Non-isotropic weighted Lp estimates for the $$ \bar \partial $$-equation on piecewise smooth strictly pseudoconvex domains

In this paper we obtain non-isotropic weighted L ^p estimates with the boundary distance weight function for the -equation on piecewise smooth strictly pseudoconvex domains under a hypothesis of complex transversality in ℂⁿ using the explicit formula of solutions by Berndtsson-Andersson. This work was supported by the Korea Research Foundation Grant funded by Korea Government (MOEHRD, Basic Research Promotion Fund) (Grant No. KRF-2005-070-C00007)     

Abelian varieties and the general Hodge conjecture

We investigate the relationship between the usual and general Hodgeconjectures for abelian varieties. For certain abelian varieties A, weshow that the usual Hodge conjecture for all powers of A implies thegeneral Hodge conjecture for A.     

Logarithmic foliations on compact algebraic surfaces

LetF be a holomorphic singular foliation with an invariant compact curveS on a compact algebraic surfaceX. In this work we prove thatF is logarithmic under some generic conditions in the singularities ofF inS.Supported by CNPq (Brazil).     

Holomorphic and algebraic vector bundles on 0-convex algebraic surfaces

LetX be a smooth complex algebraic surface such that there is a proper birational morphism/:X → Y withY an affine variety. Let X_hol be the 2-dimensional complex manifold associated toX. Here we give conditions onX which imply that every holomorphic vector bundle onX is algebraizable and it is an extension of line bundles. We also give an approximation theorem of holomorphic vector bundles on X_hol (X normal algebraic surface) by algebraic vector bundles.     

Multiplier ideal sheaves in complex and algebraic geometry

The application of the method of multiplier ideal sheaves to effective problems in algebraic geometry is briefly discussed. Then its application to the deformational invari-ance of plurigenera for general compact algebraic manifolds is presented and discussed. Finally its application to the conjecture of the finite generation of the canonical ring is explored, and the use of complex algebraic geometry in complex Neumann estimates is discussed.     

Holomorphic functions and embedded real surfaces

The paper is devoted to the study of necessary and sufficient topological conditions for an embedded real surface to lie in a strictly pseudoconvex domain on a complex surface. These results are used to construct Stein domains on algebraic manifolds and to describe envelopes of holomorphy of real surfaces in P ² and in some other complex surfaces.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 599–606, April, 1998.The author is grateful to A. Vitushkin for the statement of the problem and for constant encouragement.This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01002 and by the International Science Foundation under grant No. a97-1850.     

Determination of a Homogeneous Strictly Pseudoconvex Surface from the Coefficients of Its Normal Equation

Mathematical Notes -     

Picard and Lefschetz numbers of real algebraic surfaces

Two Picard numbers and two Lefschetz numbers are defined for a real algebraic surface. They are similar to the Picard number and the Lefschetz number of a complex algebraic surface. For these numbers, some estimates and relations in the form of inequalities are proved.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 847–852, June, 1998.     

Real separated algebraic curves, quadrature domains, Ahlfors type functions and operator theory

The aim of this paper is to inter-relate several algebraic and analytic objects, such as real-type algebraic curves, quadrature domains, functions on them and rational matrix functions with special properties, and some objects from operator theory, such as vector Toeplitz operators and subnormal operators. Our tools come from operator theory, but some of our results have purely algebraic formulation. We make use of Xia's theory of subnormal operators and of the previous results by the author in this direction. We also correct (in Section 5) some inaccuracies in the works of [D.V. Yakubovich, Subnormal operators of finite type I. Xia's model and real algebraic curves in C², Rev. Mat. Iberoamericana 14 (1998) 95-115; D.V. Yakubovich, Subnormal operators of finite type II. Structure theorems, Rev. Mat. Iberoamericana 14 (1998) 623-681] by the author.     

Smoothness and Euler characteristic of the variety of complete pairsX 23 of zero-dimensional subschemes of length 2 and 3 of algebraic surfaces

A. S. Tikhomirov’s conjecture about the smoothness of the variety of complete pairsX ₂₃ of zero-dimensional subschemes of an irreducible projective algebraic surface is verified. In the caseS=P², the topological Euler characteristic of this variety is computed. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 276–287, February, 2000.     

Essential dimension and the second serre conjecture for exceptional groups

We study the essential dimension of exceptional connected simply connected algebraic groups over algebraically closed fields. In the present paper, we find upper estimates for the essential dimensions of the groupsF ₄ ,E ₆ , andE ₇ . For the groupF ₄ , the upper estimate, thus obtained coincides with the known lower estimate. We also prove the second Serre conjecture for the groupE ₆ and for a function field. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 539–547, October, 2000.     

Cobordism,Relative Indices and Stein Fillings

In this article we build on the framework developed in Ann. Math. 166, 183–214 ([2007]), 166, 723–777 ([2007]), 167, 1–67 ([2008]) to obtain a more complete understanding of the gluing properties for indices of boundary value problems for the Spin _ℂ-Dirac operator with sub-elliptic boundary conditions. We extend our analytic results for sub-elliptic boundary value problems for the Spin _ℂ-Dirac operator, and gluing results for the indices of these boundary problems to Spin _ℂ-manifolds with several pseudoconvex (pseudoconcave) boundary components. These results are applied to study Stein fillability for compact, 3-dimensional, contact manifolds. This material is based upon work supported by the National Science Foundation under Grant No. 0603973, and the Francis J. Carey term chair. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.