Let (x, ζ,ℬt,Px) be a (standard) Markov process with state space 5 defined on the abstract space Ω. Here, xt is the sample path, ζ is the terminal time and ℬt is the smallest α-field of Ω in which xs s ≤ t are measurable. Let P’x, x ∈ S be another family of Markovian measures defined on (ℬt, Ω).

We prove a general embedding theorem for Cohen-Macaulay curves (possibly nonreduced), and deduce a cheap proof of the standard results on pluricanonical embeddings of surfaces, assuming vanishing H1(2KX) = 0.

Let k be an algebraically closed field of characteristic p≧ 0, and let X be an abelian variety over k.

The goal of this paper is to answer the following questions, when dim(X) = 1 and p≠0, posed by R. Hartshorne:

• (1) Is E(P) indecomposable, when E is an indecomposable vector bundle on X?

• (2) Is the Frobenius map F*: H1 (X, E) → H1 (X, E(p)) injective?

  We also partly answer the following question posed by D. Mumford:

• (3) Classify, or at least say anything about, vector bundles on X when dim (X) > 1.

We discuss the birational geometry of singular surfaces in positive characteristic. More precisely, we establish the minimal model program and the abundance theorem for ℚ-factorial surfaces and for log canonical surfaces. Moreover, in the case where the base field is the algebraic closure of a finite field, we obtain the same results under much weaker assumptions.

Many results of the theory of Riemann surfaces derive only from the properties of the sheaf of harmonic functions on these surfaces. It is therefore natural to try to extend these results to more comprehensive structures defined by means of a sheaf of continuous functions on a topological space which should possess the main properties of the sheaf of harmonic functions on a Riemann surface. The aim of the present paper is to generalise some known results from the theory of Riemann surfaces to spaces endowed with sheaves satisfying Brelot’s axioms [2], which we call harmonic spaces. In order to do so we had to introduce and to study the maps, associated in a natural way with this structure, called harmonic maps; they replace the analytic maps between Riemann surfaces. In this general frame we reconstruct the whole theory of Wiener compactification as well as the theory of the behaviour of analytic maps at the Wiener boundary.

The main purpose of this note is to characterize a compact Káhler-Einstein manifold in terms of curvature form. The curvature form Q is an EndT valued differential form of type (1,1) which represents the curvature class of the manifold. We shall prove that the curvature form of a Káhler metric is the harmonic representative of the curvature class if and only if the Káhler metric is an Einstein metric in the generalized sense (g.s.), that is, if the Ricci form of the metric is parallel. It is well known that a Káhler metric is an Einstein metric in the g. s. if and only if it is locally product (globally, if the manifold is simply connected and complete) of Kàhler-Einstein metrics. We obtain an integral formula, involving the integral of the trace of some operators defined by the curvature tensor, which measures the deviation of a Káhler-Einstein metric from a Hermitian symmetric metric. In the final section we shall prove the uniqueness up to equivalence of Kãhler-Einstein metrics in a simply connected compact complex homogeneous space. This result was proved by Berger in the case of a complex projective space and our proof is completely different from Berger’s.

In the present paper, we first prove the following

The notion of Henselian rings was introduced by G. Azuma-ya [l]. We concern ourselves in the present paper mainly with Henselizations of integrally closed integrity domains. Chapter I deals with general integrally closed integrity domains. As a preparation of our studies, we introduce the notion of decomposition rings analogously as in the case of fields (§1). And then we define the notions of (local) Henselian rings and Henselizations of integrally closed integrity domains, and obtain several results concerning characterizations of Henselian rings and the uniqeness of Henselizations (§2).

Complete parallel submanifolds of a real space form of constant sectional curvature k have been completely classified by Ferus [3] when k ≧ 0, and by Takeuchi [19] when k < 0. A complex space form is by definition a 2n-dimensional simply connected Hermitian symmetric space of constant holomorphic sectional curvature c and will be denoted by (c).

Let be a one-dimensional Brownian motion with constant drift µ ∈ R starting from 0. In this paper we show that

gives rise to a diffusion process and we explain how this result may be considered as an extension of the celebrated Pitman’s 2M - X theorem. We also derive the infinitesimal generator and some properties of the diffusion process and, in particular, its relation to the generalized Bessel processes.

The purpose of this paper is to prove the following two facts:

• I Every generic polarized abelian variety of odd dimension has a model rational over its field of moduli.

• II No generic principally polarized abelian variety of even dimension has a model rational over its field of moduli.

One frequently encounters (real) semisimple graded Lie algebras in various branches of differential geometry (e.g. [16], [9], [14], [18]). It is therefore desirable to study semisimple graded Lie algebras, including those which have been studied individually, in a unified way. One of our concerns is to classify (finite-dimensional) semisimple graded Lie algebras in a way that enables us to construct them.

The joint universality theorem for Lerch zeta-functions L(λl, αl, s) (1 ≤ l ≤ n) is proved, in the case when λls are rational numbers and αls are transcendental numbers. The case n = 1 was known before ([12]); the rationality of λls is used to establish the theorem for the “joint” case n ≥ 2. As a corollary, the joint functional independence for those functions is shown.

A smooth projective algebraic 3-fold V over the field C is called a Fano 3-fold if the anticanonical divisor — Kv is ample. The integer g = g(V) = ½(- Kv)3 is called the genus of the Fano 3-fold V. The maximal integer r ≧ 1 such that ϑ(— Kv)≃ ℋ r for some (ample) invertible sheaf ℋ ε Pic V is called the index of the Fano 3-fold V. Let V be a Fano 3-fold of the index r = 2 and the genus g = 21 which has the second Betti number b2(V) = 1. Then V can be embedded in P6 with degree 5, by the linear system |ℋ|, where ϑ(— Kv)≃ ℋ2 (see Iskovskih [5]). We denote this Fano 3-fold V by V5.

In this paper, we study finite symplectic actions on K3 surfaces X, that is, actions of finite groups G on X which act on H2,0(X) trivially. We show that the action on the K3 lattice H2(X, ℤ) induced by a symplectic action of G on X depends only on G up to isomorphism, except for five groups.

We have several definitions of the positivity of a vector bundle, differentiate definitions, an algebro-geometric definition, a topological definition etc. In § 1 we review the definitions and the relations between them. For a line bundle all the definitions are equivalent and every one agrees that they are reasonable. For a vector bundle, however, the definitions are not necessarily equivalent. One of the main results of this paper is the equivalence of the definitions over a complete non-singular curve. The proof is given in §2. We proved this over an elliptic curve in Umemura [18]. In this case the proof was based on Atiyah’s classification. To prove the equivalence over a curve of genus ≥ 2, the fundamental lemma is; A stable bundle of positive degree is positive in the sense of Nakano. The tool used to prove this lemma is the theory of stable bundles due to Narasimhan and Seshadri [11] —they establish a correspondence between stable bundles and certain types of irreducible unitary representations of a Fuchsian group.

Let p be a prime. If one adjoins to Q all pn-th roots of unity for n = 1,2,3, …, then the resulting field will contain a unique subfield Q∞ such that Q∞ is a Galois extension of Q with Gal (Q∞/Q) Zp, the additive group of p-adic integers. We will denote Gal (Q∞/Q) by Γ. In a previous paper [6], we discussed a conjecture relating p-adic L-functions to certain arithmetically defined representation spaces for Γ. Now by using some results of Iwasawa, one can reformulate that conjecture in terms of certain other representation spaces for Γ. This new conjecture, which we believe may be more susceptible to generalization, will be stated below.

Let M be an n-dimensional compact Kaehler manifold, TM its (holomorphic) tangent bundle and T*M its cotangent bundle. Given a complex vector bundle E over M, we denote its m-th symmetric tensor power by SmE and the space of holomorphic sections of E by Γ(E).

In 1896 G. Frobenius proved: the degree of any (absolutely) irreducible representation of a finite group divides its order. This theorem was improved by I. Schur in 1904 as follows: the degree of any irreducible representation of a finite group divides the index of its centre.

Let k be an algebraic number field and K be its normal extension of finite degree. Then the genus field K* of K over k is defined as the maximal unramified extension of K which is obtained from K by composing an abelian extension over k2). We call the degree (K*: K) the genus number of K over k.