Questions tagged [holomorphic-symplectic]
hyperkahler manifolds, complex Lagrangian submanifolds, Mukai flop, integrable systems
3
votes
1answer
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 $\Delta$. ...
1
vote
0answers
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
votes
0answers
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
votes
1answer
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 ...
1
vote
0answers
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 ...
0
votes
0answers
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 ...
1
vote
0answers
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-...
4
votes
0answers
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
votes
1answer
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-...
6
votes
1answer
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....
4
votes
0answers
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 ...
3
votes
0answers
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|_{\...
1
vote
1answer
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 ...
2
votes
0answers
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
votes
1answer
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
votes
1answer
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 $(x_i, y_i)$ ...
2
votes
1answer
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 ...
3
votes
1answer
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,...
2
votes
0answers
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 ...
1
vote
0answers
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
votes
1answer
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 ...
2
votes
0answers
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 ...
7
votes
2answers
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",...
5
votes
0answers
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 ...
1
vote
0answers
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 ...
4
votes
0answers
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 ...
0
votes
2answers
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 ...
2
votes
2answers
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 ...
3
votes
2answers
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/...
0
votes
1answer
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 ...
12
votes
1answer
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 $c_1(M)=0$ ...
3
votes
1answer
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 ...
2
votes
1answer
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 ...
6
votes
1answer
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
1answer
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 ($\...
1
vote
2answers
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 ...
2
votes
3answers
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 $\alpha$...
3
votes
0answers
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 ...
6
votes
0answers
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 ...
8
votes
0answers
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
votes
1answer
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
2answers
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 $\geq 4$...
10
votes
1answer
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
votes
0answers
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 ...
