Fibers of torus equivariant moment maps

Given a closed (possibly singular) projective variety V with a symplectic structure and a torus action, there is a moment map μ:V→Lie(T)∗. Note that the dimension of T could be much smaller than the dimension of V. How much can we say about the fibers of this moment map μ? Any references? I am most … Read more

Symplectic Hodge Maps and Mirror Symmetry

The notion of Hodge theory for symplectic manifolds seems to be getting more and more attentions in these days. See the series of papers by Yau: http://arxiv.org/abs/1011.1250 http://arxiv.org/abs/1402.0427 If we look now at mirror symmetry, is there any connection between the symplectic Hodge map on one side of the duality and the K\”ahler Hodge on … Read more

Obstructions to symplectically embedding compact manifolds of dimension 44 or higher

It is known in Li’s paper (http://arxiv.org/pdf/0812.4929v1.pdf) that in compact symplectic manifolds (X2n,ω) of dimension at least 2n≥4, an immersed symplectic surface represents a 2-homology class as long as that homology class has positive symplectic area. When 2n≥6, this immersion may even be taken to be an embedding. My question regards finding symplectically embedded submanifolds … Read more

Matsushita theorem on framed variety (X,D)

I have a question about fibrations on Irreducible log holomorphic symplectic manifolds. Lets give some introduction Motivation; A holomorphic symplectic manifold (HSM) is a 2n-dimensional compact K\”ahler manifold X which admits a non-degenerate holomorphic 2-form ω∈H0(X,Ω2). One of the examples is K3 surfaces. The non-degeneracy here means that ω∧n trivialises Ω2n=KX , i.e. c1=0. A … Read more

Maslov class of a diagonal

Let $(M,\omega)$ be a symplectic manifold. Which condition on $M$ guarantees that the diagonal of $(M \times M, (\omega,-\omega))$ has a vanishing Maslov class? $H^1(M,\mathbb{Z})=0$ is enough, but I am looking for some less restrictive condition. Answer AttributionSource : Link , Question Author : Dima Sakurov , Answer Author : Community

Singular symplectic reduction in infinite dimension

In 1991, Sjamaar and Lerman [1] introduced the notion of stratified symplectic spaces. Namely, if M is a symplectic manifold and G a Lie group acting properly (but not necessarily freely) on M and there is a moment map μ:M→g∗ for this action, then they show that the symplectic reduction M//G:=μ−1(0)/G is union a symplectic … Read more

K-stability on Fano fibration

Motivation: Let $\pi:X\to B$ be a holomorphic fibre space. By theorem 1.3 of Kawamata, if the central fibre be of the general type then all the fibres are of the general type see http://arxiv.org/pdf/math/9809091.pdf , this tells you that if the central fibre $X_0$ has Kahler-Einstein metric with negative Ricci curvature, then all the fibres … Read more

Topology of a convergent sequence of stable maps on a symplectic manifold

Let $(M,\omega)$ be a compact symplectic manifold. Let $J$ be a compatible almost complex structure. Let $g$ be the Riemannian metric corresponding to $\omega,J$. Let $f_\nu\colon C_\nu\to M$ be a sequence of $J$-holomorphic curves where $C_\nu$ are smooth closed Riemann surfaces and $f_\nu$ are injective. Assume that this sequence converges to a stable map \$f\colon … Read more

H-principle for smoothing

I’m trying to find algebraic embedding of singular curves into projective varieties that can be smoothed symplectically but not algebraically. It’s not hard (e.g. using the methods in Hartshorne-Hirschowitz “Smoothing algebraic space curve” or Hartshorne’s “Families of Curves in P3 and Zeuthen’s Problem”) to construct such examples for P3 with locally smoothable singularities (even ADE) … Read more

Lagrangian submanifold of Poisson manifolds

Let V be a finite dimensional vector space. Let ψ∈Λ2V be a (possibly degenerate) 2-vector. Then ψ defines a map V∗→V. Let U⊂V denote the image of this map. Then ψ induces a symplectic form σ on U. A subspace Z⊂V is Lagrangian with respect to ψ if Z is a Lagrangian subspace of (U,σ) … Read more