Pure parts of the mixed Hodge structures of character varieties of indivisible type

TL;DR

This paper proves the purity conjecture for the mixed Hodge structures of certain character varieties associated with punctured spheres and indivisible partitions, linking them to quiver variety cohomology.

Contribution

It establishes the isomorphism between the pure parts of the mixed Hodge structures of character varieties and the cohomology of quiver varieties in the indivisible case.

Findings

Proves the purity conjecture for character varieties of indivisible type.

Shows the pure parts of the mixed Hodge structures are isomorphic to quiver variety cohomology.

Focuses on the case where the surface is the projective line with punctures.

Abstract

We fix integers $k> 0$ and $n>0$. For a $k$-punctured Riemann surface $\Sigma \setminus \{ p_1,\ldots,p_k \}$ and a $k$-tuple $\boldsymbol{\mu}=(\mu^1,\ldots,\mu^k)$ of partitions of $n$, we can define the character variety of type $\boldsymbol{\mu}$. In this paper, we consider the case where $\Sigma=\mathbb{P}^1$ and $\boldsymbol{\mu}$ is indivisible (i.e. $\mathrm{g.c.d.}(\boldsymbol{\mu})=1$). For the case, we prove the purity conjecture due to Hausel, that is, the pure parts of the mixed Hodge structures of the character variety is isomorphic to the ordinary rational cohomology groups of the quiver variety of type $\boldsymbol{\mu}$.