Sauf pour quelques corrections ajoutées en 2010, l’article reproduit des notes, envoyées à Nagata dans les années 70, qui étaient une traduction en langage des schémas de ses articles [2] et [3] Les démonstrations sont une version “constructive” de celles de Nagata. Rendre les démonstrations “constructives” a permis de les débarrasser de l’usage de valuations. Une exposition plus détaillée et des résultats additionnels sont dans Conrad [1]. Sauf mention expresse du contraire, tous les schémas considérés sont supposés noethériens.

Except for a few corrections added in 2010, the article reproduces notes, sent to Nagata in the 1970s, which were a translation in the language of schemes of his articles [2], [3]. The proofs are a “constructive” version of his proofs. Making the proofs constructives allowed us to proceed without using valuations. A more detailed exposition and additional results are in Conrad [1]. Unless otherwise mentioned, all the schemes considered are supposed to be Noetherian.

We treat two different topics on the log minimal model program, especially for four-dimensional log canonical pairs:

 (a) finite generation of the log canonical ring in dimension four,

 (b) abundance theorem for irregular fourfolds.

We obtain (a) as a direct consequence of the existence of four-dimensional log minimal models by using Fukuda’s theorem on the four-dimensional log abundance conjecture. We can prove (b) only by using traditional arguments. More precisely, we prove the abundance conjecture for irregular $(n+1)$-folds on the assumption that the minimal model conjecture and the abundance conjecture hold in dimension $\le n$.

In this article, we introduce arithmetic linear series and give a general way to estimate them based on Yuan’s idea. As an application, we consider an arithmetic analogue of the algebraic restricted volumes.

This article is a continuation of previous work, which has the same title. Let $Y$ be an affine symplectic variety with a ${\mathbf{C}}^{\ast }$-action with positive weights, and let $\pi :X\to Y$ be its crepant resolution. Then $\pi$ induces a natural map $PDef\left(X\right)\to PDef\left(Y\right)$ of Kuranishi spaces for the Poisson deformations of $X$ and $Y$. In Part I, we proved that $PDef\left(X\right)$ and $PDef\left(Y\right)$ are both nonsingular, and this map is a finite surjective map. In this article (Part II), we prove that it is a Galois covering. Markman already obtained a similar result in the compact case, which was a motivation for this article. As an application, we construct explicitly the universal Poisson deformation of the normalization $\tilde{O}$ of a nilpotent orbit closure $\tilde{O}$ in a complex simple Lie algebra when $\tilde{O}$ has a crepant resolution.

A basic problem in the classification theory of compact complex manifolds is to give simple characterizations of complex tori. It is well known that a compact Kähler manifold $X$ homotopy equivalent to a complex torus is biholomorphic to a complex torus.

The question whether a compact complex manifold $X$ diffeomorphic to a complex torus is biholomorphic to a complex torus has a negative answer due to a construction by Blanchard and Sommese.

Their examples, however, have negative Kodaira dimension; thus it makes sense to ask whether a compact complex manifold $X$ with trivial canonical bundle which is homotopy equivalent to a complex torus is biholomorphic to a complex torus.

In this article we show that the answer is positive for complex threefolds satisfying some additional condition, such as the existence of a nonconstant meromorphic function.

We give an explicit formula for the central critical value $L(1/2,\mathrm{\pi ̂}\otimes \chi )$ of the base-change lift $\mathrm{\pi ̂}$ to an imaginary quadratic field $K$ of an automorphic representation $\pi$ as the square of a finite sum of the values of a nearly holomorphic cusp form in $\pi$ at elliptic curves with complex multiplication by $K$. As long as the transcendental factor of the value is a CM period, $\chi$ is basically any unitary arithmetic Hecke character of $K$ inducing the inverse of the central character of $\pi$.

We study the Selmer variety associated to a canonical quotient of the ${\mathbb{Q}}_p$-pro-unipotent fundamental group of a smooth projective curve of genus at least two defined over $\mathbb{Q}$ whose Jacobian decomposes into a product of abelian varieties with complex multiplication. Elementary multivariable Iwasawa theory is used to prove bounds for the dimension of the Selmer variety, which, in turn, leads to a new proof of finiteness of rational points on such curves.

This article studies a map from a global Galois cohomology group to a direct sum of quotients of local Galois cohomology groups. The kernel of such a global-to-local map is often called a Selmer group. The objective of this article is to study the cokernel of such a map. We do so in a very general context. In particular, we find various sets of assumptions which imply that a global-to-local map is surjective.

There are two types of numerically trivial involutions of an Enriques surface according to their period lattice. One is of $U\left(2\right)\perp U\left(2\right)$-type and the other is of $\left(U\right(2\left)\perp U\right(2\left)\right)$-type. An Enriques surface with an involution of $\left(U\right(2\left)\perp U\right(2\left)\right)$-type is doubly covered by a Kummer surface of product type, and such involutions are classified again into two types according to the parity of the corresponding Göpel subgroups. Involutions of odd $\left(U\right(2\left)\perp U\right(2\left)\right)$-type are constructed from the standard Cremona involutions of the quadric surface and closely related with quartic del Pezzo surfaces.