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Questions tagged [hyperkaehler]

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8
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0answers
158 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
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1answer
68 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
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1answer
99 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 ...
1
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1answer
125 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 ...
6
votes
1answer
315 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. ...
0
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0answers
74 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 ...
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0answers
86 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 ...
4
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0answers
73 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. ...
2
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0answers
81 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 $\mathcal{N}$, ...
2
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1answer
138 views

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 $\...
2
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1answer
133 views

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 ...
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1answer
156 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 ...