Questions tagged [hyperkaehler]

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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,...
Daniil Rudenko's user avatar
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 ...
Eder Moraes's user avatar
6 votes
0 answers
805 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 ...
Paul Reynolds's user avatar
5 votes
0 answers
75 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 ...
anna abasheva's user avatar
4 votes
0 answers
85 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 ...
Misha Verbitsky's user avatar
4 votes
0 answers
179 views

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, ...
red_trumpet's user avatar
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4 votes
0 answers
121 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 ...
anna abasheva's user avatar
4 votes
0 answers
130 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. ...
Sebastian's user avatar
  • 6,715
3 votes
0 answers
59 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 $r \to +\infty$. ...
Zhiqiang's user avatar
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3 votes
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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 ...
Misha Verbitsky's user avatar
3 votes
0 answers
102 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 ...
Filip's user avatar
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3 votes
0 answers
80 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 $\mathbb{C}^2/\mathbb{Z}_{/n}$?
Filip's user avatar
  • 1,617
2 votes
0 answers
126 views

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, $g(C) = 5$. ...
red_trumpet's user avatar
  • 1,071
2 votes
0 answers
72 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, ...
Misha Verbitsky's user avatar
2 votes
0 answers
159 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 $\...
user41650's user avatar
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2 votes
0 answers
93 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, ...
Davide Cesare Veniani's user avatar
2 votes
0 answers
107 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 $\mathcal{N}$, ...
Mtheorist's user avatar
  • 1,135
1 vote
0 answers
78 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 ...
LYJ's user avatar
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1 vote
0 answers
65 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,\...
Filip's user avatar
  • 1,617
1 vote
0 answers
130 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 ...
TaiatLyu's user avatar
  • 111
1 vote
0 answers
102 views

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 $\mathbb{OP}^2$. ...
Jesse Madnick's user avatar
1 vote
0 answers
61 views

Some question on defining the space of graph homology for Jacobi diagrams

I read Chapter 2.1 "The space of graph homology" from Nieper-Wißkirchen's Chern Numbers and Rozansky-Witten Invariants of Compact Hyper-Kähler Manifolds today, and have some question. I'm ...
red_trumpet's user avatar
  • 1,071
1 vote
0 answers
70 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 ...
user41650's user avatar
  • 1,922
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 ...
zenith90's user avatar
0 votes
0 answers
106 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 ...
zenith90's user avatar