\(p\)-adic period domains and toroidal partial compactifications. I
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 certain level structure \(\Gamma\) on the Galois representation at each point of \({}_ND\), obtained by a result that says that weakly admissible filtered \((\varphi, N)\)-modules are admissible. In contrast to the complex case, the natural map goes from \({}_{\Gamma, N}D\) to \({}_N D\) in the \(p\)-adic case, and locally admits a section, giving \({}_{\Gamma, N}D\) the structure of a \(p\)-adic analytic space. The key construction of the paper lies in locally attaching \(p\)-adic nilpotent orbits to \({}_{\Gamma,N}D\). In other words, one allows the monodromy operator \(N\) to deform into more non-trivial ones. The author glues the nilpotent orbits onto the original period domain \({}_{\Gamma,N}D\) and obtains certain ``\(p\)-adic log manifolds'', which are analogous to the log manifolds in the complex case. In the four examples mentioned above, the author gives an explicit description of the partial compactification. In addition to the discussion on partial compactification, the author also includes the formulation of a \(p\)-adic version of Hodge conjecture, which is well-known to the experts. The formulation is one of the motivations of studying the degeneration of \(p\)-adic Hodge structures. This paper is very well-written. In particular, it contains many examples to help readers understand his construction.
degeneration of \(p\)-adic Hodge structure
partial compactifications of \(p\)-adic period domains
scientific article; zbMATH DE number 5960764
scientific article