# Questions tagged [holomorphic-symplectic]

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

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78 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 $\Delta$. ...

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**0**answers

71 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!}\...

**3**

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53 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**

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**1**answer

60 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 ...

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64 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 ...

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73 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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101 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-...

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106 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 ...

**2**

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**1**answer

197 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-...

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**1**answer

467 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....

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69 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 ...

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78 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|_{\...

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**1**answer

172 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 ...

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168 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 $g\geq2$. ...

**18**

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**1**answer

748 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 ...

**3**

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**1**answer

314 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 $(x_i, y_i)$ ...

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**1**answer

358 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 ...

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**1**answer

289 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,...

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92 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 ...

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90 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 ...

**30**

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**1**answer

895 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 ...

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80 views

### Questions about holomorphy and zeros of the symmetric power $L$-function

Let $f$ be a primitive form of an even weight $k$ for the full modular group and let $L(Sym^rf,s)$ be the symmetric $r$th $(r\geq 2)$ power $L$-function associated to $f.$ I have three questions ...

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666 views

### Learning Quantum (Co)Homology and Landau Ginzburg Superpotential

I am learning about Quantum Homology which I have to use in my research, and I see that in many papers (For example in FOOO, "Spectral invariants with bulk, Quasimorphisms and Lagrangian Floer theory",...

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187 views

### Injective homomorphism induced by cup product in cohomology

Let $M$ be an irreducible holomorphic symplectic manifold of dimension $\geq 4$. In his paper 'A survey of Torelli and Monodromy results', Markman claims (discussion after Theorem 9.7) that the cup ...

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194 views

### A holomorphic vector bundle structure for $\Omega^{(0,1)}(M)$

For a complex manifold $M$, the complexified tangent space $\Omega^1(M)$ splits into a direct sum $\Omega^1(M) = \Omega^{(1,0)}(M) \oplus \Omega^{(0,1)}(M)$. As is well-known $\Omega^{(1,0)}(M)$ is ...

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310 views

### Complex symplectic reduction

Oddly I find about zero resources talking about "complex symplectic reduction" upon a web search. Is there anything wrong with it?
I guess maybe there are two competing settings a priori: a complex ...

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**2**answers

484 views

### When are Maslov $0$ disks non-trivial in $\pi_2(M,L)$?

My goal is to better understand the Maslov-index of pseudoholomorphic disks.
For a symplectic manifold $(M,\omega)$ and a Lagrangian submanifold $L\subset M$, the Maslov-index of a pseudoholomorphic ...

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**2**answers

354 views

### Holomorphic Line Bundles over a Homogeneous Space

Let $M=G/H$ be (compact) homogeneous complex manifold, and let $L$ be a line bundle over $M$. Can one always equip $L$ with a holomorphic structure? Can there be more then one such holomorphic ...

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618 views

### Reference Request: “Neck Stretching Procedure” (In Symplectic Field Theory)

I've been reading some papers in Symplectic Geometry which refer to something called "Stretching the neck", and give reference to Eliashberg, Givental and Hofer's SFT paper (http://arxiv.org/abs/math/...

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**1**answer

149 views

### Dominant map from hyperkahler manifolds to normal projective varieties with symplectic singularities

Let me recall some quick definitions. A projective hyperkahler manifold is a simply connected smooth projective variety $M$ such that $H^0(M,\Omega_M^2)=\mathbb C\sigma$, with $\sigma$ an everywhere ...

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**1**answer

1k views

### Why Yau's theorem implies the existence of hyperkähler metric on complex symplectic manifolds?

Every expository article on hyperkähler manifolds that I have read states without detailed proof the following fact:
It follows from Yau's theorem (i.e. a compact Kähler manifold $M$ with $c_1(M)=0$ ...

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votes

**1**answer

392 views

### Complement of Donaldson's symplectic submanifold

I am just starting to learn more symplectic geometry on Stein manifolds. I understand that an important class of Stein manifolds arises as the complement of a Donaldson's codimension two symplectic ...

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**1**answer

308 views

### differential form with empty zero locus

Hi there,
I am considering existence of holomorphic 2-form $\omega$ with empty zero locus on a complex manifold. Beside K3 and holomorphic symplectic manifold, is there any other example on general ...

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**1**answer

395 views

### Fundamental groups of symplectic leaves

Let $X = \operatorname{Spec}(R)$ be a conical Poisson variety over $\mathbb{C}$. This means
that $R$ is a non-negatively graded Poisson algebra over $\mathbb{C}$
with $R_0 = \mathbb{C}$ and that the ...

**13**

votes

**1**answer

459 views

### Energy quantization for $J$-holomorphic spheres

Let $(\mathbb{CP}^1, j, g_{\text{FS}})$ be the complex projective line with the standard complex structure and the Fubini-Study metric and let $(M,J,\omega,g)$ be an almost Kähler compact manifold ($\...

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**2**answers

740 views

### Holomorphic bundles and maps to the Grassmannian ?

Hello,
In the differentiable case it is quite easy to prove that vector bundles are equivalent to smooth maps to the Grassmannian $G_{k}(\mathbb{R}^N)$ for some integer $N>>0$. The proofs I ...

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**3**answers

509 views

### $\partial \bar{\partial}$ on a riemann surface

hallo,
i have the following question. let $M$ be a Riemann surface and $R \subset M$ be a compact totally real manifold (1 dimensional real). Furthermore assume there is a holomorphic 2-form $\alpha$...

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242 views

### Transitive action on moduli space of holomorphic curves.

If $G$ is a complex semi simple Lie group and $P$ is a parabolic subgroup of it, the quotient $G/P$ is endowed with a Kahler structure and the action of $G$ on $G/P$ is holomorphic with respect to ...

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571 views

### Are conical symplectic resolutions Mori dream spaces?

This is one of these questions where it's tempting to just leave it at the title, but let me try to define the objects in question.
A conical symplectic resolution is a projective resolution of ...

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568 views

### SFT gluing on chain level in Floer homology?

I came across a new type of gluing in the FOOO paper http://arxiv.org/abs/1002.1660 (maybe not so new to experts). The situation is as follows: one considers a relative (to lagrangian) class and ...

**7**

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**1**answer

1k views

### Zeros of holomorphic one-forms on Riemann surface

Is it true that for any point on any compact Riemann surface there exists a global holomorphic one-form, which does NOT have a zero at that point.

**3**

votes

**2**answers

280 views

### Is a terminal symplectic variety S_4?

For purposes of this question "terminal symplectic variety" means a normal variety which is symplectic (in the usual sense of symplectic singularities) and whose singular locus has codimension $\geq 4$...

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**1**answer

885 views

### Is the generic deformation of a symplectic variety affine?

Kaledin and Verbitsky have shown that symplectic varities have a remarkably nice deformation theory as symplectic varieties.
Let $X$ be a symplectic variety (a smooth quasi-projective variety over $\...

**7**

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**0**answers

211 views

### Projectivity of flops

Say I have a projective (smooth, compact) irreducible symplectic variety $X$ over $\mathbb{C}$ and I perform a Mukai flop. It is well known that if the resulting variety $\widetilde{X}$ is Kahler, it ...