Questions tagged [holomorphic-symplectic]
hyperkahler manifolds, complex Lagrangian submanifolds, Mukai flop, integrable systems
62
questions
2
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0
answers
196
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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{...
- 417
1
vote
0
answers
125
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,...
- 291
3
votes
0
answers
122
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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 ...
- 7,548
7
votes
2
answers
367
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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 ...
- 1,429
1
vote
0
answers
461
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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}^{...
- 504
1
vote
1
answer
132
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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 ...
- 1,838
7
votes
0
answers
212
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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
1
vote
0
answers
91
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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
255
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 ...
- 1,192
6
votes
0
answers
230
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,\...
- 1,838
5
votes
1
answer
501
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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 ...
- 1,838
2
votes
1
answer
317
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 ...
- 1,429
5
votes
1
answer
260
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 ...
- 371
3
votes
0
answers
97
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 ...
- 348
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
3
votes
0
answers
179
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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 ...
- 31
2
votes
1
answer
175
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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 ...
- 1,429
3
votes
1
answer
125
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 $\Delta$. ...
- 559
1
vote
0
answers
90
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!}\...
- 621
3
votes
1
answer
255
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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 ...
- 1,429
4
votes
1
answer
174
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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
0
answers
86
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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 ...
- 1,157
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
2
votes
0
answers
255
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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-...
- 175
4
votes
0
answers
171
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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 ...
- 493
3
votes
1
answer
305
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-...
- 391
9
votes
1
answer
737
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....
- 1,835
3
votes
0
answers
89
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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 ...
- 1,835
3
votes
0
answers
110
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|_{\...
- 485
2
votes
1
answer
201
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 ...
- 26.7k
2
votes
0
answers
218
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 $g\geq2$. ...
20
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 ...
- 4,651
4
votes
1
answer
483
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 $(x_i, y_i)$ ...
- 4,651
2
votes
1
answer
420
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 ...
- 53
4
votes
1
answer
457
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,...
- 191
2
votes
0
answers
103
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 ...
- 6,169
1
vote
0
answers
121
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 ...
- 1,835
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 ...
- 25.7k
2
votes
0
answers
94
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 ...
- 1,193
8
votes
2
answers
960
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",...
- 1,835
5
votes
0
answers
223
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 ...
- 381
1
vote
0
answers
264
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 ...
4
votes
0
answers
391
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 ...
- 1,307
1
vote
2
answers
697
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 ...
- 281
2
votes
2
answers
457
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 ...
6
votes
2
answers
905
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/...
- 1,835
0
votes
1
answer
186
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 ...
- 2,028
12
votes
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 $c_1(M)=0$ ...
- 1,674
4
votes
1
answer
501
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 ...
- 86