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]).

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

Si X est un ensemble mesurable au sens de Lebesgue dans un espace euclidien En et Y est un ensemble mesurable au sens de Lebesgue dans un autre espace euclidien Em, alors X × Y est mesurable dans En+m= Enx Em et sa mesure est égale au produit de la mesure de X et celle de Y. Mais il n’existe pas de telle relation simple ni pour la capacité d’ordre général ni pour la mesure de Hausdorff de dimension générale. Dans le présent mémoire nous essayons d’évaluer la capacité des ensembles produits au moyen de la capacité ou de la mesure de Hausdorff des ensembles composants.

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.

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 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.

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.

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.

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.

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.

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.

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:

The main result is a general vanishing theorem for the Dolbeault cohomology of an ample vector bundle obtained as a tensor product of exterior powers of some vector bundles. It is also shown that the conditions for the vanishing given by this theorem are optimal for some parameter values.

We first characterize the automorphism groups of Hodge structures of cubic threefolds and cubic fourfolds. Then we determine for some complex projective manifolds of small dimension (cubic surfaces, cubic threefolds, and nonhyperelliptic curves of genus 3 or 4), the action of their automorphism groups on Hodge structures of associated cyclic covers, and thus confirm conjectures made by Kudla and Rapoport in (Pacific J. Math. 260(2) (2012), 565–581).

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.

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 τ denote a point of the Siegel upper-half space be a (column) vector in Cg and m a vector with m′ resp. m″ in Zg as its first resp. second entry vectors, then the series

in which , represents an holomorphic function on the product and it is called the theta function of characteristic m.

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. Our interest is in a sort of the linearizability problem of a neighborhood of $Y$. As a higher codimensional generalization of Ueda’s result, we give a sufficient condition for the existence of a nonsingular holomorphic foliation on a neighborhood of $Y$ which includes $Y$ as a leaf with unitary-linear holonomy. We apply this result to the existence problem of a smooth Hermitian metric with semipositive curvature on a nef line bundle.

The infinite symmetric group is the discrete group of all finite permutations of the set X of all natural numbers. Among discrete groups, it has distinctive features from the viewpoint of representation theory and harmonic analysis. First, it is one of the most typical ICC-groups as well as free groups and known to be a group of non-type I. Secondly, it is a locally finite group, namely, the inductive limit of usual symmetric groups . Furthermore it is contained in infinite dimensional classical groups GL(ξ), O(ξ) and U(ξ) and their representation theories are related each other.

This paper is one step toward infinite energy gauge theory and the geometry of infinite dimensional moduli spaces. We generalize a gluing construction in the usual Yang-Mills gauge theory to an “infinite energy” situation. We show that we can glue an infinite number of instantons, and that the resulting ASD connections have infinite energy in general. Moreover they have an infinite dimensional parameter space. Our construction is a generalization of Donaldson’s “alternating method”.