Questions tagged [hyperkaehler]
The hyperkaehler tag has no usage guidance.
42
questions
3
votes
0
answers
74
views
Cup-product ($\cup$) vs. cross-product ($\times$) on the space of graph homology
I'm currently reading Nieper-Wißkirchen's Chern numbers and Rozansky-Witten invariants of compact Hyper-Kähler manifolds. He introduces the space of graph homology $\mathcal B$ as the free $K[\circ]$-...
- 1,216
1
vote
0
answers
88
views
Kähler potential for ALEs from resolving $\mathbb{C}^2/\mathbb{Z}_2$:
I am reading the famous paper of Kronheimer “The construction of ALE spaces as hyperkähler quotients”
I want to calculate explicitly the metric on the ALE spaces, obtained by a family of resolution of ...
- 43
2
votes
0
answers
73
views
Hyperkähler quotient of left $\operatorname{SU}(2)$-action on $(\mathbb{C}^2)^m \cong \mathbb{H}^m$
The natural $\operatorname{SU}(2)$-action on quaternions $\mathbb{H}\cong\mathbb{C}^2$ is hyperkähler. Extending it naturally to $\mathbb{H}^m$, one can make a hyperkähler quotient (where $\mu=(\mu_I,\...
- 1,647
1
vote
0
answers
139
views
Chainsaw quiver variety and parabolic bundle
How can we relate chainsaw quiver varieties with ADE type Nakajima quiver varieties?
We know that we can obtain ADE type quiver varieties (instantons over ALE spaces) by taking $\Gamma$ equivariant ...
- 345
5
votes
1
answer
243
views
Rozansky-Witten invariants of hyperkahler manifolds and independence of complex structure
Recently I have been learning about Rozansky-Witten invariants, mainly through Hitchin-Sawon's paper "curvature and characteristic numbers of hyperkahler manifolds" and through Justin Sawon'...
- 91
4
votes
0
answers
97
views
Existence of a rational curve in the center of a birational contraction for symplectic singularities
Let $M$ be a holomorphically symplectic
complex manifold, and $f: M \to X$
a holomorphic, birational contraction to a Stein
variety $X$, contracting a subvariety $E$
to a point, and bijective outside ...
- 9,227
1
vote
1
answer
131
views
Are horizontal divisors on abelian fibered hyperkähler manifolds proportional in $NS(X)$ up to vertical divisors?
Oguiso writes[1]
Theorem 1.1 Let $f: X \to \mathbf P^n$ be an abelian fibered HK [hyperkähler] manifold. Let $K = \mathbf C(\mathbf P^n)$ and let $A_k$ be the generic fiber of $f$. Then, $\rho(A_K)= ...
- 1,216
1
vote
1
answer
142
views
4-manifold with two compatible Kähler structures needs to be hyperkähler
In the proof of Theorem 2 of the article Four-manifolds without Einstein metrics, the author seems to be exploiting this fact:
Let $(M,g)$ be a 4-dimensional Riemannian manifold which is Kähler with ...
- 111
2
votes
0
answers
143
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Describing singular fibers of the lagrangian fibration $\mathcal M^s(0, [C], 1) \to |C|$
Let $S \to \mathbb P^2$ be a two-to-one cover branched over a sextic, i.e. $S$ is a K3-surface. Let $C \subset S$ be the preimage of a (smooth) quadric, so that by Hurwitz' formula, $g(C) = 5$. ...
- 1,216
1
vote
0
answers
114
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Is Stenzel's Ricci-flat metric on $T^*\mathbb{CP}^n$ hyperkahler?
In a well-known paper, Stenzel constructed complete Ricci-flat Kahler metrics on the total spaces of cotangent bundles of $S^n$, $\mathbb{RP}^n,$ $\mathbb{CP}^n$, $\mathbb{HP}^n$, and $\mathbb{OP}^2$.
...
- 871
7
votes
1
answer
588
views
Kähler metric with two compatible complex structures
Let $(M^4,g)$ be a complete $4$-dimensional Riemannian manifold such that two almost complex structures $I$ and $J$ are compatible with $g$ and $\nabla_g I=\nabla_g J=0$.
Can we prove that $(M,g)$ is ...
- 881
5
votes
1
answer
277
views
Can deformation equivalent Kähler manifolds always be obtained by a deformation where all the fibers are Kähler?
Given compact Kähler manifolds $X$ and $X'$ deformation equivalent over the unit disk $\Delta \subset \mathbb{C}$. More precisely, there is a proper holomorphic surjective map
\begin{align*}
\pi\...
- 423
2
votes
0
answers
81
views
3-Sasakian manifolds and contact Fano Kähler-Einstein manifolds
Let $(M,g)$ be a Riemannian manifold. The Riemannian cone
of $M$ is $C(M) = M \times {\Bbb R}^{>0}$ with the metric $t^2 g + dt\otimes dt$.
A manifold is called Sasakian if its cone is Kähler, ...
- 9,227
4
votes
0
answers
208
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Can Lagrangian fibrations have multiple fibres in codimension $1$?
I know that if $\pi: S \to \mathbb P^1$ is an elliptic fibration of a K3-surface $S$, then $\pi$ does not have multiple fibers. A proof of this can be found in Huybrechts' Lectures on K3 surfaces, ...
- 1,216
3
votes
1
answer
295
views
Does miracle flatness always fail for a non-regular base?
In [1], Huybrechts and Mauri argue that a holomorphic Lagrangian fibration $f: X \to B$ with smooth base $B$ is flat. This is an application of so called miracle flatness [2, Thm 23.1], because ...
- 1,216
1
vote
0
answers
71
views
Can this embedding to double dual EPW sextic happen?
Let $\widetilde{Y}_{A^{\perp}}$ denote the double dual EPW sextic defined by the Lagrange subspace $A\subset \bigwedge^3V_6$. If $A$ is very general, then $\widetilde{Y}_{A^{\perp}}$ is a smooth ...
- 1,982
3
votes
0
answers
65
views
Ricci deformation of hyperkahler ALE orbifold
Let $(X^2,g)$ be a hyperkahler ALE orbifold surface. Consider its Ricci deformation equation:
$$
\Delta h+2Rm(h)=0
$$
for $\text{div}_g h=\text{Tr}_gh=0$ and $h=O(r^{-\epsilon})$ as $r \to +\infty$. ...
- 881
2
votes
0
answers
163
views
Conics on Gushel-Mukai fourfold
Let $X$ be a very general Gushel-Mukai fourfold, let $\mathcal{U}$ be the tautological sub-bundle and $\mathcal{Q}$ be the tautological quotient bundle. Let $C\subset X$ be a $\rho$-conic, then $\...
- 1,982
4
votes
0
answers
194
views
Deformation to a normal cone for a holomorphically symplectic manifold
Let $X$ be a subvariety in $M$.
"Deformation to the normal cone"
is a holomorphic deformation of a neighbourhood
of $X$ in $M$ over the disk such that its central fiber
is the total space ...
- 9,227
9
votes
2
answers
724
views
Bialynicki-Birula decomposition for real analytic varieties
Let $X$ be a smooth complex algebraic variety endowed with a $\mathbb{C}^*$ action. We assume also to have an antiholomorphic involution $\sigma$ over $X$ such that it anticommutes with the action ...
3
votes
1
answer
163
views
Stuck on a computation with quaternions and moment maps
I am trying to understand an article by Gibbons, Rychenkova and Goto, called "Hyperkähler quotient construction of BPS Monopole Moduli Spaces". I will paraphrase the relevant notions and ...
- 5,118
6
votes
1
answer
257
views
The state of art of the singular Levi problem -- and hyperkähler quotients
One of the versions of the classical Levi problem asks the following:
Let $X$ be a complex manifold. Is it true that $X$ is Stein iff
$X$ admits a smooth exhaustion strictly plurisubharmonic ...
- 2,305
7
votes
0
answers
286
views
Examples of non-compact, holomorphically symplectic Kähler manifolds which are not hyperkähler
Let $(M,\omega_{1},I_{1})$ be a non-compact Kähler manifold. If $M$ admits a holomorphic symplectic form $\Omega$, is it possible M not be hyperkähler? Is there any example?
(*)Under the assumption ...
- 81
2
votes
0
answers
96
views
Monodromy operators on hyperkähler varieties
Let $X$ be a hyperkähler variety.
In an article (Conjecture 2.1) from some years ago, Markman conjectured that any monodromy operator acting trivially on $H^2(X,\mathbb Z)$ is the identity operator, ...
3
votes
0
answers
103
views
Contact 3-manifolds with hyperkahler Stein fillings?
Is there any classification result on (homeomorphism type) of contact 3-manifolds $\Sigma$ that have Stein filling $W$ that is
1. Hyperkahler (s.t. Stein structure is the Kahler part of it)
2. not ...
- 1,647
3
votes
0
answers
82
views
Hypertoric varieties in dimension 4?
Are the only smooth hypertoric varieties in real dimension 4 obtained
as minimal resolutions of type A simple singularities $\mathbb{C}^2/\mathbb{Z}_{/n}$?
- 1,647
5
votes
0
answers
77
views
Hypothetical uniqueness of an embedding of a Riemannian manifold to a compact Kähler one
Inspired by this question (Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold) I ask the following:
Suppose $X$ is a real analytic Riemannian manifold with a ...
- 2,305
4
votes
0
answers
124
views
Coordinate-free B.Feix's construction of a hyperkähler metric
In the 2001's paper 'Hyperkähler metrics on cotangent bundles' B.Feix gives a construction of a hyperkähler metric on a neighbourhood of zero section in $T^*X$ where $X$ is a real analytic Kähler ...
- 2,305
10
votes
0
answers
376
views
Hyperkähler structure on the moduli space of tetrahedra?
Consider a moduli space of geodesic tetrahedra in the hyperbolic space $\mathbb{H}^3.$ In the Klein's model the hyperbolic space can be presented as the interior of a unit ball
$$
\mathbb{H}^3=\{(x_1,...
- 4,316
4
votes
1
answer
206
views
Lagrangian cores of quiver variety in different GIT chambers
Let $\pi: \mathfrak{M}_{(\theta,0)}(Q,\text{v},\text{w}) \rightarrow \mathfrak{M}_{(0,0)}(Q,\text{v},\text{w})$ be the projective morphism between Nakajima quiver varieties when complex moment ...
- 1,647
1
vote
1
answer
238
views
Small contraction for Hyperkähler Varieties
I have the following basic question. Everything is over $\mathbb{C}$.
Let $X$ be a hyperkähler (irreducible holomorphic symplectic) variety and we consider a small contraction $f\colon X \rightarrow ...
- 175
1
vote
1
answer
284
views
compact manifold as a hyperkahler quotient of an infinite-dimensional affine space
Is it possible to obtain K3 (or any other compact
hyperkahler manifold) with its hyperkahler structure
as a hyperkahler quotient of an infinite-dimensional affine
quaternionic vector space with an ...
- 9,227
6
votes
1
answer
790
views
Are there non-projective, but algebraic, hyperkahler varieties?
Let $k$ be an algebraically closed field of characteristic zero. I am not sure what the right definition of a hyperkahler variety over $k$ is, but I think the following might be close enough.
...
- 61
0
votes
0
answers
113
views
Extending Beauville-Bogomolov orthogonal decomposition from variety to scheme
I'm seeking to understand the de Rham cohomology of a Hilbert scheme $K3^{[4]}$ of the K3 surface. By Beauville, this 8-dimensional compact manifold is Kaehler, irreducible, holomorphically symplectic ...
- 35
1
vote
0
answers
103
views
Integral cohomology on compact irreducible holomorphic symplectic manifolds
In particular, I'm attempting to understand the integral cohomology on K3^[4], a Hilbert scheme on the projective K3 surface, and whether or not this compact Kaehler manifold admits a Kaehler form ...
- 35
4
votes
0
answers
132
views
Different components of real sections for twistor spaces of hyper-Kähler manifolds
A hyper-Kähler manifold is a Riemannian manifold $(M,g)$ equipped with 3 complex structures $I,J,K$ obeying the quaternionic relations and such that $g$ is a Kähler metric for each complex structure. ...
- 6,785
2
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0
answers
109
views
How exactly are holomorphic maps with hyperkahler targets identified as triholomorphic maps?
In Appendix A of Hyperinstantons, the Beltrami Equation, and Triholomorphic Maps by Fre et al., it is described how holomorphic maps from a Riemann surface to a hyperkahler manifold $\mathcal{N}$, ...
- 1,155
3
votes
1
answer
178
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Are there hyperkahler manifolds with multi-isotropic hyperkahler submanifolds?
This is a continuation of this question. As explained in the comments, for a hyperkaehler manifold $X$, a multi-isotropic submanifold $S$ has the property that the three distinguished Kahler forms $\...
- 1,155
3
votes
1
answer
277
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Submanifold of a Hyperkahler manifold which is 'Lagrangian' w.r.t. all three symplectic structures
Let $X$ be a hyperkaehler manifold. Being hyperkaehler, it has three distinguished Kaehler forms, $\omega^u$ ($u$=1,2,3), corresponding to its three distinguished almost complex structures.
How does ...
- 1,155
1
vote
1
answer
211
views
Do these definitions of integrable quaternionic structure agree?
I have found two different definitions for integrable quaternionic structure in the literature, and I need to know if they agree with one another.
One definition that I have found (from Differential ...
- 1,155
6
votes
0
answers
818
views
Isometries of hyper-Kähler manifolds
For the purposes of this question, a hyper-Kähler manifold will be a complete connected Riemannian manifold $(\mathcal{M},g)$ whose holonomy representation is isomorphic to the natural representation ...
- 421
8
votes
2
answers
1k
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Hyper-complex and quaternionic Kähler Geometry
What is the exact relationship hyper-complex and quaternionic Kahler manifolds? From Wikipedia we get that hyper-Kahler manifolds are both hyper-complex and quaternionic Kahler. Thus, the two families ...
- 3,389