Exotic analytic structures and Eisenman intrinsic measures

Using Eisenman intrinsic measures we prove a cancellation theorem. This theorem allows to find new examples of exotic analytic structures onC ⁿ under which we understand smooth complex affine algebraic varietiers which are diffeomorphic toR ²ⁿ but not biholomorphic toC ⁿ . We also develop a new method of constructing these structures which enables us to produce exotic analytic structures onC ³ with a given number of hypersurfaces isomorphic toC ² and a family of these structures with a given number of moduli.     

A Tranversality Theorem for Holomorphic Mappings and Stability of Eisenman-Kobayashi Measures

We show that Thom's Transversality Theorem is valid for holomorphic mappings from Stein manifolds. More precisely, given such a mapping from a Stein manifold to a complex manifold and given an analytic subset of the jet space can be approximated in neighborhoods of compacts by holomorphic mappings whose -jet extensions are transversal to . As an application the stability of Eisenman-Kobayshi intrinsic -measures with respect to deleting analytic subsets of codimension is proven. This is a generalization of the Campbell-Howard-Ochiai-Ogawa theorem on stability of Kobayashi pseudodistances.

     

Non-hyperbolic complex space with a hyperbolic normalization

We construct an example of a non-hyperbolic singular projective surface whose normalization is the square of a genus 3 curve and hence, hyperbolic.

     

On a theorem of Ax

We show that in the case of affine and complete algebraic varieties over an algebraically closed field of zero characteristics any endomorphism of such a variety, that is injective on the complement to a subvariety of codimension , is an automorphism.

     

Uniform Zariski’s theorem on fundamental groups

The Zariski theorem says that for every hypersurface in a complex projective (resp. affine) space and for every generic plane in the projective (resp. affine) space the natural embedding generates an isomorphism of the fundamental groups of the complements to the hypersurface in the plane and in the space. If a family of hypersurfaces depends algebraically on parameters then it is not true in general that there exists a plane such that the natural embedding generates an isomorphism of the fundamental groups of the complements to each hypersurface from this family in the plane and in the space. But we show that in the affine case such a plane exists after a polynomial coordinate substitution. The research was partially supported by an NSA grant.     

On subelliptic manifolds

A smooth complex quasi-affine algebraic variety Y is flexible if its special group SAut(Y) of automorphisms (generated by the elements of one-dimensional unipotent subgroups of Aut(Y)) acts transitively on Y, and an algebraic variety is stably flexible if its product with some affine space is flexible. An irreducible algebraic variety X is locally stably flexible if it is a union of a finite number of Zariski open sets each of which is stably flexible. The main result of this paper states that the blowup of a locally stably flexible variety along a smooth algebraic subvariety (not necessarily equidimensional or connected) is subelliptic, and, therefore, it is an Oka manifold.     

Nuclear excitation by positron annihilation with bound electrons in a screened atomic potential

We have calculated the total cross section for the process of nuclear excitation in positron-bound electron annihilation. The calculations presented in this work use a spherically symmetric screened atomic potential. Comparisons with more approximate treatments of the process are made.     

Perturbative calculation of the cross section in double ionization by high-energy Compton scattering

In this paper we investigate double ionization in high-energy Compton scattering from the He-atom including both the shake-off mechanism and a perturbative correction to that mechanism. The correction is calculated in second-order perturbation theory and includes Coulomb electron-electron interaction in addition to the correlation in the ground state of the He-atom. Our calculations for the ratio of double to single cross section cover the range from 30 to 300 keV of impact photon energy and explain the slow convergence of the ratio towards the asymptotic value.     

Free <Emphasis Type="Bold">C</Emphasis><Subscript>+</Subscript>-actions on <Emphasis Type="Bold">C</Emphasis><Superscript>3</Superscript> are translations

Let X be a smooth contractible three-dimensional affine algebraic variety with a free algebraic C₊-action on it such that S=X//C₊ is smooth. We prove that X is isomorphic to S×C and the action is induced by a translation on the second factor. As a consequence we show that any free algebraic C₊-action on C³ is a translation in a suitable coordinate system.