2
votes
1answer
191 views

Cotangent bundle of coadjoint orbit is stein manifold?

Let me first define stein manifolds and coadjoint orbits. A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold: $X$ is holomorphically convex, ...
28
votes
1answer
683 views

What is the meaning of $(h^{11},h^{21})\to (h^{11}-240,h^{21}+240)$ in Calabi-Yau threefolds?

By browsing through the Hodge data of known Calabi-Yau threefolds, I stumbled upon an observation that frequently enough a pair of Hodge numbers $(h^{11},h^{21})$ comes together with the pair $ ...
1
vote
0answers
121 views

Explicit formula for hermitian form on coadjoint orbit of $G$ on $\mathfrak{g}^*$

Let $G$ be a compact Lie group and $\mathfrak{g}$ be its Lie algebra and $\mathfrak{g}^*$ be its dual , then I am looking for explicit formula for hermitian form on coadjoint orbit of $G$ on ...
0
votes
0answers
93 views

Dimension of the set of polarized sections

Let $M$ be a symplectic manifold. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian, i.e. Maximal isotroic, dim$P_m=n$, ...
0
votes
1answer
274 views

Symplectic structure on $Sym^kG^{\mathbb{C}} $

Let $G$ be a compact Lie group, and let $G^\mathbb{C}$ be its complexification. I am looking for a symplectic structure (without use of coordinates) on $$ Sym^kG^{\mathbb{C}}, $$ PS:Here ...
2
votes
1answer
320 views

looking for an identity for higher jet bundle $J^kM$?

We know this fact that the first jet Bundle $J^1M$ is diffeomorphic with $T^*M×\mathbb{R}$.i.e, ($J^1M=T^*M×\mathbb{R}$) Is there something like this identity for higher jet bundle $J^kM$? I editted ...
1
vote
0answers
145 views

Hyperkaehler Structures on the cotangent bundle

Let $M$ be a symplectic manifold (not Kaehler). Does there exists in a neighbourhood of the zero section in the cotangent bundle $T^{*}M$ a Hyperkaehler structure? I know that by the paper by Feix on ...
0
votes
0answers
109 views

When a hyperplane of symmetric forms is determined by a quadric hypersurface?

Let $L$ be a 2D real vector space, $L^*$ its dual, and $\{V,\omega\}$ the symplectic space with $V=L\oplus L^*$ and $\omega$ unambiguously defined by $\omega(l,\lambda):=\lambda(l)$, for all $l\in L$ ...
5
votes
2answers
318 views

Information from Moment Polytopes

Let $T$ be a compact real torus, and $X$ a Hamiltonian $T$-manifold (which you may take to be a smooth complex projective variety) with moment map $\mu:X\rightarrow\frak{t}^*$. If ...
5
votes
4answers
319 views

Prequantization and Hilbert space

In the definition of pre-quantization on a symplectic manifold $(M,\omega)$, we represent a function $f\in C^{\infty}(M)$(with Lie algebra structure) to $\hat{f}$ in the Hilbert space $L^2(M,L,\mu)$ ...
1
vote
3answers
380 views

prequantization on $TM \bigoplus T^*M$

Let $M$ be a pre-symplectic manifold.In recent years several Geometrists are working on $TM \bigoplus T^*M$ which has fascinated the complex and Poisson geometry. In recent decade also Nigel Hitchin ...
5
votes
1answer
198 views

Looking for a special rank 2 vector bundle

Let $E\to C$ be a rank $2$, degree $2g-2$, holomorphic vector bundle over a curve of genus $g$. By Riemann-Roch theorem, $$H^0(E)-H^1(E)= \deg(E)+2.(1-g)=0. $$ Question: For which $g$, there is such ...
10
votes
2answers
1k views

What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...
1
vote
1answer
239 views

Thom-Gysin Sequences and Stratifications

Let $X$ be an affine algebraic variety over $\mathbb{C}$, and let $G$ be a semisimple complex linear algebraic group acting by variety automorphisms with finitely many orbits. The decomposition of $X$ ...
0
votes
0answers
125 views

SLAGs on elliptic curves are only lines?

Let $E=\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ be an alliptic curve. Define a holomorphic 1-form $dz=dx+idy$ and a Kahler form $\omega=dx\wedge dy$. How can one prove that special Lagrangians on $E$ ...
6
votes
1answer
484 views

Darboux like theorem for non-degenerate 3-forms in 6-manifolds

we know Darboux theorem for higher-symplectic geometry is not correct in general, but is there any Darboux like theorem for non-degenerate 3-forms in 6-manifolds?
7
votes
1answer
277 views

How many “elementary” characterizations of twisted SU(2) representation varieties are known?

If $\Sigma_g$ is a genus-$g$ surface, $g \geq 2$, then let $\mathcal{M}(\Sigma_g)$ be its twisted SU(2) representation variety, i.e. $$\mathcal{M}(\Sigma_g) := \{ (A_1, B_1, \ldots, A_g, B_g) \in ...
6
votes
1answer
287 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 ...
0
votes
1answer
208 views

special Lagrangian n-Torus has Tubular neighbourhood?

Let $\imath :T^{n}\rightarrow X$ is a special Lagrangian n-Torus so that $\imath(T^{n})=L$ and all small special Lagrangian deformations of $L$ are flat then why $L$ has Tubular neighbourhood which ...
4
votes
1answer
399 views

Recognizing the stablizer of a degenerate three forms in six dimension

Define $Stab^{+}(\Omega )$={ $\phi \in GL^{+}(V)$ : $\phi^{*}\Omega=\Omega$ }. we say three-form $\Omega\in\wedge^{3}V^{*}$ is non-degenerate , if $i_X\Omega\neq 0$ for all $X\in V$-{0} Let $V\cong ...
1
vote
1answer
152 views

Enumerativity of Gromov-Witten invariants of orbifolds

For smooth Deligne-Mumford stacks, there is a well-defined Gromov-Witten theory, see http://arxiv.org/pdf/math/0103156.pdf and http://arxiv.org/pdf/math/0603151.pdf. Is there some sense, or some ...
10
votes
1answer
780 views

decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$

Do anybody know , why can we write the following decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$ 1) $E_8=(V^{\star}\otimes \wedge^{8}V^{\star})\bigoplus ...
5
votes
2answers
323 views

What information is required for SYZ mirror symmetry?

Let $X$ be a Calabi-Yau threefold. The Strominger-Yau-Zaslow conjecture suggests that $X$ should have a special Lagrangian $T^3$-fibration (when $X$ lies near a large complex structure limit) and a ...
5
votes
1answer
262 views

Questions on how SYZ conjectures is deduced from HMS conjeture.

The Strominge-Yau-Zaslow conjecture is roughly the following. Any Calabi-Yau $m$-manifold $X$ admits a special Lagrangian $T^m$ fibration (maybe at around a special point in its complex moduli space) ...
4
votes
1answer
206 views

Smooth projective toric varieties which are quotients of product of spheres and torii by a free torus action?

My question is just as in the box. Is every smooth projective toric variety diffeomorphic to a quotient of $\prod_i S^{n_i} \times T^k$ (I know torus is a one-sphere but I just wanted to make clear I ...
3
votes
1answer
232 views

Igusa invariants of genus 2 curves as Siegel modular functions?

Hi, Are the Igusa invariants (defined in Igusa's oringal paper) also classical Siegel modular forms? I read from somewhere that $\psi_4=\frac{1}{4}I_4, \quad \psi_6=\frac{1}{8}(I_2I_4-3I_6), \quad ...
5
votes
2answers
829 views

What is geometric intuition of special Lagrangian manifolds?

Let $M$ be (for example) a Calabi-Yau threefold with Kaehler form $\omega$ and holomorphic 3-form $\Omega$. We say that a submanifold $L$ of $M$ is a special Lagrangian submanifold if $L$ is ...
16
votes
3answers
997 views

Why are Gromov-Witten invariants of K3 surfaces trivial?

Why is GW invariants of K3 surfaces are trivial? My naive guess is that GW invariants are deformation invariant and you can always deform your K3 surface to non-projective one, which has no subcomplex ...
6
votes
2answers
273 views

Why is the base of SLAG fibration of CY3 expected to be $S^3$?

The SYZ conjecture roughly says that any Calabi-Yau threefold $X$ has a special Lagrnagian fibration $\pi:X\rightarrow B$ by 3-tous with section and one of its mirror partners $\check{X}$ is obtained ...
5
votes
1answer
393 views

Complex structures on a K3 surface as a hyperkähler manifold

A hyperkähler manifold is a Riemannian manifold of real dimension $4k$ and holonomy group contained in $Sp(k)$. It is known that every hyperkähler manifold has a $2$-sphere $S^{2}$ of complex ...
1
vote
0answers
125 views

Trivialization of holomorphic symplectic 2-form

Let $X$ be a holomorphic symplectic compact manifold with a fixed holomorphic 2-form $\omega$. $\omega$ yields an isomorphism $\phi:T_{X} \rightarrow \Omega_{X}$ via $$ v \mapsto \phi(v)=\omega(v,-). ...
10
votes
4answers
2k views

Geometric invariant theory for geometers

I am trying to learn "Geometric invariant theory" like it was introduced by Mumford. But I do not have a strong background in algebraic geometry since I work in geometric topology and geometry. So ...
0
votes
0answers
164 views

Image of an isotropic manifold under lagrangian correspondence is isotropic?

Is the following statement well known? Let $M,N$ be symplectic (algebraic) manifolds. Let $L \subset M \times N$ be a (smooth) Lagrangian correspondence. For a subset $X \subset M$ we denote ...
7
votes
1answer
403 views

Why is Heegaard Floer Homology defined in terms of Sym$^g\Sigma_g$ instead of Pic$^g\Sigma_g$?

Recall the definition of Heegaard Floer homology: $\Sigma_g$ is a closed surface, and $\{\alpha_1,\ldots,\alpha_g\}$ and $\{\beta_1,\ldots,\beta_g\}$ are sets of attaching circles. Then Heegaard ...
11
votes
2answers
482 views

Finding topological obstructions for a complex manifold to be Kaehler

Well, it is of the "straightforward" questions one may ask. I propose it here to see if someone could tell me more on the recent status of this quite long-standing problem. To initiate, let me give a ...
3
votes
1answer
468 views

Is P^2 important in Kontsevich's recursion formula?

There is a famous recursion formula by Kontsevich to find the number of genus zero degree $d$ curves in $\mathbb{CP}^2$ through $3d-1$ points. My question is the following: Let $S$ be a complex ...
10
votes
3answers
1k views

How far can one get with the Gross-Siebert program?

The Gross-Siebert program is said to be an algebraic analog of the SYZ conjecture and they used toric degeneration to construct a mirror dual of Calabi-Yau varieties. It seems like the singular ...
3
votes
1answer
376 views

Mirror to the dualizing sheaf

I wonder - is there a general characterization of the object in the Fukaya category that is mirror to the dualizing sheaf on the other side? This question has two cases: 1. CY 2. Non-CY In 1. what ...
5
votes
0answers
247 views

Quantum dynamics on varieties: asymptotic quantum trace distance on SHL varieties

The Question Asked Definition: the Second-Hand Lion trace distance $D_k$ Let $\mathcal{M}^{(kk)}_r$ be the set of $k\times k$ complex matrices of rank $\le r$ having unit trace norm. Then the ...
-3
votes
1answer
2k views

Quantum dynamics on varieties and Salmon Prizes

Concluding Progressive Remarks A new finding is Bates and Oeding's preprint "Toward a salmon conjecture" (arXiv:1009.6181), with its reference to the Salmon Prize. The Salmon Prize (photo of the ...
16
votes
4answers
670 views

Can you tell the volume of a symplectic manifold from the Poisson brackets?

Suppose $(X^{2n},\omega)$ is a compact symplectic manifold. Knowing the algebra $C^\infty(X)$ is equivalent to knowing the manifold $X$, and knowing the Poisson bracket ...
5
votes
2answers
762 views

Places to learn about Landau-Ginzburg models

Here is what I know about Landau-Ginzburg models: It is an important player in the story of mirror symmetry. It involves "potentials" which are functions of complex varibles, which have isolated ...
2
votes
1answer
246 views

Hamiltonian Reduction and Affine quotient

Given a smooth affine symplectic variety $V$ with an action of a connected algebraic $G$. If $\mu$ is the moment map, the define the affine quotient to be : $X = \mu^{-1}(0)// G = ...
0
votes
0answers
112 views

Non-regular (Non-coherent) subdivisions of a polygon.

There are many papers and books which study about the regular subdivision of a convex lattice polytope. My question is about "Non"-regular subdivisions of a 2-dimensional convex lattice polygon. I ...
6
votes
0answers
444 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 ...
3
votes
2answers
264 views

Obstructions to being a hyperplane section or a fibre of a Lefschetz pencil

Given a smooth projective variety $X$, when could $X$ fail to be a hyperplane section in some other variety $Y$, or fail to be the fibre of some Lefschetz pencil $\widetilde{Y} \rightarrow ...
5
votes
3answers
539 views

Kähler structure on a complex reductive group

Let $G$ be a complex reductive group, and $K$ a maximal compact subgroup (such that $K_{\mathbb{C}}=G$). By the polar decomposition theorem one has that, as manifolds, $G\cong T^*K$. The inherited ...
5
votes
1answer
302 views

Splitting principle in equivariant cohomology

The following is a weaker version of what is called splitting principle in Appendix C, page 12, see also for a lighter version Brions Eq cohom and eq intersection theory, page 6: Let $G$ be a compact ...
0
votes
1answer
312 views

Pullback of D-Modules

Hi, just a short question on the theory of D-Modules: if one has $p:X\times Y \rightarrow Y$ the projection of two smooth projective complex varieties to the second factor, then what is a ...
12
votes
1answer
508 views

Finite fundamental groups of 3-dimensional Calabi-Yau manifolds

Question. Is there an example of a compact $3$-dimensional Calabi-Yau manifold with finite fundamental group $G$ that does not admit a free action on $S^3$? This question is motivated by the ...