Morse theory and stable pairs

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, 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 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.