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Convexity package for momentum maps on contact manifolds

River Chiang and Yael Karshon

Algebraic & Geometric Topology 10 (2010) 925–977

Let a torus T act effectively on a compact connected cooriented contact manifold, and let Ψ be the natural momentum map on the symplectization. We prove that, if dimT is bigger than 2, the union of the origin with the image of Ψ is a convex polyhedral cone, the nonzero level sets of Ψ are connected (while the zero level set can be disconnected), and the momentum map is open as a map to its image. This answers a question posed by Eugene Lerman, who proved similar results when the zero level set is empty. We also analyze examples with dimT 2.

momentum map, contact manifold, torus action, convexity
Mathematical Subject Classification 2000
Primary: 53D10, 53D20
Secondary: 52B99
Received: 5 November 2009
Revised: 25 February 2010
Accepted: 2 March 2010
Published: 17 April 2010
River Chiang
Department of Mathematics
National Cheng Kung University
Tainan 701
Yael Karshon
Department of Mathematics
University of Toronto
Toronto, Ontario M5S 2E4