Maximality of moduli spaces of vector bundles on curves

TL;DR

This paper establishes that the maximality of moduli spaces of semistable vector bundles over real algebraic curves is equivalent to the maximality of the base curve, introducing a new family of maximal varieties with large dimensions.

Contribution

It proves a new criterion linking the maximality of moduli spaces to the base curve's maximality, and introduces the concept of Hodge-expressivity as a stronger property.

Findings

Moduli spaces are maximal if and only if the base curve is maximal.

Introduces Hodge-expressivity, a property stronger than maximality.

Provides examples of large-dimensional maximal varieties.

Abstract

We prove that moduli spaces of semistable vector bundles of coprime rank and degree over a non-singular real projective curve are maximal real algebraic varieties if and only if the base curve itself is maximal. This provides a new family of maximal varieties, with members of arbitrarily large dimension. We prove the result by comparing the Betti numbers of the real locus to the Hodge numbers of the complex locus and showing that moduli spaces of vector bundles over a maximal curve actually satisfy a property which is stronger than maximality and that we call Hodge-expressivity. We also give a brief account on other varieties for which this property was already known.