A support theorem for the Hitchin fibration: the case of $SL_n$

TL;DR

This paper proves that the direct image complex for the $SL_n$ Hitchin fibration is determined by its restriction to the elliptic locus, extending known results for $GL_n$, and establishes properties of the Prym group scheme.

Contribution

It establishes a support theorem for the $SL_n$ Hitchin fibration, showing the complex is determined by its restriction to the elliptic locus, and proves polarizability of the Tate module.

Findings

The direct image complex is determined by its restriction to the elliptic locus.

The Tate module of the Prym group scheme is polarizable.

Proved $ ho$-regularity for certain weak abelian fibrations.

Abstract

We prove that the direct image complex for the $D$-twisted $SL_n$ Hitchin fibration is determined by its restriction to the elliptic locus, where the spectral curves are integral. The analogous result for $GL_n$ is due to P.-H. Chaudouard and G. Laumon. Along the way, we prove that the Tate module of the relative Prym group scheme is polarizable, and we also prove $\delta$-regularity results for some auxiliary weak abelian fibrations.