To a symmetric, relatively ample line bundle on an Abelian scheme one can associate a linear combination of the determinant bundle and the relative canonical bundle, which is a torsion element in the Picard group of the base. We improve the bound on the order of this element found by Faltings and Chai. In particular, we obtain an optimal bound when the degree of the line bundle d is odd and the set of residue characteristics of the base does not intersect the set of primes p dividing d, such that p$\equiv -1$ mod(4) and p$\le 2$g$-1$, where g is the relative dimension of the Abelian scheme. Also, we show that in some cases these torsion elements generate the entire torsion subgroup in the Picard group of the corresponding moduli stack.

The purpose of this note is to present a somewhat unexpected relation between diophantine approximations and the geometric invariant theory. The link is given by Mumford's degree of contact. We show that destabilizing flags of Chow-unstable projective varieties provide systems of diophantine approximations which are better than those given by Schmidt's subspace theorem, and we give examples of these systems.

For $x\in (a_{j-1}, a_j)\ (j=1,\ldots, p+1;\ a_0\!:=-\infty, \ a_{p+1} \!:=\infty)$ the mapping $T_j\!: w=x-\sum ^p_{l=1}\lambda _l/(x-a_l)\ (\lambda _l$>$0, \ a_l\in$R) is onto R. It was shown by G. Boole in the 1850's that $\sum ^{p+1} _{j=1}[(\partial w/\partial x) ^{-1}] _{x=T^{-1}_j(w)}=1.$ We give an n-dimensional analogue of this result. The proof makes use of the Griffiths–Harris residue theorem from algebraic geometry.

Kollár's conjecture, that log canonical singularities are Du Bois, is proved in the case of Cohen–Macaulay 3-folds. This in turn is used to derive Kodaira vanishing for this class of varieties. Finally it is proved that small deformations of Du Bois singularities are again Du Bois.

Our aim is to study the cohomology groups of some coherent sheaves on a Griffiths–Schmid variety associated with an anisotropic Q-from of the unitary group SU(2,1). We define some transforms relating this cohomology to the coherent cohomology groups of some sheaves defined on certain threefolds, which are fibered in projective lines over Picard modular surfaces. In particular, we give a complete and explicit description, in terms of classical Picard modular forms, of the holomorphic (resp. anti-holomorphic) part of the 1-cohomology of the Griffiths-Schmid variety. From this, it results an explicit generating system for the part of the 2–cohomology which correspond to those automorphic representations whose archimedean component is a degenerate limit of discrete series.

The paper provides a combinatorial method to decide when the space of local systems with nonvanishing first cohomology on the complement to an arrangement of lines in a complex projective plane has as an irreducible component a subgroup of positive dimension. Partial classification of arrangements having such a component of positive dimension and a comparison theorem for cohomology of Orlik–Solomon algebra and cohomology of local systems are given. The methods are based on Vinberg–Kac classification of generalized Cartan matrices and study of pencils of algebraic curves defined by mentioned positive dimensional components.