2010/02/16 by Richard A. Wentworth, Wentworth, Richard A., Graeme Wilkin +1
Mathematics · #14D20 #53D20 #58E15 #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:14D20 #msc:53D20 #msc:58E15
paper · pdf · doi:10.48550/arxiv.1002.3124
34 pages, 1 figure. Corrected proof of Lemma 3.12
arxiv created 2010/06/26 · arxiv updated 2010/06/29
We study the Morse theory of the Yang-Mills-Higgs functional on the space of pairs (A,Φ), where A is a unitary connection on a rank 2 hermitian vector bundle over a compact Riemann surface, and Φ is a holomorphic section of (E, dA"). 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 Poincaré polynomial of the space of τ-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.