Line bundles on rigid spaces in thev-topology

Abstract

AbstractFor a smooth rigid spaceXover a perfectoid field extensionKof$\mathbb {Q}_p$, we investigate how thev-Picard group of the associated diamond$X^{\diamondsuit }$differs from the analytic Picard group ofX. To this end, we construct a left-exact ‘Hodge–Tate logarithm’ sequence$$\begin{align*}0\to \operatorname{Pic}_{\mathrm{an}}(X)\to \operatorname{Pic}_v(X^{\diamondsuit})\to H^0(X,\Omega_X^1)\{-1\}. \end{align*}$$We deduce some analyticity criteria which have applications top-adic modular forms. For algebraically closedK, we show that the sequence is also right-exact ifXis 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 properX, we use this to construct thep-adic Simpson correspondence of rank one.