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Tame circle actions

Authors: Susan Tolman and Jordan Watts
Journal: Trans. Amer. Math. Soc. 369 (2017), 7443-7467
MSC (2010): Primary 53D20; Secondary 53D05, 53B35
Published electronically: June 13, 2017
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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

Jordan Watts
Affiliation: Department of Mathematics, University of Colorado Boulder, Boulder, Colorado 80309

Keywords: Tamed symplectic form, symplectic reduction, blow-up, symplectic cutting, holomorphic action, Hamiltonian action, K\"ahler manifold, moment map
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.
Article copyright: © Copyright 2017 American Mathematical Society

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