Line bundles on rigid spaces in the<i>v</i>-topology
Ben Heuer
Abstract
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 $$\begin{align*}0\to \operatorname{Pic}_{\mathrm{an}}(X)\to \operatorname{Pic}_v(X^{\diamondsuit})\to H^0(X,\Omega_X^1)\{-1\}. \end{align*}$$ We deduce some analyticity criteria which have applications 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.