Hodge symmetry for rigid varieties via log hard Lefschetz

TL;DR

This paper proves Hodge symmetry for certain smooth proper rigid analytic spaces over non-archimedean fields, utilizing cases of Kato's log hard Lefschetz conjecture, and extends previous results without moduli space techniques.

Contribution

It establishes Hodge symmetry under new conditions by proving cases of Kato's log hard Lefschetz conjecture for specific log schemes.

Findings

Hodge symmetry holds for spaces with projective reduction in specified cases.

Kato's log hard Lefschetz conjecture is proven for H^1 and combinatorial type log schemes.

A new proof of Hodge symmetry for H^1 avoids moduli space methods.

Abstract

Motivated by a question of Hansen and Li, we show that a smooth and proper rigid analytic space $X$ with projective reduction satisfies Hodge symmetry in the following situations: (1) the base non-archimedean field $K$ is of residue characteristic zero, (2) $K$ is $p$-adic and $X$ has good ordinary reduction, (3) $K$ is $p$-adic and $X$ has "combinatorial reduction."' We also reprove a version of their result, Hodge symmetry for $H^1$, without the use of moduli spaces of semistable sheaves. All of this relies on cases of Kato's log hard Lefschetz conjecture, which we prove for $H^1$ and for log schemes of "combinatorial type."