## Tame circle actions

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- by Susan Tolman and Jordan Watts PDF
- Trans. Amer. Math. Soc.
**369**(2017), 7443-7467 Request permission

## 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 $e^t \cdot x$ increases as $t \in \mathbb {R}$ increases.

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## Additional Information

**Susan Tolman**- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
- Email: stolman@math.uiuc.edu
**Jordan Watts**- Affiliation: Department of Mathematics, University of Colorado Boulder, Boulder, Colorado 80309
- MR Author ID: 901837
- Email: jordan.watts@colorado.edu
- Received by editor(s): September 29, 2015
- Received by editor(s) in revised form: June 17, 2016
- Published electronically: June 13, 2017
- Additional Notes: The first author was partially supported by National Science Foundation Grant DMS-1206365

The second author thanks the University of Illinois at Urbana-Champaign for their support. Moreover, this manuscript was significantly improved by suggestions from an anonymous referee; the authors extend their thanks. - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 7443-7467 - MSC (2010): Primary 53D20; Secondary 53D05, 53B35
- DOI: https://doi.org/10.1090/tran/7113
- MathSciNet review: 3683114