Let $Ω$ be a non-singular syplectic form on some vector space $V$. Denote by $S^{n}_{k}(Ω)$ the set of all $k$-dimensional planes $s$ in $V$ such that the restriction of $Ω$ onto $s$ is singular. For the cases when $k=2,n-2$ a simple geometric characteristic of $S^{n}_{k}(Ω)$ will be described.
Removal of singularities and Gromov compactness for symplectic vortices
证明了紧致Hamilton流形中能量一致有界的辛涡旋的规范等价类的模空间允许多稳涡旋的Gromov紧化。这扩展了Mundet i Riera和田的结果为圆行动的情况下，任意紧李群。我们的论点依赖于一个先验估计的旋涡，使我们能够应用技术所使用的McDuff和Salamon在他们的证明格罗莫夫紧pseudoholomorphic曲线。作为中间结果，我们证明了一个可去奇点定理。
Singular oscillatory integrals in equivariant cohomology. Residue formulae for basic differential forms on general symplectic manifolds
设$M$是辛流形，$G$是作用在$M$上的连通紧李群.本文研究了用基本微分形式表示的$M$的等变上同调，并利用奇点的分解和驻相原理，通过某些留数公式将其与Marsden-Weinstein约化空间的上同调联系起来.当$ M $是一个紧的辛流形或$G$ -流形的切丛时，Jeffrey，Kirwan等人对一般等变闭形式和Ramacher对基本微分形式分别导出了类似的留数公式.
We construct a family version of symplectic Floer cohomology for magnetic cotangent bundles, without any restrictions on the magnetic form, using the dissipative method for compactness introduced in \cite{Groman2015}. As an application, we deduce that if $N$ is a closed manifold and $ σ$ is a magnetic form that is not weakly exact, then the $ π_1$-sensitive Hofer-Zehnder capacity of any compact set in the magnetic cotangent bundle determined by $ σ$ is finite.
Symplectic forms on the space of embedded symplectic surfaces and their reductions
Let (M,ω) be a symplectic manifold , and (Σ,σ) a closed connected symplectic 2-manifold. We construct a weakly symplectic form {ω^{D}}_{(Σ, σ)} on the space of immersions Σ \to M that is a special case of Donaldson's form. We show that the restriction of {ω^{D}}_{(Σ,σ)} to any orbit of the group of Hamiltonian symplectomorphisms through a symplectic embedding (Σ,σ) \to (M,ω) descends to a weakly symplectic form ω^ D_{\red} on the quotient by Sympl(Σ,σ), and that the obtained symplectic space is a symplectic quotient of the subspace of symplectic embeddings S_{e} (Σ,σ) with respect to the Sympl…
We set up a topological framework for degenerations of symplectic manifolds into singular spaces paying a special attention to the behavior of Lagrangian manifolds and their (holomorphic) membranes. We show that degenerations into singular toric varieties provide a source of exotic Lagrangian tori.
We consider the canonical contact structures on links of rational surface singularities with reduced fundamental cycle. These singularities can be characterized by their dual resolution graphs: the graph is a tree, and the weight of each vertex is no greater than its negative valency. In a joint work with Starkston, we previously showed that if the weight of each vertex in the graph is at most -5, the contact structure has a unique symplectic filling (up to symplectic deformation and blow-up). The proof was based on a symplectic analog of de Jong-van Straten's description of smoothings of thes…
In this work, we study some convex cones associated to isotropic representations of symmetric spaces. We explain the inequalities that describe them by means of cohomological conditions. In particular, we study the singular Horn cone which is the counterpart of the classical Horn cone, where the eigenvalues of Hermitian square matrices are replaced by the singular values of rectangular matrices.
We prove a theorem on singular symplectic cotangent bundle reduction in the Fréchet setting and apply it to Yang-Mills-Higgs theory with special emphasis on the Higgs sector of the Glashow-Weinberg-Salam model. For the latter model we give a detailed description of the reduced phase space and show that the singular structure is encoded in a finite-dimensional Lie group action.
In the present paper, we consider a smooth $C^\infty$ symplectic classification of Lagrangian fibrations near cusp singularities, parabolic orbits and cuspidal tori. We show that for these singularities as well as for an arrangement of singularities known as a flap, which arises in the integrable subcritical Hamiltonian Hopf bifurcation, the action variables form a complete set of $C^\infty$ symplectic invariants. We also give a symplectic classification for parabolic orbits in the real-analytic case. Namely, we prove that a complete symplectic invariant in this case is given by a real-analyti…
