Let h1(u),…, hk(u) be Laurent polynomials of m-variables and let

be a non-degenerate complete intersection variety. Such an intersection variety appears as an exceptional divisor of a resolution of non-degenerate complete intersection varieties with an isolated singularity at the origin (Ok4]).

Let M be a compact boundaryless C∞-manifold, and let Diff (M) be the space of C1-diffeomorphisms of M endowed with the C1-topology. An Axiom A diffeomorphism is said to satisfy the strong transversality condition if for every x∊M, TxM = TxW3(x) + TxWu(x).

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.

First published online by Duke University Press 15 January 2010, subsequently published online by Cambridge University Press 11 January 2016, doi:10.1017/S0027763000009818

This note is to be a supplement of the preceeding paper in the journal by Matsushima, settling a question raised by him. In his paper he associates a holomorphic vector bundle over a complex torus to a holomorphic representation of what he calls Heisenberg group. We shall show that a simple holomorphic vector bundle is determined in this manner if and only if the associated projective bundle admits an integrable holomorphic connection. A theorem by Morikawa ([3], Theorem 1) is the motivation of this problem and is somewhat strengthened by our result.

There is in the literature a lot of determinant formulae involving the relative class number of an imaginary abelian field. Usually such a formula contains a factor which is equal to zero for many fields and so it gives no information about the class number of these fields. The aim of this paper is to show a way of obtaining most of these formulae in a unique fashion, namely by means of the Stickelberger ideal. Moreover some new and non-vanishing formulae are derived by a modification of Ramachandra’s construction of independent cyclotomic units.

Let A be a self-injective algebra over an algebraically closed field. We study the stable dimension of A, which is the dimension of the stable module category of A in the sense of Rouquier. Then we prove that A is representation-finite if the stable dimension of A is zero.

For a finite set S of primes of a number field K and for σ1,…, σe ∈ Gal(K) we denote the field of totally S-adic numbers by Ktot,S and the fixed field of σ1,…,σe in Ktot,S by Ktot,S(σ). We prove that foralmost all σ ∈ Gal(K)e the absolute Galois group of Ktot,S(σ) is the free product of and a free product of local factors over S.

Let E/F be an unramified cyclic extension of local non-archimedean fields, G a connected reductive group over F, K(F) (resp. K(E)) a hyper-special maximal compact subgroup of G(F) (resp. G(E)), and H(F) (resp. H(E)) the Hecke convolution algebra of compactly-supported complex-valued K(F) (resp. G(E))-biinvariant functions on G(F) (resp. G(E)). Then the theory of the Satake transform defines (see § 2) a natural homomorphism H(E) → H(F), θ→f. There is a norm map N from the set of stable twisted conjugacy classes in G(E) to the set of stable conjugacy classes in G(F); it is an injection (see [Ko]). Let Ω‱(x, f) denote the stable orbital integral of f in H(F) at the class x, and Ω‱(y, θ) the stable twisted orbital integral of θ in H(E) at the class y.

The main purpose of the present paper is to give the details of the results announced in the P. J. A. note [11], establishing some global theorems on rigidity for a certain class of isometric imbeddings.

Let {(Sn, Tn), n = 0, 1, 2, …} be a two-dimensional random walk with stationary independent increments starting at the origin 0. Throughout the paper we always assume the following condition:

In this paper we consider a subfield K in a cyclotomic field km of conductor m such that [km: K] = 2 in the cases of m = lpn with a prime p, where l = 4 or p > l = 3. Then the theme is to know whether the ring of integers in K has a power basis or does not.

Associated with a Hamiltonian holomorphic vector field on a compact Kähler manifold, a nice functional on a space of Kähler metrics will be constructed as an integration of the bilinear pairing in [FM] contracted with the Hamiltonian holomorphic vector field. As applications, we have functionals whose critical points are extremal Kähler metrics or “Kähler-Einstein metrics” in the sense of [M4], respectively. Finally, the same method as used by [G1] allows us to obtain, from the convexity of , the uniqueness of “Kähler-Einstein metrics” on nonsingular toric Fano varieties possibly with nonvanishing Futaki character.

We formulate a Beilinson-Bernstein-type derived equivalence for a quantized enveloping algebra at a root of 1 as a conjecture. It says that there exists a derived equivalence between the category of modules over a quantized enveloping algebra at a root of 1 with fixed regular Harish-Chandra central character and the category of certain twisted D-modules on the corresponding quantized flag manifold. We show that the proof is reduced to a statement about the (derived) global sections of the ring of differential operators on the quantized flag manifold. We also give a reformulation of the conjecture in terms of the (derived) induction functor.

Recently I. Gelfand and M. Neumark [2] have determined the types of irreducible unitary representations of the group G1 of linear transformations of the straight line. The analogous result is obtained for the group G2 of transformations z → az + b in the complex-number plane , where a and b run over all complex numbers with the exception of a = 0, which may be considered as the group of all sense-preserving similar transformations in the two-dimensional euclidean space E2.

Let be a free abelian group of finite rank, let Γ be a sub-semigroup of satisfying certain finiteness conditions, and let be a (Γ × Z2)-graded Lie superalgebra. In this paper, by applying formal differential operators and the Laplacian to the denominator identity of , we derive a new recursive formula for the dimensions of homogeneous subspaces of . When applied to generalized Kac-Moody superalgebras, our formula yields a generalization of Peterson’s root multiplicity formula. We also obtain a Freudenthal-type weight multiplicity formula for highest weight modules over generalized Kac-Moody superalgebras.

We prove a Noether-Lefschetz type theorem for varieties of r-planes in complete intersections. We then use it to study the Abel-Jacobi map of planes on a smooth cubic fivefold.

1. Introduction. Let G be a rank 3 permutation group of even order on a finite set X, |X| = n, and let Δ and Γ be the two nontrivial orbits of G in X×X under componentwise action. As pointed out by Sims [6], results in [2] can be interpreted as implying that the graph = (X, Δ) is a strongly regular graph, the graph theoretical interpretation of the parameters k, l, λ and μ of [2] being as follows: k is the degree of , λ is the number of triangles containing a given edge, and μ is the number of paths of length 2 joining a given vertex P to each of the l vertices ≠ P which are not adjacent to P. The group G acts as an automorphism group on and on its complement = (X,Γ).

Let F be a non-Archimedean locally compact field, of residual characteristic p, and let D be a finite-dimensional central division F-algebra. Let ℓ be a prime number different from p. In this article, generalizing the results of [GJ], we associate, to each ℓ-modular smooth irreducible representation π of GLm(D), two invariants L(T,π), ε(T,π,ψ), where T is an indeterminate and ψ is a nontrivial character of F.