# Questions tagged [holomorphic-symplectic]

hyperkahler manifolds, complex Lagrangian submanifolds, Mukai flop, integrable systems

73
questions

3
votes

0
answers

71
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 $K[\circ]$-...

2
votes

0
answers

89
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 $x\in X$.
...

1
vote

0
answers

88
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 $\...

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,\...

3
votes

0
answers

95
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 ...

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 ...

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 ...

3
votes

0
answers

182
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 ...

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 ...

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 ...

4
votes

0
answers

199
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, ...

2
votes

0
answers

236
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{...

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,...

3
votes

0
answers

190
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
votes

2
answers

471
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
vote

0
answers

1k
views

### Explicit construction of Fubini Study Metric

I have a question about a remark on Fubini Study metric on $\mathbb{CP}^n$
from Notes on canonical Kähler metrics
on page 8 is remarked (Example 2.12 4.):
Fix a Hermitian innerproduct on $\mathbb{C}^{...

1
vote

1
answer

244
views

### Visualising the moduli space of stable disks with interior marked point and 4 marked point on the boundary

Is there any nice description/picture of the moduli space of stable disks with 1 interior marked point and 4 marked points on the boundary?
I'm expecting it to be a 3-dimensional polytope, because ...

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 ...

1
vote

0
answers

104
views

### Action on cohomology by automorphisms of ihs manifolds

For all known deformation types of irreducible holomorphic symplectic manifolds (which I call K3, K3$^n$, Kum$^n$, OG$^3$, OG$^5$, the exponent being half the complex dimension), it is known that the ...

3
votes

1
answer

282
views

### Rank 3 Lagrangian vector bundles on an elliptic curve

Let $k$ be an algebraically closed field of characteristic zero (feel free to assume $k= \mathbb{C}$) and $E$ an elliptic curve over $k$ with identity $P \in E(k)$.
I am interested in certain ...

6
votes

0
answers

268
views

### Is there an symplectic field theory compactness theorem applicable in the context of Floer cohomology of a symplectomorphism?

Is there any reference in the literature about results regarding symplectic field theory (SFT) compactness for a neck-stretch in the context of Floer homology of a symplectomorphism $\phi \colon (M,\...

6
votes

1
answer

685
views

### Bubbling off of a pseudo holomorphic sphere on surface with cylindrical ends

I need some clarification about the reason why we have a sphere bubbling off in the situation described by Seidel in The Symplectic Floer Homology of a Dehn Twist.
I’ll try to summarize to the best ...

2
votes

1
answer

561
views

### Two Lagrangian submanifolds with clean intersections

Having two closed exact Lagrangian submanifolds $L_1$ and $L_2$ that intersect cleanly inside a Liouville manifold $M$ with $c_1(M)=0,$ is there (with possibly some other conditions) any relation ...

5
votes

1
answer

320
views

### Locally affine varieties and du Val singularities

Let me start with an apologetic disclaimer: I am very far from an algebraic geometer, so this question might be crudely formulated.
I have a specific question about du Val singularities, but while ...

3
votes

0
answers

115
views

### Reference for "holomorphic contact geometry"

Just like holomorphic symplectic geometry is a complexification of real symplectic geometry, I am wondering is there any good survey paper or book talking about holomorphic version of real contact ...

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 ...

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 $\mathbb{C}^2/\mathbb{Z}_{/n}$?

3
votes

0
answers

235
views

### Symplectic Chern class of holomorphic symplectic manifold

I've posted this question already on math.stackexchage. Unfortunately, it did not receive any answers even though there was a bounty on it. So maybe somebody of you could help me. I apologize if this ...

2
votes

1
answer

197
views

### Core components of quiver varieties as fiber bundles of flag varieties

Is there an example of Nakajima quiver variety of type A which has all core components smooth, such that at least one of them is NOT an iterated fibre bundle of flag manifolds (i.e. a space obtained ...

3
votes

1
answer

148
views

### The singularties of the dicriminant loci of the Lagrangian fibration

Let $X$ be a holomorphic symplectic variety of dimension $2n$ and $\pi: X \to \mathbb{P}^n$ be a Lagrangian fibration. It is known that $\pi$ is smooth outside of the discrimiant divisor $\Delta$. ...

1
vote

0
answers

93
views

### One-point partition

Let following be the partition function in infinitely many variables $x_i$ the linear coordinates on the vector space $V$.
$$
\mathcal{Z}=exp\Big(\sum_{\substack{g\geq 0\\n\geq 1}}\frac{h^{g-1}}{n!}\...

4
votes

1
answer

343
views

### Equivariant quantum cohomology of conical symplectic resolutions

There is a couple of papers on this [Braverman, Maulik, Oblomkov, Okounkov, Pandharipande] where authors calculate quantum cohomology for various conical symplectic resolutions. The language in these ...

4
votes

1
answer

203
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
vote

0
answers

94
views

### Effective classes in toric Kähler manifolds

In an article about toric manifolds, I have seen the following notions, which I don't understand. We view a symplectic toric manifold $(M,\omega)$ as a Kähler manifold with Kähler form $\omega$, and ...

0
votes

0
answers

111
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 ...

2
votes

0
answers

350
views

### Relation between Beauville-Bogomolov form and Intersection Product on Hilbert scheme of K3 surfaces

I am learning about Hilbert scheme of points $S^{[n]}$ on projective K3 surfaces S. Since these are hyperkähler varieties, the second cohomology $H^2(S^{[n]},\mathbb{Z})$ is endowed with the non-...

5
votes

0
answers

224
views

### Lagrangian foliation for a holomorphic symplectic manifold

I am interested in gathering as many examples as possible for Lagrangian foliations of holomorphically symplectic manifolds $(X, \omega)$, where $X$ is a $2n$-dimensional complex manifold equipped ...

3
votes

1
answer

376
views

### Disconnecting the Lagrangian Grassmannian

Let $(V, \omega)$ be a symplectic vector space of dimension 2n. This has a Lagrangian Grassmannian $\Lambda(V)$ of Lagrangian subspaces of $V$. Now consider the following subvariety: Fix a half-...

9
votes

1
answer

881
views

### Deligne Mumford Compactification of Moduli Space Of Annuli

I am reading Abouzaid's paper "A geometric criterion for generating the Fukaya category" (https://arxiv.org/abs/1001.4593), and it is claimed there, without proof, in section C.4 in the appendix (pp....

3
votes

0
answers

95
views

### Can one choose a sufficiently generic path of a.c.s such that only "codimension 1" bubbling occurs?

Consider a symplectic manifold $(M,\omega)$ of dimension $2n$ (closed or open with bounded geometry). Let $L\subset M$ be a compact Lagrangian submanifold (not necessarily connected).
Consider two ...

3
votes

0
answers

134
views

### Lagrangian subvariety in coadjoint orbit

In Chriss and Ginzburg's Representation Theory and Complex Geometry, Theorem 3.3.6 says that
"Let $\mathbb{O}$ be a coadjoint orbit in $\mathfrak{g}^*$. Let $x\in \mathbb{O}$ be such that $x|_{\...

2
votes

1
answer

256
views

### Irreducibility of holomorphic symplectic quotients

Let a connected algebraic group $G$ (over $\mathbb C$, say) act Hamiltonianly on an algebraic symplectic variety $M$, with moment map $\Phi: M\to \mathfrak g^*$. In the example I care about, vaguely ...

2
votes

0
answers

259
views

### Holomorphic symplectic form on the moduli space of Higgs bundles

I have the following problem: consider the moduli space $\mathcal{M}:=\mathcal{M}_X(n, 0)$ of semistable Higgs bundles of rank $n$ and degree $0$ on a compact Riemann surface $X$ of genus $g\geq2$. ...

21
votes

1
answer

1k
views

### Why symplectic geometry gives Poisson geometry

One way I've learned to understand Poisson geometry is to consider it as symplectic geometry with no open conditions - i.e. no condition of nondegeneracy. This idea can be applied to many other ...

6
votes

1
answer

626
views

### Kahler version of Darboux's Theorem

In symplectic geometry, Darboux's theorem says that locally, any symplectic manifold of dimension $2n$ looks like symplectic Euclidean space (that is, there is some set of coordinates $(x_i, y_i)$ ...

2
votes

1
answer

477
views

### Local deformations of Lagrangian submanifolds in holomorphic symplectic manifold and their intersections

Let $Y\subset X$ be a Lagrangian submanifold in a holomorphic symplectic manifold $X$. We know that there exists a local moduli space $M$, which parametrizes lagrangian submanifolds in $X$(there are ...

4
votes

1
answer

568
views

### What is the relation between holomorphic blow-up and symplectic blow-up?

McDuff has shown us exactly how the symplectic blow-up procedure along a symplectic submanifold affects the symplectic structure in the ambient space, i.e., if $\omega$ is the original symplectic form,...

2
votes

0
answers

108
views

### Symplectic structure moduli of simple bundles on hyper-Kaehler manifolds

Let $S$ be a K3 or Abelian surface and let $M_{S}$ be a moduli of stable bundles on $S$. Then, Mukai proves that there $M_{S}^{H}$ has a symplectic structure. Indeed, let $\mathcal{F}$ be the ...

1
vote

0
answers

138
views

### Shape of the bubbling limit of holomorphic discs

I will present my question in the specifics I encountered it, so maybe some of the details are irrelevant for the desired conclusion.
Consider $(S^2\times S^2,\omega_{std})$ the product of two ...

35
votes

1
answer

1k
views

### Finding the octonionic analog of the K3 surface, via (almost) hyperkahler geometry?

The K3 manifold is an amazing object in mathematics which plays an important role in several fields ranging from the study of smooth 4-manifolds to algebraic geometry to differential geometry and ...