Tame Circle Actions

TL;DR

This paper extends key theories of Hamiltonian circle actions to complex manifolds with tamed symplectic forms, enabling new constructions and answering open questions in symplectic geometry.

Contribution

It generalizes standard Hamiltonian circle action theories to tamed symplectic forms, facilitating novel constructions and solutions to open problems.

Findings

Theory extends to tamed symplectic forms with isolated fixed points

Constructs a non-Hamiltonian symplectic circle action with 32 fixed points

Provides new insights into moment map behavior in complex manifolds

Abstract

In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the K\"ahler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. However, we also believe that the setting we work in is intrinsically interesting, and elucidates the key role played by the following fact: the moment image of $e^t \cdot x$ increases as $t \in \mathbb{R}$ increases.