论文标题
几何$ p $ -adic simpson在等级一
A geometric $p$-adic Simpson correspondence in rank one
论文作者
论文摘要
For any smooth proper rigid space $X$ over a complete algebraically closed extension $K$ of $\mathbb Q_p$ we give a geometrisation of the $p$-adic Simpson correspondence of rank one in terms of analytic moduli spaces: The $p$-adic character variety is canonically an étale twist of the moduli space of topological torsion Higgs line bundles over the Hitchin base.这也消除了指数的选择。关键的想法是将双方与$ V $ - 线捆绑包的模量空间联系起来：我们开发了$ v $ -sheaves的拓扑扭转子藏理论，并将其应用于diamantine $ v $ v $ -picard -picard functor of Arxiv：2103.16557。 作为这种几何通信的应用，我们研究了Faltings提出的$ p $ - 亚洲非亚伯式霍奇理论中的一个主要开放问题，即，希格斯捆绑包将对应于$ p $ - adic-adic simpson通讯下的连续表示。 We answer this question in rank one by describing the essential image of the continuous characters $π^{\acute{e}t}_1(X)\to K^\times$ in terms of moduli spaces: For projective $X$ over $K=\mathbb C_p$, it is given by Higgs line bundles with vanishing Chern classes like in complex geometry, but in general we show that the correct condition is the strictly更强有力的假设是，基础线束是拓扑组$ \ mathrm {pic}（x）$中的拓扑扭转元素。
For any smooth proper rigid space $X$ over a complete algebraically closed extension $K$ of $\mathbb Q_p$ we give a geometrisation of the $p$-adic Simpson correspondence of rank one in terms of analytic moduli spaces: The $p$-adic character variety is canonically an étale twist of the moduli space of topological torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of $v$-line bundles: We develop a theory of topological torsion subsheaves of $v$-sheaves and apply this to the diamantine $v$-Picard functor of arXiv:2103.16557. As an application of this geometric correspondence, we study a major open question in $p$-adic non-abelian Hodge theory raised by Faltings, namely which Higgs bundles will correspond to continuous representations under the $p$-adic Simpson correspondence. We answer this question in rank one by describing the essential image of the continuous characters $π^{\acute{e}t}_1(X)\to K^\times$ in terms of moduli spaces: For projective $X$ over $K=\mathbb C_p$, it is given by Higgs line bundles with vanishing Chern classes like in complex geometry, but in general we show that the correct condition is the strictly stronger assumption that the underlying line bundle is a topological torsion element in the topological group $\mathrm{Pic}(X)$.
