Morse theory and stable pairs

Richard Wentworth, Graeme Peter Desmond Wilkin

Research output: Chapter in Book/Report/Conference proceedingChapter


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.
Original languageEnglish
Title of host publicationVariation problems in differential geometry
Subtitle of host publicationLondon Mathematical Society Lecture Note Series
EditorsRoger Bielawski, Kevin Houston, Martin Speight
PublisherCambridge University Press
Number of pages39
ISBN (Print)9780521282741
Publication statusPublished - 1 Dec 2011

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