\(p\)-adic period domains and toroidal partial compactifications. I
type: Item
instance of: scholarly article
title: \(p\)-adic period domains and toroidal partial compactifications. I
zbMATH Open document ID: 1238.14015
DOI: 10.1215/21562261-1299900
author: Kazuya Kato
published in: Kyoto Journal of Mathematics
publication date: 2011-10-21T00:00:00Z
review text: The goal of this paper is to study the degeneration of \(p\)-adic Hodge structures, by giving certain partial compactifications of \(p\)-adic period domains. The construction is based on analogy with the degeneration of complex Hodge structures, as studied by the author in a series of joint work with \textit{C. Nakayama} and \textit{S. Usui} [Proc. Japan Acad. Ser. A Math. Sci. 86, 107--112 (2010; Zbl 1209.14008); Kyoto J. Math. 51, No. 1, 149--261 (2011; Zbl 1233.14007); in: Algebraic analysis and around in honor of Professor Masaki Kashiwara's 60th birthday. Tokyo: Mathematical Society of Japan. Advanced Studies in Pure Mathematics 54, 187-222 (2009; Zbl 1179.14008)]. (The construction considered in this paper is closer to the first aforementioned paper.) The author first explains four examples of compactifications of complex period domains of mixed Hodge structures: elliptic curves, upper half plane, universal elliptic curves, and dilog functions. Then the author constructs the analogous compactifications of \(p\)-adic period domains of mixed Hodge structures in general and also more explicitly in these four cases. To be more precise, let \(K\) be a mixed characteristic complete discrete valuation field with perfect residue field \(k\). Let \(W(k)\) be the ring of Witt vectors of \(k\) and \(\varphi\) the arithmetic Frobenius on \(K_0 = W(k)[\frac{1}{p}]\). The author fixes a \((\varphi, N)\)-module over \(K_0\) together with a weight filtration, that is a finite dimensional vector space \(H\) over \(K_0\) endowed with a \(\varphi\)-semilinear bijection of \(H\), a \(K_0\)-linear map \(N: H \to H(-1)\), and an increasing exhaustive filtration \(W_\bullet\) such that \(\varphi(W_w) \subset W_w\) and \(N(W_w) \subset W_w\). The subspace \({}_ND\) of admissible filtrations is an open rigid analytic subspace of \(\check{D}\), the space of all filtrations on \(H\) with fixed Hodge-Tate weights. The author then considers the space \({}_{\Gamma,N}D\) parametrizing a cer…
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14F40
zbMATH DE Number: 5960764
zbMATH Keywords: \(p\)-adic Hodge theory
zbMATH Keywords: degeneration of \(p\)-adic Hodge structure
zbMATH Keywords: partial compactifications of \(p\)-adic period domains
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/21562261-1299900
OpenAlex ID: W2092393856
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description: scientific article; zbMATH DE number 5960764