Morse theory and stable pairs

TL;DR

This paper analyzes the Morse theory of the Yang-Mills-Higgs functional on pairs over Riemann surfaces, establishing perfect stratifications, cohomology surjections, and computing equivariant Poincaré polynomials, thus extending known results to rank 2 cases.

Contribution

It introduces a perfect Morse stratification for rank 2 pairs, proves Kirwan surjectivity in this context, and computes the equivariant Poincaré polynomial, extending previous work on symmetric products.

Findings

Perfect Morse stratification for rank 2 pairs.

Kirwan surjectivity for the space of pairs.

Computed equivariant Poincaré polynomial, recovering Thaddeus's result.

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 Poincar\'e 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.