The present note is intended to be a supplement to [9], in which the following is proven: Let V be a smooth projective variety over the field of complex numbers C, T a smooth quasi-projective variety, Z a cycle in T × V of codimension p. If Z(t) is l-cube equivalent to zero for general t e T, then, setting r = dim V − p,

vanishes for l′ < l, where {tZ} is the correspondence defined by Z.

The general problem of rebuilding the threefolds X endowed with a given ample divisor H, possibly non-effective, is closely related to the study of the complete linear system |KX + H| adjoint to H. Many powerful results are known about |KX + H|, for instance when the linear system | H | contains a smooth surface or, more particularly, when H is very ample (e.g. see Sommese [S1] and [S2]). From this point of view we study some properties of |KX + H |, which turn out to be very useful in the description of the threefolds X polarized by an ample divisor H whose arithmetic virtual genus g(H) is sufficiently low.

In [5, 6, 7] I introduced the concept of Einstein-Hermitian vector bundle. Let E be a holomorphic vector bundle of rank r over a complex manifold M. An Hermitian structure h in E can be expressed, in terms of a local holomorphic frame field s1, …, sr of E, by a positive-definite Hermitian matrix function (hij) defined by

Soit X un groupe abélien localement compact, non-compact, séparé et dénombrable à l’infini. On désignera par ξ une mesure de Haar fixée sur X. Pour les noyaux de convolution réels sur X, le principe semi-complet du maximum est fondamental dans la théorie du potentiel. Soit N un noyau de convolution réel sur X vérifiant le principe semi-complet du maximum (désigné par Nє(PSM)). Pour déterminer l’allure de N à l’infini, la N-réduite ηN, δ de N à l’infini δ joue un rôle essentiel. Pour une exhaustion de compacts de est décroissante et est une mesure de Radon réelle sur X ou bien, pour une fonction finie et continue dans X à support compact quelconque, est la iV-réduite de N sur CKn. Dans ce cas, est indépendant du choix de Posons

In this paper we consider the Cauchy problem for a hyperbolic system with characteristics of variable multiplicity and construct a certain solution whose wave front set propagates precisely along the so-called “broken null bicharacteristic flow”, in other words, along the admissible trajectory if we use the terminology of [6].

In this paper we show, for an example of hyperbolic system, that the wave front set in Gevrey class of a certain solution propagates precisely along “generalized broken null-bicharacteristic flow” defined by Kumano-go-Taniguchi [7] and Wakabayashi [10] [11].

Let Λ be the Leech lattice which is an even unimodular lattice with no vectors of squared length 2 in 24-dimensional Euclidean space R24. Then the Mathieu Group M24 is a subgroup of the automorphism group .0 of Λ and the action on Λ of M24 induces a natural permutation representation of M24 on an orthogonal basis For , let Λm be the sublattice of vectors invariant under m:

The method to find a system of fundamental units were given by G. F. Voronoi in the case of purely cubic fields (see B. N. Delone and D. K. Faddeev [1]), and by K. K. Biilebic [2] in general case of algebraic number fields except for quadratic fields. But they are rather complicated for direct calculation. On the other hand, in some special cases, units can be calculated by the Jacobi-Perron algorithm which is a generalization of the continued fractional expansion (see, for example, L. Bernstein [3]). In this paper, we will show the new method to calculate a system of independent units by multiplying several numbers. We will give some examples for purely cubic fields in Section 3.