Wentworth, Richard and Wilkin, Graeme Peter Desmond (2011) Morse theory and stable pairs. In: Bielawski, Roger, Houston, Kevin and Speight, Martin, (eds.) Variation problems in differential geometry. Cambridge University Press , pp. 142-180.
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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Copyright, Publisher and Additional Information: | This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details. |
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Institution: | The University of York |
Academic Units: | The University of York > Faculty of Sciences (York) > Mathematics (York) |
Depositing User: | Pure (York) |
Date Deposited: | 18 Sep 2019 14:30 |
Last Modified: | 06 Dec 2023 10:08 |
Status: | Published |
Publisher: | Cambridge University Press |
Refereed: | No |
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