Hodge symmetry for rigid varieties via $\log$ hard Lefschetz

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.”

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This work is a part of the project KAPIBARA supported by the funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 802787).

Received 22 January 2021

Accepted 4 July 2021

Published 21 June 2023