All Questions
Tagged with holomorphic-symplectic ag.algebraic-geometry
29 questions
2
votes
0
answers
91
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 $x\in X$.
...
- 178
1
vote
0
answers
96
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 $\...
- 178
3
votes
0
answers
110
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 ...
- 1,677
3
votes
0
answers
187
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
254
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 ...
- 325
4
votes
0
answers
128
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
234
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, ...
- 1,286
2
votes
0
answers
241
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
168
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
4
votes
0
answers
202
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 ...
- 9,237
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
283
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 ...
- 984
5
votes
1
answer
323
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 ...
- 390
2
votes
1
answer
202
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 ...
- 1,677
3
votes
1
answer
150
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 $\Delta$. ...
- 569
0
votes
0
answers
118
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
363
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-...
- 175
2
votes
0
answers
272
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 $g\geq2$. ...
2
votes
1
answer
483
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 ...
- 63
2
votes
0
answers
112
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 ...
- 7,300
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 ...
- 27.5k
5
votes
0
answers
260
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 ...
- 401
1
vote
1
answer
221
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 ...
- 2,108
2
votes
1
answer
408
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 ...
- 583
7
votes
1
answer
561
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 ...
- 2,537
6
votes
0
answers
659
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 ...
- 44.7k
3
votes
2
answers
358
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 $\geq 4$...
- 44.7k
11
votes
1
answer
1k
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 $\...
- 44.7k
7
votes
0
answers
244
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 ...
- 14.7k