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

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 $X^{♢}$ differs from the analytic Picard group of $X$. To this end, we construct a left-exact "Hodge--Tate logarithm" sequence \[0\to \mathrm{Pic}_{\mathrm{an}}(X)\to \mathrm{Pic}_v(X^\diamondsuit)\to H^0(X,Ω_X^1)\{-1\}.\] 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.

added section on analyticity criteria, generalised setting to any perfectoid base field

Cited by in corpus (3)

• v-vector bundles on $p$-adic fields and Sen theory via the Hodge-Tate stack
• The small $p$-adic Simpson correspondence in terms of moduli spaces
• On the $v$-Picard group of Stein spaces