Geometry of the logarithmic Hodge moduli space (with an Appendix joint with Siqing Zhang)

TL;DR

This paper establishes the smoothness and geometric properties of the logarithmic Hodge moduli space over various bases, and explores the implications for etale cohomology and perverse filtrations, advancing understanding of these moduli spaces in algebraic geometry.

Contribution

It proves the smoothness and geometric integrality of the logarithmic Hodge moduli space over arbitrary bases and fields, and derives cohomological isomorphisms and filtrations.

Findings

Smoothness over the affine line for the moduli space

Geometric integrality when the base is a field

Specialization isomorphisms in etale cohomology

Abstract

We show the smoothness over the affine line of the Hodge moduli space of logarithmic t-connections of coprime rank and degree on a smooth projective curve with geometrically integral fibers over an arbitrary Noetherian base. When the base is a field, we also prove that the Hodge moduli space is geometrically integral. Along the way, we prove the same results for the corresponding moduli spaces of logarithmic Higgs bundles and of logarithmic connections. We use smoothness to derive specialization isomorphisms on the etale cohomology rings of these moduli spaces; this includes the special case when the base is of mixed characteristic. In the special case where the base is a separably closed field of positive characteristic, we show that these isomorphisms are filtered isomorphisms for the perverse filtrations associated with the corresponding Hitchin-type morphisms.