Morse theory and stable pairs

Richard Wentworth; Graeme Peter Desmond Wilkin

Morse theory and stable pairs

Richard Wentworth, Graeme Peter Desmond Wilkin

Mathematics

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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.

Original language	English
Title of host publication	Variation problems in differential geometry
Subtitle of host publication	London Mathematical Society Lecture Note Series
Editors	Roger Bielawski, Kevin Houston, Martin Speight
Publisher	Cambridge University Press
Chapter	8
Pages	142-180
Number of pages	39
Volume	394
ISBN (Print)	9780521282741
Publication status	Published - 1 Dec 2011

Bibliographical note

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Wentworth Wilkin Stable Pairs
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Cite this

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title = "Morse theory and stable pairs",

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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N2 - 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.

AB - 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.

M3 - Chapter

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EP - 180

BT - Variation problems in differential geometry

A2 - Bielawski, Roger

A2 - Houston, Kevin

A2 - Speight, Martin

PB - Cambridge University Press

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