论文标题
$ V $ - 学术中的刚性空间上的线束
Line bundles on rigid spaces in the $v$-topology
论文作者
论文摘要
对于完美的固体空间$ x $，$ \ mathbb q_p $上的$ k $ k $ k $ k $ k $ k $ of MATHBB q_p $，我们调查了相关钻石$ x^\ diamondsuit $的$ v $ - picard group与$ x $的分析PICARD组不同。为此，我们构造了一个左侧的“ hodge- tate对数”序列\ [0 \ to \ mathrm {pic} _ {\ mathrm {an}}}（x）\ to \ to \ mathrm {pic} _v（pic} _v（x^\ diamondssuit）\ to h^to h^0（x^\ diamond） 我们推断出一些分析性标准，这些标准具有$ p $ - ad的模块化表格。 对于代数关闭的$ k $，我们表明，如果$ x $适当或一维，则该序列也是正确的。相比之下，我们表明，对于仿射空间$ \ mathbb a^n $，hodge-tate Googarithm的图像恰好由封闭的差异组成。 随之而来的是，可以将$ v $ - 线束的分裂解释为希格斯捆绑包。对于适当的$ x $，我们使用它来构建排名第一的$ p $ adic simpson通信。
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 \[0\to \mathrm{Pic}_{\mathrm{an}}(X)\to \mathrm{Pic}_v(X^\diamondsuit)\to H^0(X,Ω_X^1)\{-1\}.\] 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.
