Questions tagged [hyperkaehler]
The hyperkaehler tag has no usage guidance.
28
questions
3
votes
1
answer
165
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 ...
- 585
1
vote
0
answers
62
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 ...
- 1,516
3
votes
0
answers
47
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 $r \to +\infty$. ...
- 407
2
votes
0
answers
143
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 $\...
- 1,516
3
votes
0
answers
122
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 ...
- 7,528
8
votes
2
answers
620
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
140
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 ...
- 3,594
6
votes
1
answer
199
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 ...
- 2,011
7
votes
0
answers
212
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
84
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
92
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 ...
- 1,429
3
votes
0
answers
66
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 $\mathbb{C}^2/\mathbb{Z}_{/n}$?
- 1,429
5
votes
0
answers
70
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 ...
- 2,011
4
votes
0
answers
109
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 ...
- 2,011
10
votes
0
answers
330
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,...
- 4,044
4
votes
1
answer
174
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 ...
- 1,429
1
vote
1
answer
161
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 ...
- 165
1
vote
1
answer
242
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 ...
- 7,528
5
votes
1
answer
584
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.
...
- 51
0
votes
0
answers
90
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
99
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
113
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. ...
- 6,290
2
votes
0
answers
96
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 $\mathcal{N}$, ...
- 1,033
2
votes
1
answer
155
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 $\...
- 1,033
2
votes
1
answer
194
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 ...
- 1,033
1
vote
1
answer
190
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 ...
- 1,033
6
votes
0
answers
723
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 ...
- 381
8
votes
2
answers
1k
views
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 ...
- 3,199