Invariant hypersurfaces of endomorphisms of projective varieties

We consider surjective endomorphisms f of degree >1 on projective manifolds X   of Picard number one and their f⁻¹$f^{-1}$-stable hypersurfaces V, and show that V   is rationally chain connected. Also given is an optimal upper bound for the number of f⁻¹$f^{-1}$-stable prime divisors on (not necessarily smooth) projective varieties.     

Galois coverings of the product of the projective lines by abelian surfaces

We consider the quotient space of an abelian surface by a finite subgroup of the automorphism group. We classify the analytic representation of the group and the branch divisor of the natural projection to the quotient space, where the quotient space is isomorphic to the product of the projective lines.     

On endomorphisms of projective bundles

 Let X be a projective bundle. We prove that X admits an endomorphism of degree >1 and commuting with the projection to the base, if and only if X trivializes after a finite covering. When X is the projectivization of a vector bundle E of rank 2, we prove that it has an endomorphism of degree >1 on a general fiber only if E splits after a finite base change. Received: 16 September 2002 / Revised version: 15 November 2002 Published online: 3 March 2003     

Koszul algebras and finite Galois coverings

The relationships between Koszulity and finite Galois coverings are obtained, which provide a construction of Koszul algebras by finite Galois coverings.     

Positive Galois coverings of self-injective algebras

In the article, we study the structure of Galois coverings of self-injective artin algebras with infinite cyclic Galois groups. In particular, we characterize all basic, connected, self-injective artin algebras having Galois coverings by the repetitive algebras of basic connected artin algebras and with the Galois groups generated by positive automorphisms of the repetitive algebras.     

Quantitative Siegel's theorem for Galois coverings

It is known that Siegels theorem on integral points is effective for Galoiscoverings of the projective line. In this paper we obtain a quantitative version of this result, giving an explicit upper bound for the heights of S-integral K-rational points in terms of the number field K, the set of places S and the defining equation of the curve.Our main tools are Bakers theory of linear forms in logarithms and thequantitative Eisenstein theorem due to Schmidt, Dwork and van der Poorten.     

Torsion theories and Galois coverings of topological groups

For any torsion theory in a homological category, one can define a categorical Galois structure and try to describe the corresponding Galois coverings. In this article we provide several characterizations of these coverings for a special class of torsion theories, which we call quasi-hereditary. We describe a new reflective factorization system that is induced by any quasi-hereditary torsion theory. These results are then applied to study various examples of torsion theories in the category of topological groups.     

Mumford coverings of the projective line

A Mumford covering of the projective line over a complete non-archimedean valued field K is a Galois covering X? P¹_K X\rightarrow {\bf P}^1_K such that X is a Mumford curve over K. The question which finite groups do occur as Galois group is answered in this paper. This result is extended to the case where P¹_K {\bf P}^1_K is replaced by any Mumford curve over K.     

Equidistribution speed for endomorphisms of projective spaces

Let f be a non-invertible holomorphic endomorphism of mathbbP^k{mathbb{P}^{k}}, f ⁿ its iterate of order n and μ the equilibrium measure of f. We estimate the speed of convergence in the following known result. If a is a Zariski generic point in mathbbP^k{mathbb{P}^{k}}, the probability measures, equidistributed on the preimages of a under f ⁿ , converge to μ as n goes to infinity.     

Galois coverings of selfinjective algebras by repetitive algebras

In the representation theory of selfinjective artin algebras an important role is played by selfinjective algebras of the form where is the repetitive algebra of an artin algebra and is an admissible group of automorphisms of . If is of finite global dimension, then the stable module category of finitely generated -modules is equivalent to the derived category of bounded complexes of finitely generated -modules. For a selfinjective artin algebra , an ideal and , we establish a criterion for to admit a Galois covering with an infinite cyclic Galois group . As an application we prove that all selfinjective artin algebras whose Auslander-Reiten quiver has a non-periodic generalized standard translation subquiver closed under successors in are socle equivalent to the algebras , where is a representation-infinite tilted algebra and is an infinite cyclic group of automorphisms of .

     

On the Galois coverings of a cluster-tilted algebra

We study the module category of a certain Galois covering of a cluster-tilted algebra which we call the cluster repetitive algebra. Our main result compares the module categories of the cluster repetitive algebra of a tilted algebra C and the repetitive algebra of C, in the sense of Hughes and Waschbüsch.