Basic Kirwan injectivity and its applications

Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem, and use it to study Hamiltonian torus actions on transversely K\"ahler foliations. Among other things, we prove a foliated version of the Carrell-Liberman theorem. As an immediate consequence, this confirms a conjecture raised by Battaglia and Zaffran on the basic Hodge numbers of symplectic toric quasifolds. As an aside, we also present a symplectic approach to the calculation of basic Betti numbers of symplectic toric quasifolds.Comment: 18 pages, comments welcom

The convexity package for Hamiltonian actions on conformal symplectic manifolds

Consider a Hamiltonian action of a compact connected Lie group on a conformal symplectic manifold. We prove a convexity theorem for the moment map under the assumption that the action is of Lee type, which establishes an analog of Kirwan's convexity theorem in conformal symplectic geometry.Comment: 31 pages, 1 figure. Appendix on conformal presymplectic manifolds added. Minor mistakes correcte

Symplectic structures on stratified pseudomanifolds

The purpose of this paper is to investigate the definition of symplectic structure on a smooth stratified pseudomanifold in the framework of local \C^{\infty}-ringed space theory. We introduce a sheaf-theoretic definition of symplectic form and cohomologically symplectic structure on smooth stratified pseudomanifolds. In particular, we give an indirect definition of symplectic form on the quotient space of a smooth $G$-stratified pseudomanifold. Based on the structure theorem of singular symplectic quotients by Sjamaar--Lerman, we show that the singular reduced space $M_{0}=\mu^{-1}(0)/G$ of a symplectic Hamiltonian $G$-manifold $(M,\omega,G,\mu)$ admits a natural (indirect) symplectic form and a unique cohomologically symplectic structure.Comment: The local potentials of the induced Kahler metric on the Kahler quotient are continuous in general. This was overlooked in the proof of Theorem 1 in version2. The definition is corrected. Comments are welcome, 40 page