(Konno) There is an isomorphism of Z-algebras
φ : Z[x1 , . . . , xm ]/J −→ H ∗ (X ; Z) that sends xs to c1 (Ls ) for1 ≤ s ≤ m.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

Topological K-ring presentation
Let X be a toric hyperKähler manifold
H = {H1 , . . . , Hm }— associated smooth hyperplanearrangement
Ideal J 0 in Z[x1 , . . . , xm ] generated byY\

xs whenevers∈IY

Hs = ∅, I ⊆ [1, m]s∈Ihu,vs i(1 − xs )−s|hu,vs i>0Y(1 − xs )−hu,vs i , u ∈ M.

s|hu,vs i<0

Theorem([U]) There is an isomorphism of Z-algebras
φ : Z[x1 , . . . , xm ]/J 0 −→ K ∗ (X ) that sends xs to 1 − [Ls ] for1 ≤ s ≤ m.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

Topological K-ring presentation
Let X be a toric hyperKähler manifold
H = {H1 , . . . , Hm }— associated smooth hyperplanearrangement
Ideal J 0 in Z[x1 , . . . , xm ] generated byY\

xs whenevers∈IY

Hs = ∅, I ⊆ [1, m]s∈Ihu,vs i(1 − xs )−s|hu,vs i>0Y(1 − xs )−hu,vs i , u ∈ M.

s|hu,vs i<0

Theorem([U]) There is an isomorphism of Z-algebras
φ : Z[x1 , . . . , xm ]/J 0 −→ K ∗ (X ) that sends xs to 1 − [Ls ] for1 ≤ s ≤ m.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

Topological K-ring presentation
Let X be a toric hyperKähler manifold
H = {H1 , . . . , Hm }— associated smooth hyperplanearrangement
Ideal J 0 in Z[x1 , . . . , xm ] generated byY\

xs whenevers∈IY

Hs = ∅, I ⊆ [1, m]s∈Ihu,vs i(1 − xs )−s|hu,vs i>0Y(1 − xs )−hu,vs i , u ∈ M.

s|hu,vs i<0

Theorem([U]) There is an isomorphism of Z-algebras
φ : Z[x1 , . . . , xm ]/J 0 −→ K ∗ (X ) that sends xs to 1 − [Ls ] for1 ≤ s ≤ m.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

Topological K-ring presentation
Let X be a toric hyperKähler manifold
H = {H1 , . . . , Hm }— associated smooth hyperplanearrangement
Ideal J 0 in Z[x1 , . . . , xm ] generated byY\

xs whenevers∈IY

Hs = ∅, I ⊆ [1, m]s∈Ihu,vs i(1 − xs )−s|hu,vs i>0Y(1 − xs )−hu,vs i , u ∈ M.

s|hu,vs i<0

Theorem([U]) There is an isomorphism of Z-algebras
φ : Z[x1 , . . . , xm ]/J 0 −→ K ∗ (X ) that sends xs to 1 − [Ls ] for1 ≤ s ≤ m.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

Topological K-ring presentation
Let X be a toric hyperKähler manifold
H = {H1 , . . . , Hm }— associated smooth hyperplanearrangement
Ideal J 0 in Z[x1 , . . . , xm ] generated byY\

xs whenevers∈IY

Hs = ∅, I ⊆ [1, m]s∈Ihu,vs i(1 − xs )−s|hu,vs i>0Y(1 − xs )−hu,vs i , u ∈ M.

s|hu,vs i<0

Theorem([U]) There is an isomorphism of Z-algebras
φ : Z[x1 , . . . , xm ]/J 0 −→ K ∗ (X ) that sends xs to 1 − [Ls ] for1 ≤ s ≤ m.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

Idea of proofs

Although X is non-compact in general it is homotopy
equivalent to its “core” Core(X ) which is a finite union ofcompact toric submanifolds. (Core(X ) is a strong
deformation retract of X )We can apply the Atiyah Hirzebruch spectral sequencewhich degenerates in this setting since the integral odd
cohomology vanishes.
E2p,q = H p (X , K q (pt)) ⇒ K p+q (X ).

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

Idea of proofs

Although X is non-compact in general it is homotopy
equivalent to its “core” Core(X ) which is a finite union ofcompact toric submanifolds. (Core(X ) is a strong
deformation retract of X )We can apply the Atiyah Hirzebruch spectral sequencewhich degenerates in this setting since the integral odd
cohomology vanishes.
E2p,q = H p (X , K q (pt)) ⇒ K p+q (X ).

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

Idea of proofs

We show (by methods similar to that used for toric andtorus manifolds) that K ∗ (X ) is generated by the theisomorphism classes of the complex line bundles whose
first Chern classes generate the cohomology ring. (Since
H 2 (X ; Z) generates H ∗ (X ; Z).)
The presentation for K ∗ (X ) again follows from that of thecohomology ring similar to the case of toric manifolds.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

Idea of proofs

We show (by methods similar to that used for toric andtorus manifolds) that K ∗ (X ) is generated by the theisomorphism classes of the complex line bundles whose
first Chern classes generate the cohomology ring. (Since
H 2 (X ; Z) generates H ∗ (X ; Z).)
The presentation for K ∗ (X ) again follows from that of thecohomology ring similar to the case of toric manifolds.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

Cotangent bundle of complex projective space

ExampleThe cotangent bundle of the complex projective space T ∗ (CPn )is a toric hyperKähler manifold associated to the hyperplanearrangement H = {H1 , . . . , Hn , Hn+1 } in Rn consisting of
Hj = {(a1 , . . . , an ) | aj = −1}
for 1 ≤ j ≤ n and Hn+1 = {(a1 , . . . , an ) | a1 + · · · + an = 1}.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

K-ring of T ∗ (CPn )ExampleJ 0 is the ideal in Z[x1 , . . . , xn+1 ] generated by
the monomial x1 · x2 · · · xn+1 since I = [1, n + 1] is the only
subset such that H1 ∩ · · · ∩ Hn+1 = ∅and the n relations (1 − xj ) − (1 − xn+1 ) for 1 ≤ j ≤ ncorresponding to the basis e1∗ , . . . , en∗ .

Z[x]/(1 − x)n+1 −→ K ∗ (X ) where x 7→ 1 − [Ln+1 ] defines
an isomorphism of Z-algebras.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

K-ring of T ∗ (CPn )ExampleJ 0 is the ideal in Z[x1 , . . . , xn+1 ] generated by
the monomial x1 · x2 · · · xn+1 since I = [1, n + 1] is the only
subset such that H1 ∩ · · · ∩ Hn+1 = ∅and the n relations (1 − xj ) − (1 − xn+1 ) for 1 ≤ j ≤ ncorresponding to the basis e1∗ , . . . , en∗ .

Z[x]/(1 − x)n+1 −→ K ∗ (X ) where x 7→ 1 − [Ln+1 ] defines
an isomorphism of Z-algebras.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

K-ring of T ∗ (CPn )ExampleJ 0 is the ideal in Z[x1 , . . . , xn+1 ] generated by
the monomial x1 · x2 · · · xn+1 since I = [1, n + 1] is the only
subset such that H1 ∩ · · · ∩ Hn+1 = ∅and the n relations (1 − xj ) − (1 − xn+1 ) for 1 ≤ j ≤ ncorresponding to the basis e1∗ , . . . , en∗ .

Z[x]/(1 − x)n+1 −→ K ∗ (X ) where x 7→ 1 − [Ln+1 ] defines
an isomorphism of Z-algebras.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

K-ring of T ∗ (CPn )ExampleJ 0 is the ideal in Z[x1 , . . . , xn+1 ] generated by
the monomial x1 · x2 · · · xn+1 since I = [1, n + 1] is the only
subset such that H1 ∩ · · · ∩ Hn+1 = ∅and the n relations (1 − xj ) − (1 − xn+1 ) for 1 ≤ j ≤ ncorresponding to the basis e1∗ , . . . , en∗ .

Z[x]/(1 − x)n+1 −→ K ∗ (X ) where x 7→ 1 − [Ln+1 ] defines
an isomorphism of Z-algebras.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

K-ring of T ∗ (CPn )ExampleJ 0 is the ideal in Z[x1 , . . . , xn+1 ] generated by
the monomial x1 · x2 · · · xn+1 since I = [1, n + 1] is the only
subset such that H1 ∩ · · · ∩ Hn+1 = ∅and the n relations (1 − xj ) − (1 − xn+1 ) for 1 ≤ j ≤ ncorresponding to the basis e1∗ , . . . , en∗ .

Z[x]/(1 − x)n+1 −→ K ∗ (X ) where x 7→ 1 − [Ln+1 ] defines
an isomorphism of Z-algebras.

V. UmaK-theory of toric hyperKähler manifolds

Motivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

References
R. Bielawski and A. Dancer, The geometry and topology of
toric hyperKähler manifolds, Comm. Anal. Geom. 8 (2000),
no. 4, 727–760.
T. Hausel and B. Sturmfels, Toric HyperKähler Varieties,
Documenta Math. 7, (2002), 495–534.
H. Konno: Cohomology rings of toric hyperKähler
manifolds, Internat. J. Math.11 (2000), no. 8, 1001–1026.
S. Kuroki, Equivariant cohomology distinguishes thegeometric structures of toric hyperKähler manifolds,
Proceedings of the Steklov Institute of Mathematics, 275(1) (2011), 251-283.
V. Uma, K-theory of hyperKähler toric manifolds
arXiv.1808.03008
V. UmaK-theory of toric hyperKähler manifolds
Help | Advanced SearchMotivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

References
R. Bielawski and A. Dancer, The geometry and topology of
toric hyperKähler manifolds, Comm. Anal. Geom. 8 (2000),
no. 4, 727–760.
T. Hausel and B. Sturmfels, Toric HyperKähler Varieties,
Documenta Math. 7, (2002), 495–534.
H. Konno: Cohomology rings of toric hyperKähler
manifolds, Internat. J. Math.11 (2000), no. 8, 1001–1026.
S. Kuroki, Equivariant cohomology distinguishes thegeometric structures of toric hyperKähler manifolds,
Proceedings of the Steklov Institute of Mathematics, 275(1) (2011), 251-283.
V. Uma, K-theory of hyperKähler toric manifolds
arXiv.1808.03008
V. UmaK-theory of toric hyperKähler manifolds
Help | Advanced SearchMotivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

References
R. Bielawski and A. Dancer, The geometry and topology of
toric hyperKähler manifolds, Comm. Anal. Geom. 8 (2000),
no. 4, 727–760.
T. Hausel and B. Sturmfels, Toric HyperKähler Varieties,
Documenta Math. 7, (2002), 495–534.
H. Konno: Cohomology rings of toric hyperKähler
manifolds, Internat. J. Math.11 (2000), no. 8, 1001–1026.
S. Kuroki, Equivariant cohomology distinguishes thegeometric structures of toric hyperKähler manifolds,
Proceedings of the Steklov Institute of Mathematics, 275(1) (2011), 251-283.
V. Uma, K-theory of hyperKähler toric manifolds
arXiv.1808.03008
V. UmaK-theory of toric hyperKähler manifolds
Help | Advanced SearchMotivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

References
R. Bielawski and A. Dancer, The geometry and topology of
toric hyperKähler manifolds, Comm. Anal. Geom. 8 (2000),
no. 4, 727–760.
T. Hausel and B. Sturmfels, Toric HyperKähler Varieties,
Documenta Math. 7, (2002), 495–534.
H. Konno: Cohomology rings of toric hyperKähler
manifolds, Internat. J. Math.11 (2000), no. 8, 1001–1026.
S. Kuroki, Equivariant cohomology distinguishes thegeometric structures of toric hyperKähler manifolds,
Proceedings of the Steklov Institute of Mathematics, 275(1) (2011), 251-283.
V. Uma, K-theory of hyperKähler toric manifolds
arXiv.1808.03008
V. UmaK-theory of toric hyperKähler manifolds
Help | Advanced SearchMotivation
Basic Construction
Our Results

K-ring of toric hyperKähler manifolds

References
R. Bielawski and A. Dancer, The geometry and topology of
toric hyperKähler manifolds, Comm. Anal. Geom. 8 (2000),
no. 4, 727–760.
T. Hausel and B. Sturmfels, Toric HyperKähler Varieties,
Documenta Math. 7, (2002), 495–534.
H. Konno: Cohomology rings of toric hyperKähler
manifolds, Internat. J. Math.11 (2000), no. 8, 1001–1026.
S. Kuroki, Equivariant cohomology distinguishes thegeometric structures of toric hyperKähler manifolds,
Proceedings of the Steklov Institute of Mathematics, 275(1) (2011), 251-283.
V. Uma, K-theory of hyperKähler toric manifolds
arXiv.1808.03008
V. UmaK-theory of toric hyperKähler manifolds
Help | Advanced Search