Equivariant formality of Hamiltonian transversely symplectic foliations

Abstract

Consider the Hamiltonian action of a compact connected Lie group on a transversely symplectic foliation which satisfies the transverse hard Lefschetz property. We establish an equivariant formality theorem and an equivariant symplectic dδ-lemma in this setting. As an application, we show that if the foliation is also Riemannian, then there exists a natural formal Frobenius manifold structure on the equivariant basic cohomology of the foliation.

Basic Kirwan injectivity and its applications

“…Proof. The first assertion that F is Riemannian is shown in [14,Lemma 5.1]. We need only to show that F is a Killing foliation, and that the action of G is foliate.…”

“…They discovered that when the action is clean, components of a moment map must be Morse-Bott functions, and extended Atiyah-Guillemin-Sternberg-Kirwan convexity theorem to clean Hamiltonian actions. Lin and Yang [14] studied Hamiltonian actions on a transversely symplectic foliation from Hodge theoretic viewpoint, and established in this setup the equivariant formality result for the equivariant basic cohomology. In the current paper, building on the equivariant formality result in [14] and the Borel localization result in [15], we establish the following foliated version of the Kirwan injectivity theorem in symplectic geometry.…”

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.…”