All Questions
Tagged with complex-geometry sg.symplectic-geometry
158 questions
1
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55
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Real-holomorphic Hamiltonian vector fields
Consider a Kähler manifold with complex structure $J$. Is there a characterization of real-valued functions $H$ for which the corresponding Hamiltonian vector field $X_H$ is real-holomorphic, that is, ...
- 121
1
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0
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51
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Moduli space of curves away from singular subsets
Recently, I'm interested in the moduli spaces of curves in a (possibly noncompact) complex orbifold (resp. symplectic and almost complex) away from singularities.
More specifically, I'm interested in ...
- 285
1
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0
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151
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Blowing up $\mathbb{CP}^2$ nine times and exactness of symplectic form
Consider the (symplectic) blow up $\operatorname{Bl}_k(\mathbb{CP}^2)$ of $\mathbb{CP}^2$ at $k$ points. I have heard that for $k=1,2,\ldots,8$ the size of the balls been blown up can be choosen in ...
- 191
3
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0
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59
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Complex structures compatible with a symplectic toric manifold
Let $(M^{2m},\omega)$ be a compact symplectic manifold with equipped with an effective Hamiltonian torus $\mathbb T^m$ action.
Suppose $J_0$ and $J_1$ are two $\mathbb T^m$-invariant compatible ...
- 1,441
7
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1
answer
508
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Are holomorphic Lagrangians locally graphs?
Let $(M, \omega)$ be a holomorphic symplectic manifold of (complex) dimension $2n$. Let $x$ be a point in $M$. My understanding from the discussion and answers to this MO question is that there exists ...
- 27.5k
2
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219
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Manifold whose symplectic structure of the cotangent bundle is intrinsically different from any symplectic structure arising from $\mathbb{C}^n$
Inspired by this question Symplectic structure of $TS^{n-1}$ we ask:
What is an example of a manifold $M$ whose cotangent bundle $T^*M$ is an Stein manifold but the canonical symplectic ...
- 366
4
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1
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254
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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
5
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1
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232
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Fujiki class $\mathcal C$ with a symplectic structure
Recall that a compact complex manifold $X$ is said to be in Fujiki class $\mathcal C$ if there is a proper modification $\mu:\tilde X\to X$ such that $\tilde X$ is a compact Kähler manifold. If $X$ ...
- 471
4
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1
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197
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Family of Dolbeault operators on complex vector bundles over $\mathbb{CP}^1$
Let $\pi \colon E \rightarrow \mathbb{CP}^1$ be a complex vector bundle. It is a well-known fact that a Dolbeault operator on $\pi\colon E \rightarrow \mathbb{CP}^1$ gives a holomorphic structure on $...
- 369
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119
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Examples and classification of holomorphic strips in $(\mathbb{C}\mathbb{P}^n,\mathbb{R}\mathbb{P}^n)$
Consider an exact isotopy $\phi_t$ of $\mathbb{C}\mathbb{P}^n$ such that $\phi_1(\mathbb{R}\mathbb{P}^n)\pitchfork \mathbb{R}\mathbb{P}^n$. When trying to compute the Lagrangian Floer cohomology of $(\...
- 791
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1
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415
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Hyperkahler and symplectic complex geometry: reference?
I would need some references regarding symplectic and hyperkahler (complex) geometry. My background is mostly from algebraic geometry and I know a little bit the basics on Kahler manifolds.
I would be ...
- 1,061
7
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1
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291
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Cotangent bundles of surfaces as varieties
As far as I understand, it is easy to see (and find in the literature) that the affine variety
$$z_1^2+z_2^2+z_3^2=1$$
with the restriction of the standard $\omega_{std}$ of $\mathbb{C}^3$ is ...
- 203
3
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200
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Are there known examples of almost complex manifolds admitting neither a symplectic nor a complex structure? [closed]
I have seen the the example of $S^6$ being touted around here and there but it does not seem to be generally confirmed that there is no complex structure on it.
user125834
3
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0
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114
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Existence of uniformly bounded Darboux chart
In Donaldson's paper Symplectic submanifolds and almost-complex geometry, he mentioned that for each point $p$ in a compact almost-Kähler manifold $(V,\omega ,J)$, there exists a Darboux chart $\...
- 341
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1
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301
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Boundary Maslov index of holomorphic disks in Calabi-Yau manifolds
Let $u \colon \Sigma^2 \to M^{2n}$ be a holomorphic disk (so $\Sigma = \{z \in \mathbb{C} \colon |z| \leq 1\}$) in a compact Calabi-Yau manifold $M$ of real dimension $2n$ with boundary on a ...
- 871
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1
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479
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Do holomorphic symplectic manifolds admit (high codimension) embeddings in some standard space?
Per the Whitney embedding theorem, any manifold $M$ can be embedded into a sufficiently high dimensional Euclidean space.
According to Gromov's h-principle for contact embeddings, any contact manifold ...
- 8,723
1
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1
answer
156
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Deform a complex structure fixing marked points
Let $\Sigma$ be a closed orientable surface of genus $g$ with $m$ marked points $x=\{x_1, \ldots, x_m\}$ and $j_0$ denote a complex structure on $\Sigma$. Take a neighborhood $U$ of the isomorphism ...
- 369
2
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100
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Symplectic form on $\Omega^0(X,End(E))$
Let $E\rightarrow X$ be a holomorphic vector bundle over a Kahler manifold. Is there a natural symplectic form on the space $\Omega^0(X,End(E))$ ?
For example on $\Omega^1(X,End(E))$ we have the ...
- 789
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79
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Symplectic structure on the space of complexes of holomorphic vector bundles
Let $E\rightarrow X$ be a holomorphic vector bundle over a complex manifold. Denote by $Dol(E)$ the space of holomorphic structures on $E$. Fix any Hermitian metric $h$ on $E$ and denote by $\mathcal{...
- 789
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540
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Polarizations in algebraic and symplectic geometry
In context of Abelian varieties there are a couple of equivalent ways to
introduce the polarization of a algebraic variety. One way is to
choose a line bundle $\mathcal{L}$ which satisfies certain ...
- 6,038
2
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1
answer
243
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Extension of a holomorphic vector bundle on a nodal curve
I am reading a paper on holomorphic curves and stuck in an argument about extension of a given holomorphic vector bundle over a nodal curve.
Let $C$ be a nodal curve without closed componets and $E$ a ...
- 369
3
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0
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71
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Holomorphic homeomorphisms
Let $M$ be a connected closed smooth manifold. Consider the group $\mathrm{Homeo}(M)$ of homeomorphisms $M\to M$ endowed with the $C^0$-topology.
If $M$ has a symplectic structure some people study ...
user164740
4
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0
answers
104
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Non-isomorphic compact Kähler manifolds not containing submanifolds biholomorphic to their conjugates
Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$.
Assume there is a diffeomorphism $\nu:M\to N$ ...
user164740
2
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1
answer
191
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Non-symplectomorphic isometric compact Kähler manifolds
Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$.
Assume there is a diffeomorphism $\phi:M\to N$...
user164740
13
votes
1
answer
493
views
Non-isomorphic compact Kähler manifolds that are biholomorphic, symplectomorphic and isometric
Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$.
Assume there is a diffeomorphism $\nu:M\to N$ ...
user164740
8
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0
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315
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Singularities of a morphism from a smooth projective variety to an abelian variety
Let $f: X\to A$ be a (flat) morphism from a smooth complex projective variety $X$ to an abelian variety $A$. Consider the following natural diagram:
$$T^*X\overset{df}{\longleftarrow}X\times H^0(A, \...
- 1,081
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119
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Organizing mirror pairs
At a maximally vague and naive level, mirror symmetry asks the following question: given a complex manifold $(X, I)$, is there a symplectic manifold $(M, \omega)$ and an equivalence between the ...
- 3,020
1
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1
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108
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Chart in $1$-parameter family of Lagrangians in a Kähler manifold
Let $(X,\omega,J)$ be a complex $n$-dimensional Kähler manifold ($\omega$ Kähler form, $J$ complex structure) and $L \subset X$ be a closed real-analytic Lagrangian submanifold. Furthermore, let $L_{t}...
- 11
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548
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Holomorphic Weinstein Lagrangian neighborhood theorem
The Weinstein Lagrangian neighborhood theorem says that if $(M,\omega)$ is a symplectic manifold and $L\subset M$ is a Lagrangian submanifold, then there are neighbourhoods $U$ of $L$ in $M$, and $U'$ ...
- 779
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152
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Almost complex structure commuting with symplectomorphism
Let $(V,\omega)$ be a symplectic vector space with symplectic form $\omega$. Furthermore, let $\varphi : V \rightarrow V$ be a linear symplectomorphism. Consider the set
$$
\mathcal{I}_{\varphi} := \{ ...
- 49
8
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1
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638
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Automorphism group of compact hyperkähler manifolds
Let $M$ be a compact simply-connected hyperkähler manifold, and let
$$
\mathrm{Aut}(M)
$$
be the automorphism group of $M$, i.e. the group of tri-holomorphic diffeomorphisms preserving the metric.
...
- 779
4
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250
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Quotients of Kähler manifolds
Let $X$ be a Kähler manifold and $G$ a complex semisimple Lie group acting freely on $X$ by biholomorphisms and such that the Riemannian metric is preserved by a maximal compact subgroup $K$ of $G$. ...
- 41
2
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0
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115
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A non-Kaehler manifold complex and symplectic in exactly one way
Does there exist a closed connected smooth manifold that admits exactly one (up to biholomorphism) integrable complex structure and exactly one (up to symplectomorphism and rescaling) symplectic ...
- 21
8
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1
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363
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Independence of Duistermaat-Heckman measure
Suppose that a compact Kähler manifold $(X,\omega)$ has a real torus acting on it by symplectomorphisms in a Hamiltonian way (the torus is not necessarily of maximal rank). Then for any smooth ...
- 185
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2
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695
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Kronheimer's results on ALE spaces as hyperkahler quotients
Background: In his two papers from late 80s Kronheimer proved that any 4-dimensional ALE space is given by a hyperkahler quotient, say $X_{{\zeta_\mathbb{R}},{\zeta_\mathbb{C}}}(Q)$ where Q is a ...
- 1,677
7
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1
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458
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Large isometry groups of Kaehler manifolds
Let $M$ be a closed simply-connected Kaehler manifold that is not isomorphic to a product of lower-dimensional Kaehler manifolds. Pick an orientation for $\mathbb{C}$; this endows $M$ with an ...
user136269
3
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0
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238
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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
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2
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520
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Multiple mirrors phenomenon from SYZ and HMS perspective
There is a set of ideas called mirror symmetry which, roughly speaking, relates symplectic and complex geometry of Calabi--Yau manifolds. There are also extensions to Fano and general type varieties ...
- 375
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2
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995
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Holomorphic version of Darboux's theorem
I would like to ask if there is a holomorphic version of Darboux's theorem. More concretely, given a holomorphic symplectic manifold $(X, \omega)$ is there a local holomorphic symplectomorphism from $(...
- 609
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0
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355
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First Chern Class of Contact Structure which is not Torsion
Let $(M,\xi)$ be a closed connected $3-$dimensional contact manifold with contact structure $\xi$. It is known that the first Chern class $c_{1}(\xi)$ defines an element in $H^{2}(M;\mathbb{Z})$ and ...
- 39
6
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2
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508
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Uniqueness of a compatible Kahler-Einstein structure on a symplectic manifold?
$\require{AMScd}$
Preliminaries: Let $(X,\omega,J)$ be a closed Kahler manifold. That is, $X$ is a closed $2n$-manifold, $\omega$ is a symplectic form and $J$ is a compatible (integrable) complex ...
- 1,231
14
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2
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1k
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Relation between mirror symmetry, homological mirror symmetry, and SYZ conjecture
I'm very new to mirror symmetry, and have a hard time establishing a broad overview of the subject. In particular I do not understand the precise relation between the following three conjectures:
...
- 2,788
9
votes
2
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725
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Two homeomorphic non-diffeomorphic complex manifolds
Does there exist a closed topological manifold supporting two non-diffeomorphic smooth structures both of which admit a compatible complex structure? Also the same question, but for symplectic ...
- 287
8
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1
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669
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Beilinson-Drinfeld quantization and stable bundles
To motivate this question, I'm going to try and explain some background notions. This won't be absolutely necessary for experts, but I want to be vaguely honest about where this question comes from. ...
- 3,020
17
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0
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770
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What are hyperkähler metrics used for?
It seems that a lot of effort has been devoted to endow holomorphic-symplectic manifolds with hyperkähler metrics. It started with Calabi [4] with $T^*\mathbb{CP}^n$. Other examples include coadjoint ...
- 179
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2
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612
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Complex Analytic Structure on Moduli Space of Stable Maps
Suppose $(X,\omega,J)$ is a compact Kähler manifold, and $\beta\in H_2(X,\mathbb Z)$ is given. Then, we can form the space $\overline{\mathcal M}:=\overline{\mathcal M}_{0,0}(X,\beta)$ of stable maps $...
- 1,761
4
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0
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153
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Dimension of linear complex-symplectic reduction
Let $(V,\omega)$ be a finite-dimensional complex-symplectic vector space and $G$ be a complex reductive group acting linearly on $V$ by preserving $\omega$. Then, there is a moment map
$$\mu:V\to\...
- 41
1
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0
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145
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Biholomorphic maps between cotangent bundles with non-standard complex structures
Let $X$ be a compact Kähler manifold. Let $\omega_i$ (i=1,2) be Kähler forms on $X$. Assume that $\psi:X\rightarrow X$ is a diffeomorphism such that $\psi^*\omega_2=\omega_1$.
Recall that each $\...
- 404
5
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0
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227
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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 ...
- 609
10
votes
2
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526
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Two smooth tangent almost complex curves in a $4$-manifold
I would like to know if following is correct.
Statement. Suppose we have a smooth (i.e., $C^\infty$) almost complex structure on $\mathbb R^4$ and $C_1, C_2$ are two $J$-holomorphic curves passing ...
- 14.3k