Morse theory and stable pairs

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Title of host publicationVariation problems in differential geometry
DatePublished - 1 Dec 2011
Pages142-180
Number of pages39
PublisherCambridge University Press
EditorsRoger Bielawski, Kevin Houston, Martin Speight
Volume394
Original languageEnglish
ISBN (Print)9780521282741

Abstract

We study the Morse theory of the Yang-Mills-Higgs functional on the space of pairs (A; \Phi), where A is a unitary connection on a rank 2 hermitian vector bundle over a compact Riemann surface, and \Phi is a holomorphic section of (E; d_A). We prove that a certain explicitly defined substratification of the Morse stratification is perfect in the sense of G-equivariant cohomology, where G denotes the unitary gauge group. As a consequence, Kirwan surjectivity holds for pairs. It also follows that the twist embedding into higher degree induces a surjection on equivariant cohomology. This may be interpreted as a rank 2 version of the analogous statement for symmetric products of Riemann surfaces. Finally, we compute the G-equivariant Poincare polynomial of the space of semistable pairs. In particular, we recover an earlier result of Thaddeus. The analysis provides an interpretation of the Thaddeus flips in terms of a variation of Morse functions.

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