The purpose of this paper is to introduce a-functions and b-functions of prehomogeneous vector spaces in the original way of M. Sato and give a proof of the structure theorem of them. All the results were obtained by M. Sato when he constructed the theory of prehomogeneous vector spaces in 60’s. However he did not write a paper on his outcomes at that time. His theory was distributed through his lectures and informal seminars. Only small number of people could know it. The only publication left for us is a mimeographed note of his lecture [Sa-Sh1] written by T. Shintani in Japanese.

Let Gn (resp. Γn) be the real symplectic (resp. Siegel modular) group of degree n. The Siegel cusp form is a holomorphic function on the Siegel upper half plane which satisfies functional equations relative to Γn and vanishes at the cusps.

Class number one problem for imaginary quadratic fields was solved in 1966 by A. Baker and H.M. Stark independently. However, the problem for real quadratic fields is still unsolved. It seems to us that one of the most essential difficulties of the problem for real quadratic fields comes from deep connection of the class number with the fundamental unit.

The notion of a ray class field, which is fundamental in Takagi’s class field theory, has no immediate analogon in the function field case. The reason for this lies in the lacking of a distinguished maximal order. In this paper I overcome this difficulty by a relative version of the notion of ray class fields to be defined for every holomorphy ring of the field. The prototype for this new notion is M. Rosen’s definition of a Hilbert class field for function fields [6].

Let k be an algebraic number field of finite degree, and p be a fixed rational prime. We denote the set of all the non-Archimedian prime divisors of k by S0(k) and the set of all the real Archimedian ones by (k). Put and S = S0 ∪ S∞, and define a subgroup of the unit group (k) of k by

Throughout this note, A denotes a commutative local Noetherian ring with maximal ideal m and M a finitely generated A-module with dim (M) = d. Let x1, …, xd be a system of parameters (s.o.p. for short) for M and I the ideal of A generated by x1, …, xd.

In our earlier paper [8] we discussed Fano manifolds X that are of the form X = ℙ() with a rank-2 vector bundle on a surface S. Here we study a more general situation of Fano manifolds, ruled over the complex projective plane P2 as Pr-1-bundles, i.e., being of the form ℙ() with -a bundle of rank r ≥ 3 on P2.

The defect relation gives the best-possible estimate, where f is a linearly non-degenerate holomorphic curve in Pn(C) and H1, …, Hq are hyperplanes in Pn(C) which are in general position. However, the case of moving hyperplanes has ever got only n(n + 1) instead of n + 1 (Stoll [4]) and it has not yet been known whether this bound is best-possible or not. In this paper we shall give some particular cases which have the bound n + 1.

A formal quantization of Poisson algebras was discussed by several authors (see for instance Drinfel’d [D]). A formal Lie algebra generated by homotopy classes of loops on a Riemann surface ∑ was obtained by W. Goldman in [G], and its Poisson algebra was quantized, in the sense of Drinfel’d, by Turaev in [T].

Let S be a three-dimensional regular local ring and let I be a prime ideal in S of height two. This paper is motivated by the question of when I is a set-theoretic complete intersection and when the symbolic Rees algebra S(I) = ⊕n≥0I(n)tn is Noetherian. The connection between the two problems is given by a result of Cowsik which says that the Noetherian property of S(I) implies that I is a set-theoretic complete intersection ([1]).

We will study holomorphic mappings

from a connected complex manifold M of dimension m to a projective algebraic manifold N of dimension n. Assume first that N is of general type, i.e.

where KN→N is the canonical bundle of N. If KN is positive, then N is of general type.

This announcement is a continuation of Hu [3]. Our results improve Theorem 1 of [3], but the latter is needed in the proof of the former.

Let f: M → N be a holomorphic mapping from a connected complex manifold M of dimension m to a projective algebraic manifold N of dimension n.

In this paper we study the ideal boundaries of surfaces admitting total curvature as a continuation of [Sy2] and [Sy3]. The ideal boundary of an Hadamard manifold is defined to be the equivalence classes of rays. This equivalence relation is the asymptotic relation of rays, defined by Busemann [Bu]. The asymptotic relation is not symmetric in general. However in Hadamard manifolds this becomes symmetric. Here it is essential that the manifolds are focal point free.

In this paper we extend the work on affine immersions [N-Pi]-1 to the case of affine immersions between complex manifolds and lay the foundation for the geometry of affine Kähler immersions. The notion of affine Kähler immersion extends that of a holomorphic and isometric immersion between Kähler manifolds and can be contrasted to the notion of holomorphic affine immersion which has been established in the work of Dillen, Vrancken and Verstraelen [D-V-V] and that of Abe [A].