Let A$^′$be a complete characteristic (0,p) discrete valuation ring with absolute ramification degree e and a perfect residue field. We are interested in studying the category${\mathcal F}$${\mathcal F}$$_A$$_′$of finite flat commutative group schemes over A$^′$withp-power order. When e= 1, Fontaine formulated the purely ’linear algebra‘ notion of a finite Honda system over A$^′$and constructed an anti-equivalence of categories between${\mathcal F}$${\mathcal F}$$_A$$_′$and the category of finite Honda systems over A$^′$ when p< 2. We generalize this theory to the case e≤ − 1.

The purpose of this paper is to show how generalizations of generic vanishing theorems to a${\mirrored Q}$-divisor setting can be used to study the geometric properties of pluritheta divisors on a principally polarized Abelian variety (PPAV for short).

Recently, differential geometric properties of embedded projective varieties have gained increasing interest. In this note, we consider plane algebraic curves equipped with the Fubini–Study metric from${\mirrored P}$$_2$(${\mirrored C}$) and give an estimate for the diameter in terms of the degree, initiated in a paper by F. A. Bogomolov.

Let K be a quadratic extension of a field k which is either local field or a finite field. Let G be an algebraic group over k. The aim of the present paper is to understand when a representation of G(K) has a G(k) invariant linear form. We are able to accomplish this in the case when G is the group of invertible elements of a division algebra over k of odd index if k is a local field, and for general connected groups over finite fields.