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.

We determine the universal deformation ring, in the sense of Mazur, of a residual representation $\bar \rho :G_K\to {\rm GL}_2(k)$, where k is a finite field of characteristic p and K is a local field of residue characteristic p. As one might hope for, but is not proven in the global case, the deformation ring is a complete intersection, flat over W(k), with the exact number of equations given by the dimension of $H^2(G_K,{\rm ad}_{\bar \rho})$. We then go on to determine the ordinary locus inside the deformation space and, using ideas of Mazur, apply this to compare the universal and the universal ordinary deformation spaces. Provided that the universal ring for ordinary deformations with fixed determinant is finite flat over W(k), as was shown in many cases by Diamond, Fujiwara, Taylor–Wiles and Wiles, we show that the corresponding universal deformation ring – with no restriction of ordinariness or fixed determinant – is a complete intersection, finite flat over W(k) of the dimension conjectured by Mazur, provided that the restriction of $\det (\bar \rho)$ to the inertia subgroup is different from the inverse cyclotomic character.

We investigate the Chow groups of zero cycles of products of curves over a p-adic field by means of the Milnor K-groups of their Jacobians as introduced by Somekawa. We prove some finiteness results for CH0(X)/m for X a product of curves over a p-adic field.

For a finite morphism f : X → Y of smooth varieties such that f maps X birationally onto X′=f(X), the local equations of f are obtained at the double points which are not triple. If $\cal C$ is the conductor of X over X′, and $D = Sing(X') ⊂ X'$, $Δ ⊂ X$ are the subschemes defined by $\cal C$, then D and Δ are shown to be complete intersections at these points, provided that $\cal C$ has “the expected” codimension. This leads one to determine the depth of local rings of X′ at these double points. On the other hand, when $\cal C$ is reduced in X, it is proved that X′ is weakly normal at these points, and some global results are given. For the case of affine spaces, the local equations of X′ at these points are computed.

For a domain Ω of ${\mathbb C}^N$ we introduce a fairly general and intrinsic condition of weak q-pseudoconvexity, and prove, in Theorem 4, solvability of the $\bar \partial$-complex for forms with $C^\infty (\bar \Omega)$-coefficients in degree $\geq q+1$. All domains whose boundary have a constant number of negative Levi eigenvalues are easily recognized to fulfill our condition of q-pseudoconvexity; thus we regain the result of Michel (with a simplified proof). Our method deeply relies on the $L^2$-estimates by Hörmander (with some variants). The main point of our proof is that our estimates (both in weightened-$L^2$ and in Sobolev norms) are sufficiently accurate to permit us to exploit the technique by Dufresnoy for regularity up to the boundary.

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.

The number N of rational points on an algebraic curve of genus g over a finite field ${\mathbb F}_q$ satisfies the Hasse–Weil bound $N \leq q + 1 +2g\sqrt {q}$. A curve that attains this bound is called maximal. With $g_0 =\textstyle {1\over 2}(q -\sqrt {q})$ and $g_1= \textstyle {1\over 4}(\sqrt {q}- 1)^2$, it is known that maximalcurves have $g= g_0$ or $g \leq g_1$. Maximal curves with $g=g_0$ or $g_1$ have been characterized up to isomorphism. A natural genus to be studied is $g_2 =\textstyle {1 \over 8}(\sqrt {q}- 1)(\sqrt {q}- 3),$ and for this genus there are two non-isomorphic maximal curves known when $\sqrt {q}\equiv 3\;({\rm mod}\; {4})$. Here, a maximal curve with genus $g_2$ and a non-singular plane model is characterized as a Fermat curve of degree $ {1\over 2}(\sqrt {q} + 1)$.

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.

Let Rk(n) denote the number of representations of a natural number n as the sum of three cubes and a kth power. In this paper, we show that R3(n) [Lt ] n5/9+ε, and that R4(n) [Lt ] n47/90+ε, where ε > 0 is arbitrary. This extends work of Hooley concerning sums of four cubes, to the case of sums of mixed powers. To achieve these bounds, we use a variant of the Selberg sieve method introduced by Hooley to study sums of two kth powers, and we also use various exponential sum estimates.

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.

In this paper, we study polynomial structures by starting on the Lie algebra level, then passing to Lie groups to finally arrive at the polycyclic-by-finite group level. To be more precise, we first show how a general solvable Lie algebra can be decomposed into a sum of two nilpotent subalgebras. Using this result, we construct, for any simply connected, connected solvable Lie group G of dim n, a simply transitive action on Rn which is polynomial and of degree ≤ n3. Finally, we show the existence of a polynomial structure on any polycyclic-by-finite group Γ, which is of degree ≤ h(Γ)3 on almost the entire group (h (Γ) being the Hirsch length of Γ).

We describe a general method for calculating equivariant Euler characteristics. The method exploits the fact that the γ-filtration on the Grothendieck group of vector bundles on a Noetherian quasi-projective scheme has finite length; it allows us to capture torsion information which is usually ignored by equivariant Riemann–Roch theorems. As applications, we study the G-module structure of the coherent cohomology of schemes with a free action by a finite group G and, under certain assumptions, we give an explicit formula for the equivariant Euler characteristic $\chi ({\cal O}_X)={\rm H}^0(X, {\cal O}_X)-{\rm H}^1(X, {\cal O}_X)$ in the Grothendieck group of finitely generated ${\bf Z}[G]$-modules, when X is a curve over ${\bf Z}$ and G has prime order.

Let A be a finite-dimensional k-algebra over algebraically closed field k and $m{\rm mod}\, A$ be the category of finite-dimensional left A-modules. We show that a module M in ${\rm mod}\, A$ degenerates to another module N in ${\rm mod}\, A$ if and only if there is an exact sequence $0\to N\to M\oplus Z\to Z\to 0$ in ${\rm mod}\, A$ for some A-module Z. Moreover, we give a description of minimal degenerations of finite-dimensional A-modules.

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.