The symplectic vortex equations and invariants of Hamiltonian group actions

2001/11/15 by Kai Cieliebak, Ana Rita Gaio, A. Rita Gaio +6 · 2 citations
Mathematics · #37Jxx #53Dxx #Advanced Algebra and Geometry #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Symplectic Geometry (math.SG) #math.SG #msc:37Jxx #msc:53Dxx

paper · pdf · doi:10.48550/arxiv.math/0111176

87 pages

openalex publication_date 2001/11/15 · arxiv created 2002/10/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we define invariants of Hamiltonian group actions for central regular values of the moment map. The key hypotheses are that the moment map is proper and that the ambient manifold is symplectically aspherical. The invariants are based on the symplectic vortex equations. Applications include an existence theorem for relative periodic orbits, a computation for circle actions on a complex vector space, and a theorem about the relation between the invariants introduced here and the Seiberg--Witten invariants of a product of a Riemann surface with a two-sphere.

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