On real moduli spaces over M-curves

TL;DR

This paper investigates the topology of real moduli spaces of rank 2 stable bundles over M-curves with maximal real structures, extending previous calculations of their cohomology to genus 2 cases.

Contribution

It computes the integral cohomology of the real moduli space for genus 2 M-curves, generalizing Thaddeus' work on complex moduli spaces.

Findings

Calculated $H^* ( ext{N}', ext{Z})$ for genus 2 M-curves

Extended Thaddeus' approach to real moduli spaces

Provided new topological insights into real algebraic curves

Abstract

Let $F$ be a genus $g$ curve and $\sigma: F \to F$ a real structure with the maximal possible number of fixed circles. We study the real moduli space $\N' = \Fix (\sigma^{#})$ where $\sigma^{#}: \N \to \N$ is the induced real structure on the moduli space $\N$ of stable holomorphic bundles of rank 2 over $F$ with fixed non-trivial determinant. In particular, we calculate $H^* (\N',\mathbb Z)$ in the case of $g = 2$, generalizing Thaddeus' approach to computing $H^* (\N,\mathbb Z)$.