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LOGARITHMIC DE RHAM COMPARISON FOR OPEN RIGID SPACES

Part of: Compact analytic spaces Arithmetic problems. Diophantine geometry Cycles and subschemes

Published online by Cambridge University Press:  25 September 2019

SHIZHANG LI  and
XUANYU PAN
SHIZHANG LI
Affiliation:
Department of Mathematics, Columbia University, MC 4406, 2990 Broadway, New York, NY 10027, USA; shanbei@math.columbia.edu
XUANYU PAN
Affiliation:
Institute of Mathematics, AMSS, Chinese Academy of Sciences, 55 ZhongGuanCun East Road, Beijing 100190, China Max Planck Institute for Mathematics, Vivatsgasse 7, Bonn, 53111, Germany; pan@amss.ac.cn, panxuanyu@mpim-bonn.mpg.de

Abstract

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In this note, we prove the logarithmic $p$-adic comparison theorem for open rigid analytic varieties. We prove that a smooth rigid analytic variety with a strict simple normal crossing divisor is locally $K(\unicode[STIX]{x1D70B},1)$ (in a certain sense) with respect to $\mathbb{F}_{p}$-local systems and ramified coverings along the divisor. We follow Scholze’s method to produce a pro-version of the Faltings site and use this site to prove a primitive comparison theorem in our setting. After introducing period sheaves in our setting, we prove aforesaid comparison theorem.

MSC classification

Primary: 14G22: Rigid analytic geometry 14C30: Transcendental methods, Hodge theory , Hodge conjecture
Secondary: 14G20: Local ground fields 32J27: Compact Kähler manifolds
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e32
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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