Tame circle actions

Susan Tolman, Jordan Watts

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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 Kähler 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 et · x increases as t ∈ ℝ increases.

Original languageEnglish
Pages (from-to)7443-7467
Number of pages25
JournalTransactions of the American Mathematical Society
Volume369
Issue number10
DOIs
StatePublished - 2017

Keywords

  • Blow-up
  • Hamiltonian action
  • Holomorphic action
  • Key words and phrases
  • Kähler manifold
  • Moment map
  • Symplectic cutting
  • Symplectic reduction
  • Tamed symplectic form

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