A p$p$-adic Simpson correspondence for smooth proper rigid varieties by Ben Heuer ISBN 101007/S00222025013214 instant download

A p$p$-adic Simpson correspondence for smooth proper rigid varieties by Ben Heuer ISBN 101007/S00222025013214 instant download

Abstract

For any smooth proper rigid analytic space X over a complete algebraically closedextension of Qp, we construct a p-adic Simpson correspondence: an equivalenceof categories between vector bundles on Scholze’s pro-étale site of X and Higgsbundles on X. This generalises a result of Faltings from smooth projective curves toany higher dimension, and further to the rigid analytic setup. The strategy is new, andis based on the study of rigid analytic moduli spaces of pro-étale invertible sheaveson spectral varieties.Mathematics Subject Classification 14G22 · 14G45 · 14D221 Introduction1.1 Main resultLet K be a complete algebraically closed extension of Qp. The goal of this article isto prove the following global p-adic Simpson correspondence.Theorem 1.1 (Theorem 5.1) Let X be a smooth proper rigid space over K. Thenchoices of a B+dR/ξ 2-lift X of X and of an exponential Exp for K induce an exacttensor equivalenceSX,Exp :{︁pro-étale vector bundles on X}︁ ∼→ {︁Higgs bundles on X}︁−which is natural in the datum of the pair (X,X).Here an exponential for K is a continuous splitting of the p-adic logarithmlog: 1 + mK → K, and the two sides of the are defined as follows: