In this text we study using stack–theoretic techniques the crystalline structure on the de Rham cohomology of a complete smooth variety over a p–adic field. Such a structure is known to exists by the CdR–conjecture of Fontaine now proven independently by Faltings, Niziol, and Tsuji, and is intimately tied to the action of Galois on the p–adic étale cohomology. The paper contains two main parts....

In 1535, when Jacques Cartier landed in Canada for the second time, he sailed up the Saint-Lawrence, and spent the winter at Stadaconé. His ships were frozen in the ice for five months. A strange disease attacked the indigenous population, and then the French sailors. In a two month period, almost all Cartier's men became sick. Facing such a tragic situation, the explorer carried out an examina...

In this article we prove that for any complete toric variety, and for any Cartier divisor, the ring of global sections of multiples of the line bundle associated to the divisor is finitely generated.

We consider the operation of intersecting with a locally principal Cartier divisor (i.e., a Cartier divisor which is principal on some neighborhood of its support). We describe this operation explicitly on the level of cycles and rational equivalences and as a corollary obtain a formula for rational equivalence between intersections of two locally principal Cartier divisors. Such canonical rati...

For a one-dimensional family of abelian varieties equipped with principal theta divisors a canonical limit is constructed as a pair consisting of a reduced projective variety and a Cartier divisor on it. Properties of such pairs are established.