Basic Kirwan injectivity and its applications

Abstract

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 Kähler 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.

Dolbeault cohomology of complex manifolds with torus action

“…Using the representation of Z K as an intersection of quadrics, Ishida showed in [12,Theorem 8.1] that all generators of the Dolbeault cohomology ring are of type (1,1). Recently, Lin and Yang proved a more general result [16,Theorem 5.2] describing the Dolbeault cohomology of transverse Kähler foliations admiting a Hamiltonian torus action.…”