Algebraic independence of the periods of abelian integrals

Abelian integrals of the first and the second kind are proved to have two algebraically independent periods. Some corollaries concerning the algebraic independence of the values of Euler's beta and gamma functions at rational points are derived.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 681–691, November, 1996.     

Inversion of potentials in rn with the aid of Gauss-Weierstrass integrals

Mathematical Notes -     

Inversion of singular integrals with multiplicative Cauchy kernel and infinite integration domain

We consider the simplest singular integral equations of the first kind with multiplicative Cauchy kernel and with integration either over the three-dimensional domain (0,+∞) × (0,+∞) × (0,+∞) or over the entire three-dimensional Euclidean space. For these equations, we construct general solutions in the Hölder class and find statements of uniquely solvable problems.     

Almost-affable abelian groups

In this paper we show that a direct summand of a simply presented mixed abelain group is an almost affable group. As a consequence, the classification theorem due to the author is extended to the largest possible class.     

Universal abelian groups

We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example:Theorem:For n≧2, there is a purely universal separable p-group in ℵ _n if, and only if, . Partially supported by the United States-Israel Binational Science Foundation. Publication number 455.     

Characterization of abelian varieties

We prove that any smooth complex projective variety X with plurigenera P ₁(X)=P ₂(X)=1 and irregularity q(X)=dim(X) is birational to an abelian variety. Oblatum 26-V-1999 & 13-VI-2000?Published online: 11 October 2000     

Quasi-minimal abelian groups

An abelian group is said to be quasi-minimal (purely quasi-minimal, directly quasi-minimal) if it is isomorphic to all its subgroups (pure subgroups, direct summands, respectively) of the same cardinality as . Obviously quasi-minimality implies pure quasi-minimality which in turn implies direct quasi-minimality, but we show that neither converse implication holds. We obtain a complete characterisation of quasi-minimal groups. In the purely quasi-minimal case, assuming GCH, a complete characterisation is also established. An independence result is proved for directly quasi-minimal groups.

     

Graphically abelian groups

We define a group G to be graphically abelian if the function g?g⁻¹ induces an automorphism of every Cayley graph of G. We give equivalent characterizations of graphically abelian groups, note features of the adjacency matrices for Cayley graphs of graphically abelian groups, and show that a non-abelian group G is graphically abelian if and only if G=E×Q, where E is an elementary abelian 2-group and Q is a quaternion group.     

Coloring abelian groups

A sufficient (and necessary, if n=2) condition for the existence of a particular kind of n-coloring of an abelian group is given, and applied to show that (a) the real line is colorable with two colors so that the distance 1 is forbidden for one color, and the distance s>0 for the other, or so that both 1 and s are forbidden for both colors, if and only if s is not the ratio of an odd and an even integer; (b) the chromatic number of Q² and Q³ is 2, but that of Qⁿ is greater than 2 for n>3.     

Syzygies of abelian varieties

We prove a conjecture of R. Lazarsfeld on the syzygies (of the homogeneous ideal) of abelian varieties embedded in projective space by multiples of an ample line bundle. Specifically, we prove that if is an ample line on an abelian variety, then satisfies the property as soon as . The proof uses a criterion for the global generation of vector bundles on abelian varieties (generalizing the classical one for line bundles) and a criterion for the surjectivity of multiplication maps of global sections of two vector bundles in terms of the vanishing of the cohomology of certain twists of their Pontrjagin product.