We express the cohomology of the complement of a real subspace arrangement of diagonal linear subspaces in terms of the Betti numbers of a minimal free resolution. This leads to formulas for the cohomology in some cases, and also to a cohomology vanishing theorem valid for all arrangements.

This paper is concerned with representations of split orthogonal and quasi-split unitary groups over a nonarchimedean local field which are not generic, but which support a unique model of a different kind, the generalized Bessel model. The properties of the Bessel models under induction are studied, and an analogue of Rodier‘s theorem concerning the induction of Whittaker models is proved for Bessel models which are minimal in a suitable sense. The holomorphicity in the induction parameter of the Bessel functional is established. Local coefficients are defined for each irreducible supercuspidal representation which carries a Bessel functional and also for a certain component of each representation parabolically induced from such a supercuspidal. The local coefficients are related to the Plancherel measures, and their zeroes are shown to be among the poles of the standard intertwining operators.

We study the Lie algebra of derivations of the coordinate ring of affine toric varieties defined by simplicial affine semigroups and prove the following results:

Such toric varieties are uniquely determined by their Lie algebra if they are supposed to be Cohen–Macaulay of dimension [ges ] 2 or Gorenstein of dimension =1.

In the Cohen–Macaulay case, every automorphism of the Lie algebra is induced from a unique automorphism of the variety.

Every derivation of the Lie algebra is inner.

This paper addresses questions involving the sharpness of Vojta‘s conjecture and Vojta‘s inequality for algebraic points on curves over number fields. It is shown that one may choose the approximation term mS(D,-) in such a way that Vojta‘s inequality is sharp in Theorem 2.3. Partial results are obtained for the more difficult problem of showing that Vojta‘s conjecture is sharp when the approximation term is not included (that is, when D=0). In Theorem 3.7, it is demonstrated that Vojta‘s conjecture is best possible with D=0 for quadratic points on hyperelliptic curves. It is also shown, in Theorem 4.8, that Vojta‘s conjecture is sharp with D=0 on a curve C over a number field when an analogous statement holds for the curve obtained by extending the base field of C to a certain function field.

It is shown that if the fundamental group of a normal algebraic variety, respectively Zariski open subset of a compact Kähler manifold, is solvable with a faithful linear representation over Q, respectively polycyclic, then it is virtually nilpotent.

In this paper we study – for a semistable scheme – a comparison map between its log-syntomic cohomology and the p-adic étale cohomology of its generic fiber. The image can be described in terms of what Bloch and Kato call the local points of the underlying motive. The results extend a proven conjecture of Schneider which treats the good reduction case. The proof uses the theory of logarithmic schemes, some crystalline cohomology theories defined on them and various techniques in p-adic Hodge theory, in particular the Fontaine–Jannsen conjecture proven by Kato and Tsuji as well as Fontaine‘s rings of p-adic periods and their properties. The comparison result may become useful with respect to cycle class maps.

We show that the Kodaira dimension of the moduli space of polarized K3 surfaces of degree 2n in non negative if n = 42, 43, 51, 53, 55, 57, 59, 61, 66, 67, 69, 74, 83, 85, 105, 119 or 133. We use an automorphic form associated with the fake monster Lie algebra constructed by Borcherds.

In this paper, we prove an analogue of the result known as Mazur‘s Principle concerning optimal levels of mod [ell ] Galois representations. The paper is divided into two parts. We begin with the study (following Katz–Mazur) of the integral model for certain Shimura curves and the structure of the special fibre. It is this study which allows us to generalise, in the second part of this paper, Mazur‘s result to totally real fields of odd degree.

Classifying spaces and moduli spaces are constructed for two invariants of isolated hypersurface singularities, for the polarized mixed Hodge structure on the middle cohomology of the Milnor fibre, and for the Brieskorn lattice as a subspace of the Gauß–Manin connection. The relations between them, period mappings for μ-constant families of singularities, and Torelli theorems are discussed.

We show that unipotent overconvergent isocrystals are algebraic and that the category of unipotent overconvergent isocrystals has a Frobenius automorphism. We also prove a structure theorem for unipotent overconvergent F-isocrystals over an open subset of the line, analogous to Dieudonné–Manin decomposition theorem for F-isocrystals.

The classical limit of the scaled elliptic algebra${\mathcal A}$ħ,η ($\widehat{\mathfrak sl}$2) is investigated. The limiting Lie algebra is described in two equivalent ways: as a central extension of the algebra of generalized automorphic sl2 valued functions on a strip and as an extended algebra of decreasing automorphic sl2 valued functions on the real line. A bialgebra structure and an infinite-dimensional representation in the Fock space are studied. The classical limit of elliptic algebra ${\mathcal A}$q,p ($\widehat{\mathfrak sl}$2) is also briefly presented.

Let X be an m dimensional smooth projective variety with a Kähler metric. We construct a metrized line bundle $\cL$ with a rational section s over the product $\cC$p(X)× $\cC $q(X) of Chow varieties $\cC$p(X), $\cC$q(X) such that $\[{1\over (m-1)!}\log|s(A,B)|^2=\langle A, B\rangle \]$ for disjoint A, B. That gives an answer to a part of Barry Mazur‘s proposal in a private communication to Bruno Horris about the Archimedean height pairing 〈 A, B〉 on a smooth projective variety X.

Cet article constitue une partie importante de ma thèse. Je l‘ai écrit sous la direction avisée de Henri Carayol, à qui je dois un très grand merci car ce sont ses conseils et son optimisme qui m‘ont permis d‘en venir à bout. Je tiens également à remercier Laurent Clozel pour toutes ses remarques, utiles et pertinentes.

Fix a prime number p [ges ] 5 and a positive integer N prime to p. We consider the projective limits of p-adic étale cohomology groups of the modular curves X1(Npr) and Y1(Npr) (r [ges ] 1), which are denoted by ESp(N) ${\biib Z}$p and GES p(N)${\biib Z}$p, respectively. Let e* ′ be the projector to the direct sum of the ωi-eigenspaces of the ordinary part, for i [nequiv] 0, −1 mod p−1. Our main result states that e* ′ GESp (N)${\biib Z}$p has a good p-adic Hodge structure, which can be described in terms of λ-adic modular forms, extending the previously known result for e*′ ESp (N)${\biib Z}$p. We then apply the method of Harder and Pink to the Galois representation on e*′ ESp(N) ${\biib Z}$p to construct large unramified abelian p-extensions over cyclotomic ${\bib Z}$p-extensions of abelian number fields.

We define the notion of a morphism of generalized semi-stable type, which is a generalization of the notion of a semistable degeneration over a curve. We partially generalize Steenbrink‘s results on the limit of Hodge structures to the case of such a morphism. As an application we prove the E1-degeneration of the relative Hodge–De Rham spectral sequence for this case.

Let A=(aij) be an orthogonal matrix (over R or C) with no entries zero. Let B= (bij) be the matrix defined by bij= 1/ai j. M. Kontsevich conjectured that the rank of B is never equal to three. We interpret this conjecture geometrically and prove it. The geometric statement can be understood as variants of the Castelnuovo lemma and Brianchon‘s theorem.

We examine connections between A-hypergeometric differential equations and the theory of integer programming. In the first part, we develop a ’hypergeometric sensitivity analysis‘ for small variations of constraint constants with creation operators and b-functions. In the second part, we study the indicial polynomial (b-function) along the hyperplane xi=0 via a correspondence between the optimal value of an integer programming problem and the roots of the indicial polynomial. Gröbner bases are used to prove theorems and give counter examples.

Recently, L. Rozansky and E. Witten associated to any hyper-Kähler manifold X a system of ’weights‘ (numbers, one for each trivalent graph) and used them to construct invariants of topological 3-manifolds. We give a simple cohomological definition of these weights in terms of the Atiyah class of X (the obstruction to the existence of a holomorphic connection). We show that the analogy between the tensor of curvature of a hyper-Kähler metric and the tensor of structure constants of a Lie algebra observed by Rozansky and Witten, holds in fact for any complex manifold, if we work at the level of cohomology and for any Kähler manifold, if we work at the level of Dolbeault cochains. As an outcome of our considerations, we give a formula for Rozansky–Witten classes using any Kähler metric on a holomorphic symplectic manifold.