The cohomology theory of associative algebras has been developed by G. liochschild [1], [2], [3], and the 1-, 2-, and 3-dimensional cohomology groups have been interpreted with reference to classical notions of structure in his papers. Recently M. Ikeda has obtained, by a detailed analysis of Hochschild’s modules, an interesting structural characterization of the class of algebras whose 2-dimensional cohomology groups are all zero [5].

In the present work we classify the relatively minimal 3-dimensional quasihomogeneous complex projective varieties under the assumption that the automorphism group is not solvable. By relatively minimal we understand varieties X having at most ℚ-factorial terminal singularities and allowing an extremal contraction X → Y where dim Y < 3.

Let M be an n-dimensional Hadamard manifold, that is, a complete simply connected C∞ Riemannian manifold with nonpositive sectional curvatures. Making use of geodesic rays, Eberlein and O’Neill [11] constructed a compactification = MS(∞) of M which gives a homeomorphism of (M, S(∞)) with the Euclidean pair (Bn, Sn-1). In this paper we shall study the asymptotic Dirichlet problem for the Laplace-Beltrami operator, which is stated as follows:

L’objet de ce mémoire est essentiellement la recherche et l’étude de structures algébriques permettant de traduire le Calcul propositionnel et le Calcul des prédicats restreint du premier ordre. Nous avons guidé et limité cette recherche en imposant aux ensembles algébriques introduits de satisfaire simultanément aux trois conditions:

• — donner une représentation du Calcul Logique sous sa forme la plus classique, c’est à dire avant l’obtention d’un anneau booléien par passage au quotient. Ce sera le rôle des structures “prébooléiennes”.

• — posséder un ensemble d’opérations traduisant un procédé de décision tel que celui des tableaux sémantiques, qui correspondent ici aux “tableaux de réduction”. —avoir un caractère universel dans leur construction à partir des atomes.

Let S be a locally compact (not compact) Hausdorff space satisfying the second axiom of countability and let ℬ be the σ field of all Borel subsets of S and let A be the σ-field of all the subsets of S which, for each finite measure μ defined on (S, A), are in the completed σ field of ℬ relative to μ. We denote by C0 the Banach space of continuous functions vanishing at infinity with the uniform norm and Bk the space of bounded A-measurable functions with compact support in S.

We defined a prime decomposition symbol [d1 d2, p] in a previous paper [3], and characterized in [4] the set of rational primes p which are decomposed completely in a non-abelian central extension which is of degree 8 in substance. The explicit value of the symbol was determined by using a solution of certain ternary quadratic diophantine equation. The solution corresponds to a square root of an ideal class of the principal genus of a quadratic field. This is translated to a problem in classes of integral quadratic forms, namely to find a form whose duplication is a given one contained in a principal genus. An explicit method to find the form is given by Gauss in [5, Art. 286], which is due to his ternary form reduction.

In this paper we shall determine KSO-groups for 4-dimensional CW-complexes by their cohomology rings. We denote by KSO(X) the group of orientable stable vector bundles over X. In 1959 A. Dold and H. Whitney [1] gave the classification of SO(n)-bundles over a 4-complex. It seems, however, to the authors that group structures of them are unknown. We shall give another definition of the difference bundles defined in [1], and we determine the group structure of KSO(X).

Let p be a fixed prime integer, and G a finite p-group. For a subgroup H of G, we denote the centralizer of H in G by CG(H). The commutator subgroup of G is denoted by [G, G].

Using the geometry of an almost del Pezzo threefold, we show that the moduli space ${\mathcal{S}}_{g,1}^{0,\text{hyp}}$ of genus $g$ one-pointed ineffective spin hyperelliptic curves is rational for every $g\geqslant 2$.

Let Γ be a ring with unit element and let Λ be the Ore extension of Γ with respect to a derivation d of Γ [4, 3]. It is shown in [3] that l.gl. dim Λ ≦ 1 + l.gl. dim Γ. It is not in general possible to replace this inequality by equality.

The infinite dimensional Laplacian operator can be discussed in connection with the infinite dimensional rotation group ([1]). Our interest centers entirely on observing how each one-parameter subgroup of the infinite dimensional rotation group contributes to the determination of the Laplacian operator.

In this paper we consider the following nonlinear parabolic variational inequality; u(t) ∈ D(Φ) for all where Δp is the so-called p-Laplace operator and Φ is a proper, lower semicontinuous functional. We have obtained two results concerning to solutions of this problem. Firstly, we prove a few regularity properties of solutions. Secondly, we show the continuous dependence of solutions on given data u0 and f.

We show that a (Spinq-style) twistor space admits a canonical Spin structure. The adiabatic limits of η-invariants of the associated Dirac operator and of an intrinsically twisted Dirac operator are then investigated.

The purpose of this paper is to study the dimension formula for cusp forms of weight one, following the series of Hiramatsu [2] and Hiramatsu-Akiyama [3]. We define as usual the subgroup Γ0(N) of SL2(Z) by

.

We describe the divisor class group and the graded canonical module of the multisection ring T (X;D1,…, Ds) for a normal projective variety X and Weil divisors D1,…, Ds on X under a mild condition. In the proof, we use the theory of Krull domain and the equivariant twisted inverse functor.

We let the notations be as in [3]. Then, in the category 6 of all bordered Riemann surfaces, the following inclusion diagram holds [3, Theorem 9]:

Let X be an algebraic complex projective surface equipped with the euclidean topology and E a rank 2 topological vector bundle on X. It is a classical theorem of Wu ([Wu]) that E is uniquely determined by its topological Chern classes . Viceversa, again a classical theorem of Wu ([Wu]) states that every pair (a, b) ∈ (H (X, Z), Z) arises as topological Chern classes of a rank 2 topological vector bundle. For these results the existence of an algebraic structure on X was not important; for instance it would have been sufficient to have on X a holomorphic structure. In [Sch] it was proved that for algebraic X any such topological vector bundle on X has a holomorphic structure (or, equivalently by GAGA an algebraic structure) if its determinant line bundle has a holomorphic structure. It came as a surprise when Elencwajg and Forster ([EF]) showed that sometimes this was not true if we do not assume that X has an algebraic structure but only a holomorphic one (even for some two dimensional tori (see also [BL], [BF], or [T])).

Let E and E′ be C∞ vector bundles over a C∞ manifold M. Denote by Γ(E) (resp. by Γ(E′) the vector space of C∞ cross-sections of E (resp. of E′) over M. Take a linear differential operator of the first order D: Γ(E) → Γ(E′) induced by a vector bundle mapping σ(D): jl(E) ′ E′, where Jk(E) denotes the vector bundle of k-jets of cross-sections of E.