Questions tagged [holomorphic-symplectic]
hyperkahler manifolds, complex Lagrangian submanifolds, Mukai flop, integrable systems
73
questions
3
votes
0
answers
65
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 $K[\circ]$-...
- 1,141
2
votes
0
answers
88
views
Embeddings of symplectic singularities into smooth manifolds
Let $X$ be a symplectic variety with terminal singularities of dimension $2n$, $\sigma\in H^0(X^{reg},\Omega^2_{X^{reg}})$ a holomorphic symplectic form. Pick a neighborhood $U$ of a point $x\in X$.
...
- 153
1
vote
0
answers
79
views
Birational deformations of holomorphic symplectic manifolds
Let $X$ and $X'$ be birational holomorphic symplectic manifolds. Then the birational morphism between them identifies $H^2(X)$ with $H^2(Y)$. The period space of $X$ is defined to be a subset of $\...
- 153
2
votes
0
answers
72
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
3
votes
0
answers
92
views
additive vs multiplicative quiver/hypertoric varieties - properties
It is a standard fact that a smooth Nakajima quiver variety / hypertoric variety $X$ has the following properties:
It is holomorphic symplectic $(X,\omega)$, in fact 1'. hyperkahler
It has a ...
- 1,647
4
votes
0
answers
87
views
A couple of questions about the moduli space of annuli with some marked points on the boundary components
I'm trying to work out an answer for my previous question and I'm stuck with the following issue:
In the paper Deformations of Bordered Riemann surfaces and associahedral polytopes by Devadoss, Heath ...
- 2,018
18
votes
2
answers
2k
views
Is this entire function a square?
Let $f$ be the entire function on $\mathbb C$ defined by
$$
f(z)=\frac{z-\sin z}{z}.
\tag{1}\label{1}$$
It is easy to see that $f$ is positive on $\mathbb R^*$ and has a zero of order 2 at 0.
Does ...
- 15.7k
3
votes
0
answers
180
views
Relative automorphism groups of holomorphic Lagrangian fibrations
Let $p\colon X\to B$ be a Lagrangian fibration on an irreducible holomorphic symplectic manifold over $\mathbb C$. Assume that the base $B$ is smooth, hence $\mathbb P^n$. Define the relative tangent ...
- 2,305
4
votes
1
answer
245
views
When does a holomorphic symplectic manifold compactify to a Poisson manifold?
Let $X$ be a complex manifold endowed with a holomorphic closed 2-form $\omega$ whose associated map $\omega : TX \to T^*X$ is invertible. Can we always embed $X$ as an open subset of a compact ...
- 315
4
votes
0
answers
127
views
Are algebraic symplectic manifolds locally exact?
my guess is "no", even in the etale topology.
Are there interesting examples of algebraic symplectic manifolds which are locally exact in Zariski or etale topology? What about Hilbert ...
- 807
4
votes
0
answers
195
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, ...
- 1,141
2
votes
0
answers
233
views
Cohomology of Beauville–Mukai varieties
The rational second cohomology of the Hilbert scheme on a K3 surface $S$ are spanned by $H^2(S,\mathbb{Q})$ plus the class of the exceptional divisor. The mapping $H^2(S, \mathbb{Q}) \to H^2(\mathrm{...
- 427
1
vote
0
answers
164
views
Does the blow-up preserve symplectic structure?
Let $X \subset \mathbb P(\mathfrak g)$ be an adjoint variety for a simple complex Lie algebra $\mathfrak g$ appearing in the last line of the Freudenthal Magic Square, that is $\mathfrak g \in \{F_4,...
- 381
3
votes
0
answers
189
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 ...
- 9,157
7
votes
2
answers
467
views
Symplectic resolutions amongst cotangent bundles
It is known that a generalized flag variety $X=T^*(G/P)$ is a (symplectic) resolution of singularities of its affinization $X^\text{aff}\mathrel{:=}\operatorname{Spec}(H^0(X,\mathcal{O}_X))$. In type ...
- 1,647