Line bundles on rigid spaces in the $v$-topology

TL;DR

This paper explores the relationship between the $v$-Picard group and the analytic Picard group of smooth rigid spaces over perfectoid fields, introducing a Hodge--Tate logarithm sequence with applications to $p$-adic modular forms and the $p$-adic Simpson correspondence.

Contribution

It constructs a Hodge--Tate logarithm sequence relating $v$-line bundles and analytic line bundles, revealing new insights into $p$-adic geometry and Higgs bundles.

Findings

The sequence is exact for proper or one-dimensional spaces over algebraically closed fields.

For affine space, the image of the logarithm is exactly the closed differentials.

Up to splitting, $v$-line bundles can be viewed as Higgs bundles.

Abstract

For a smooth rigid space $X$ over a perfectoid field extension $K$ of $\mathbb Q_p$, we investigate how the $v$-Picard group of the associated diamond $X^\diamondsuit$ differs from the analytic Picard group of $X$. To this end, we construct a left-exact "Hodge--Tate logarithm" sequence \[0\to \mathrm{Pic}_{\mathrm{an}}(X)\to \mathrm{Pic}_v(X^\diamondsuit)\to H^0(X,\Omega_X^1)\{-1\}.\] We deduce some analyticity criteria which have applications to $p$-adic modular forms. For algebraically closed $K$, we show that the sequence is also right-exact if $X$ is proper or one-dimensional. In contrast, we show that for the affine space $\mathbb A^n$, the image of the Hodge--Tate logarithm consists precisely of the closed differentials. It follows that up to a splitting, $v$-line bundles may be interpreted as Higgs bundles. For proper $X$, we use this to construct the $p$-adic Simpson correspondence of rank one.