Vanishing theorems and universal coverings of projective varieties

This article contains a new argument which proves vanishing of the first cohomology for negative vector bundles over a complex projective variety if the rank of the bundle is smaller than the dimension of the base. Similar argument is applied to the construction of holomorphic functions on the universal covering of the complex projective variety .Comment: 6 pages, AMSTe

Special elliptic fibrations

We construct examples of elliptic fibrations of orbifold general type (in the sense of Campana) which have no etale covers dominating a variety of general type.Comment: 13 pages, LaTe

Lagrangian subvarieties of abelian fourfolds

We construct examples of algebraic surfaces with interesting fundamental groups.Comment: 26 pages, LaTe

On uniformly rational varieties

We investigate basic properties of uniformly rational varieties, i.e. those smooth varieties for which every point has a Zariski open neighborhood isomorphic to an open subset of A^n. It is an open question of Gromov whether all smooth rational varieties are uniformly rational. We discuss some potential criteria that might allow one to show that they form a proper subclass in the class of all smooth rational varieties. Finally we prove that small algebraic resolutions and big resolutions of nodal cubic threefolds are uniformly rational.Comment: 18 page

Co-fibered products of algebraic curves

We give examples of failure of the existence of co-fibered products in the category of algebraic curves.Comment: 7 page

Essential dimension, stable cohomological dimension, and stable cohomology of finite Heisenberg groups

We compare the notions of essential dimension and stable cohomological dimension of a finite group G, prove that the latter is bounded by the length of any normal series with cyclic quotients for G, and show that, however, this bound is not sharp by showing that the stable cohomological dimension of the finite Heisenberg groups H_p, p any prime, is equal to two.Comment: 18 page

Universal spaces for unramified Galois cohomology

We construct and study universal spaces for birational invariants of algebraic varieties over algebraic closures of finite fields.Comment: 34 page

Weak Hironaka theorem

The purpose of this note is to give a simple proof of the following theorem: Let $X$ be a normal projective variety over an algebraically closed field $k$, \op{char} k = 0 and let $D \subset X$ be a proper closed subvariety of $X$. Then there exist a smooth projective variety $M$, a strict normal crossings divisor $R \subset M$ and a birational morphism $f : M \to X$ with $f^{-1} D = R$. The method of proof is inspired by A.J. de Jong alteration ideas. We also use a multidimensional version of G.Belyi argument which allows us to simplify the shape of a ramification divisor. By induction on the dimension of $X$ the problem is reduced to resolving toroidal singularities. This process however is too crude and does not permit any control over the structure of the birational map $f$. A different proof of the same theorem was found independently by D. Abramovich and A.J. de Jong. The approach is similar in both proofs but they seem to be rather different in details.Comment: 11 pages, minor corrections, version revised for publication LATEX 2

Noether's problem and descent

We study Noether's problem from the perspective of torsors under linear algebraic groups and descent.Comment: 14 page

Commuting elements in Galois groups of function fields

We study abelian subgroups of Galois groups of function fields.Comment: 49 pages, LaTe