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 analogue of the Carrell–Liberman theorem. As an application, this confirms a conjecture raised by Battaglia–Zaffran on the basic Hodge numbers of symplectic toric quasifolds. Our methods also allow us to present a symplectic approach to the calculation of the basic Betti numbers of symplectic toric quasifolds.

Cohomological localization for transverse Lie algebra actions on Riemannian foliations

“…Naturally one wonders if the Kirwan surjectivity and injectivity theorem could also be generalized to Hamiltonian actions on pre-symplectic manifolds. In [LY19] and [LY23], using symplectic Hodge theoretic techniques, Lin and Yang had the Kirwan injectivity theorem generalized to the case of a Hamiltonian torus action on a pre-symplectic manifold that satisfies the transverse Hard Lefschetz property.…”

“…In view of these results, our general results will also apply to a large class of Hamiltonian torus actions on symplectic quasifolds. Indeed, the Kirwan injectivity theorem has already been used in [LY23] to derive an explicit combinatorial formula for the basic Betti numbers and basic Hodge numbers of toric quasifolds.…”