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Alexander Rashkovskii

Publications and source records attributed to Alexander Rashkovskii.

35 records · Page 2Linked to original sources
arXiv2012

Powers of ideals and convergence of Green functions with colliding poles

Let us have a family of ideals of holomorphic functions vanishing at N distinct points of a complex manifold, all tending to a single point. As is known, convergence of the ideals does not guarantee the convergence of the pluricomplex Green functions to the Green function of the limit ideal; moreover, the existence of the limit of the Green functions was unclear. Assuming that all the powers of the ideals converge to some ideals, we prove that the Green functions converge, locally uniformly away from the limit pole, to a function which is essentially the upper envelope of the scaled Green functions of the limits of the powers. As examples, we consider ideals generated by hyperplane sections of a holomorphic curve near its singular point. In particular, our result explains recently obtained asymptotics for 3-point models.

math.CV↗
arXiv2012

Extreme plurisubharmonic singularities

A plurisubharmonic singularity is extreme if it cannot be represented as the sum of non-homothetic singularities. A complete characterization of such singularities is given for the case of homogeneous singularities (in particular, those determined by generic holomorphic mappings) in terms of decomposability of certain convex sets in $\Rn$. Another class of extreme singularities is presented by means of a notion of relative type.

math.CV↗
arXiv2012

Multi-circled singularities, Lelong numbers, and integrability index

By comparing Green functions of multi-circled plurisubharmonic singularities in the n-domensional complex space to their indicators, we obtain formulas for the higher Lelong numbers and integrability index for such singularities and extend Howald's result on multiplier ideals for monomial ideals to multi-circled singularities. This also leads to an elementary proof of the relations between the k-th Lelong numbers and the integrability index. For k=1 and arbitrary plurisubharmonic functions the inequality is due to Skoda, and for k=n and any plurisubharmonic function with isolated singularity the relation is due to Demailly. We also describe multi-circled functions for which the inequalities are equalities. By a reduction to Demailly's inequality we prove these inequalities in the general case of plurisubharmonic functions as well. In addition, we get a description of all plurisubharmonic singularities whose integrability index is given by the lower bound in Skoda's inequality.

math.CV↗
arXiv2011

Special polyhedra for Reinhardt domains

We show that every bounded hyperconvex Reinhardt domain can be approximated by special polynomial polyhedra defined by homogeneous polynomial mappings. This is achieved by means of approximation of the pluricomplex Green function of the domain with pole at the origin.

math.CV↗
arXiv2011

Limits of multipole pluricomplex Green functions

Let $S_ε$ be a set of $N$ points in a bounded hyperconvex domain in $C^n$, all tending to 0 as$ε$ tends to 0. To each set $S_ε$ we associate its vanishing ideal $I_ε$ and the pluricomplex Green function $G_ε$ with poles on the set. Suppose that, as $ε$ tends to 0, the vanishing ideals converge to $I$ (local uniform convergence, or equivalently convergence in the Douady space), and that $G_ε$ converges to $G$, locally uniformly away from the origin; then the length (i.e. codimension) of $I$ is equal to $N$ and $G \ge G_I$. If the Hilbert-Samuel multiplicity of $I$ is strictly larger than $N$, then $G_ε$ cannot converge to $G_I$. Conversely, if the Hilbert-Samuel multiplicity of $I$ is equal to $N$, (we say that $I$ is a complete intersection ideal), then $G_ε$ does converge to $G_I$. We work out the case of three poles; when the directions defined by any two of the three points converge to limits which don't all coincide, there is convergence, but $G > G_I$.

math.CV↗
arXiv2010

Classical and new loglog-theorems

We present integral variants of results due to Carleman, Wolf, Levinson, Sjoberg, and Matsaev on majorants of analytic functions. Main ingredient is a complete description for radial projections of harmonic measures of strictly star-shaped domains in the plane.

math.CV↗
arXiv2009

An extremal problem for generalized Lelong numbers

We look for pointwise bounds on a plurisubharmonic function near its singularity point, given the value of its generalized Lelong number with respect to a plurisubharmonic weight. To this end, an extremal problem is considered. In certain cases, the problem is solved explicitly.

math.CV↗
arXiv2008

A remark on amoebas in higher codimensions

It is shown that tube sets over amoebas of algebraic varieties (and, more generally, of almost periodic holomorphic chains) of dimension q are q-pseudoconcave in the sense of Rothstein. This is a direct consequence of a representation of such sets as supports of positive closed currents.

math.CV↗
arXiv2008

Analyticity and propagation of plurisubharmonic singularities

A variant of Siu's analyticity theorem is proved for relative types of plurisubharmonic functions. Some results on propagation of plurisubharmonic singularities and maximality of pluricomplex Green functions with analytic singularities are derived.

math.CV↗
arXiv2007

Relative types and extremal problems for plurisubharmonic functions

We study properties of relative types of plurisubharmonic functions with respect to maximal plurisubharmonic weights. It is shown that they give a general form for upper semicontinuous, tropically additive functionals on plurisubharmonic singularities. We consider some extremal problems whose solutions are Green-like functions that give best possible bounds on a plurisubharmonic function, given the values of its types relative to some of (or all) the weights. An analyticity theorem is proved for the upperlevel sets for the types with respect to exponentially Hölder continuous weights, which leads to a result on propagation of plurisubharmonic singularities.

math.CV↗
arXiv2001

Total Masses of Mixed Monge-Ampere Currents

Monge-Ampere currents generated by plurisubharmonic functions of logarithmic growth are studied. Upper bounds for their total masses are obtained in terms of growth characteristics of the functions. In particular, this gives a plurisubharmonic version of D.Bernstein's theorem on the number of zeros of polynomial mappings in terms of the Newton polyhedra, and a representation for Demailly's generalized degrees of plurisubharmonic functions.

math.CV↗
arXiv2000

Lelong numbers with respect to regular plurisubharmonic weights

Generalized Lelong numbers of plurisubharmonic functions with respect to plurisubharmonic weights (due to Demailly) are specified for weights with multicircled asymptotics. Explicit formulas for these values are obtained in terms of the directional Lelong numbers of the functions and the Newton diagrams of the weights. An extension of Demailly's approximation theorem is proved as well.

math.CV↗
arXiv1999

Indicators for Plurisubharmonic Functions of Logarithmic Growth

A notion of indicator for a plurisubharmonic function u of logarithmic growth in C^n is introduced and studied. It is applied to evaluation of the total Monge-Ampère measure (dd^cu)^n({C}^n). Upper bounds for the measure are obtained in terms of growth characteristics of u. When u=log|f| for a polynomial mapping f with isolated zeros, the indicator generates the Newton polyhedron of f whose volume bounds the number of the zeros.

math.CV↗
arXiv1999

Newton Numbers and Residual Measures of Plurisubharmonic Functions

We study the masses charged by $(dd^cu)^n$ at isolated singularity points of plurisubharmonic functions $u$. It is done by means of the local indicators of plurisubharmonic functions. As a consequence, bounds for the masses are obtained in terms of the directional Lelong numbers of $u$, and the notion of the Newton number for a holomorphic mapping is extended to arbitrary plurisubharmonic functions. We also describe the local indicator of $u$ as the logarithmic tangent to $u$.

math.CV↗
arXiv1999

Local Indicators for Plurisubharmonic Functions

A notion of local indicator for a plurisubharmonic function is introduced. The indicator is a certain plurisubharmonic function in the unit polydisc, which controls the behavior of the considered function near a fixed point of its singularity, as well as the residual Monge-Ampere mass of the function at this point. A pluricomplex Green function with respect to given indicators is constructed and applied to the Dirichlet problem for the Monge- Ampere operator in the class of plurisubharmonic functions with isolated singularities.

math.CV↗