Semistable Varieties Research Articles

Syntomic cohomology and p-adic regulators for varieties over p-adic fields

We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over $p$-adic rings extends uniquely to a cohomology theory for varieties over $p$-adic fields that satisfies $h$-descent. This new cohomology - syntomic cohomology - is a Bloch-Ogus cohomology theory, admits period map to \'etale cohomology, and has a syntomic descent spectral sequence (from an algebraic closure of the given field to the field itself) that is compatible with the Hochschild-Serre spectral sequence on the \'etale side. In relative dimension zero we recover the potentially semistable Selmer groups and, as an application, we prove that Soul\'e's \'etale regulators land in the potentially semistable Selmer groups. Our construction of syntomic cohomology is based on new ideas and techniques developed by Beilinson and Bhatt in their recent work on $p$-adic comparison

On p-adic differential equations on semistable varieties II

This paper is a complement to the paper “On p-adic differential equations on semistable varieties” by Di Proietto. Given an open variety over a DVR with semistable reduction, the author constructed in that paper a fully faithful algebraization functor from the category of certain log overconvergent isocrystals on the special fiber to the category of modules with regular integrable connection on the generic fiber. In this paper, we prove that, with convenable hypothesis, this functor is a tensor functor whose essential image is closed under extensions and subquotients. As a consequence, we can find suitable Tannakian subcategories of log overconvergent isocrystals and of modules with regular integrable connection on which the algebraization functor is an equivalence of Tannakian categories.

On $$p$$ -adic differential equations on semistable varieties

In this paper we prove a comparison theorem between the category of certain modules with integrable connection on the complement of a normal crossing divisor of the generic fiber of a proper semistable variety over a DVR and the category of certain log overconvergent isocrystals on the special fiber of the same open.

Log Structures on Generalized Semi-Stable Varieties

In this paper, a class of morphisms which have a kind of singularity weaker than normal crossing is considered. We construct the obstruction such that the so-called semi-stable log structures exists if and only if the obstruction vanishes. In the case of no power, if the obstruction vanishes, then the semi-stable log structure is unique up to a unique isomorphism. So we obtain a kind of canonical structure on this family of morphisms.

Universal log structures on semi-stable varieties

Given a morphism of schemes which is flat, proper, and "fiber-by-fiber semi-stable'', we study the problem of extending the morphism to a morphism of fine log schemes, which is log smooth, integral, and vertical. The problem is rephrased in terms of a functor on the category of fine log schemes over the base, and the main result of the paper is that this functor is representable by a fine log scheme whose underlying scheme maps naturally to the base by a monomorphism of finite type. In the course of the proof, we also generalize results of Kato on the existence of log structures of embedding and semi-stable type.

Semi-stability and base change

Let X and Y be regular strictly semi-stable varieties over a discrete valuation ring R and let \( R \subseteqq {\widetilde {R}} \) be a finite ramified extension of discrete valuation rings. We explicitly give desingularization procedures for the base change \( X \times _R {\widetilde {R}} \) and for the product \( X \times _R Y \).

Generalized Chow Forms and Reduction of Semi-Stable Varieties

In this note, by using the method of S. Zhang [1], we obtain the local version of a theorem of BGS [2] which links Faltings heights of projective varieties with the Philippon heights for the corresponding generalized Chow points. By the stable reduction theorem of S. Zhang [1], we prove that Chow semistabilities and generalized Chow semistabilities are the same.

Generalized Chow Forms and Reduction of Semi-Stable Varieties

Generalized Chow Forms and Reduction of Semi-Stable Varieties

Endomorphisms of Semi-Stable Abelian Varieties Over Number Fields

(End Jo(N)) (0 Q of all endomorphisms of Jo(N) is equal to the algebra generated by the Hecke operators, regarded as Q-endomorphisms of J0(N). Our initial proof that this is indeed the case combined P. Deligne's theorem that Jo(N) is semi-stable over Q (in the sense of [6]) with a rather elaborate argument involving l-adic representations. This argument gave a proof that all endomorphisms of a semi-stable abelian variety over Q are rational whenever the variety has a certain endomorphism structure (similar to that of [4, Section 5]). With Shimura's help we were able to simplify the argument and at the same time make it more general. Eventually the principle emerged that all endomorphisms of a semi-stable variety are unramifted. (See (1.1) and (1.3) for precise statements.) Since Q has no unramified extensions, it follows that every endomorphism of a semi-stable abelian variety over Q is rational (i.e., defined over Q). We apply these results to Jo(N) in Section 3 and to certain other modular varieties in Section 4. The applications require an auxiliary result (based on an idea of W. Casselman) concerning the endomorphism algebra of an abelian variety with many real endomorphisms; the result is proved in Section 2.