The Hodge--Poincar\'e polynomial of the moduli spaces of stable vector bundles over an algebraic curve

TL;DR

This paper derives formulas for the Hodge--Poincaré polynomial of moduli spaces of stable vector bundles over algebraic curves, especially when rank and degree are not coprime, with explicit results for rank 2 and even degree.

Contribution

It provides new explicit formulae for the Hodge--Poincaré polynomial of moduli spaces of stable vector bundles in non-coprime cases, extending previous results.

Findings

Formulas for the Hodge--Poincaré series of quotients under reductive group actions.

Explicit computation for rank 2 bundles with even degree.

Application to moduli spaces of stable vector bundles.

Abstract

Let X be a nonsingular complex projective variety that is acted on by a reductive group $G$ and such that $X^{ss} \neq X_{(0)}^{s}\neq \emptyset$. We give formulae for the Hodge--Poincar\'e series of the quotient $X_{(0)}^s/G$. We use these computations to obtain the corresponding formulae for the Hodge--Poincar\'e polynomial of the moduli space of properly stable vector bundles when the rank and the degree are not coprime. We compute explicitly the case in which the rank equals 2 and the degree is even.