L^2-Cohomology and complete Hamiltonian manifolds

TL;DR

This paper extends Frankel's theorem to certain non-compact K"ahler manifolds by leveraging Hodge theory, providing new conditions under which Hamiltonian actions with fixed points occur.

Contribution

It introduces a metatheorem that generalizes Frankel's theorem to non-compact manifolds where Hodge theory applies, with concrete examples.

Findings

Frankel's theorem holds on non-compact manifolds under Hodge theory conditions

The paper identifies specific situations where the assumptions are satisfied

Provides a framework for Hamiltonian actions on non-compact K"ahler manifolds

Abstract

A classical theorem of Frankel for compact K\"ahler manifolds states that a K\"ahler S^1-action is Hamiltonian if and only if it has fixed points. We prove a metatheorem which says that when Hodge theory holds on non-compact manifolds, then Frankel's theorem still holds. Finally, we present several concrete situations in which the assumptions of the metatheorem hold.