Stable pairs and Morse theory

R. Wentworth and G. Wilkin

We study the Morse theory of the Yang-Mills-Higgs functional on the space of pairs (A, s), where A is a unitary connection on a rank 2 hermitian vector bundle over a compact Riemann surface, and s is a holomorphic section of E. 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 tau-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.