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Sergey Ivashkovich

Publications and source records attributed to Sergey Ivashkovich.

7 recordsLinked to original sources
arXiv2022

Discrete and Continuous Versions of the Continuity Principle

The goal of this paper is to present a certain generalization of the classical Kontinuitätssatz of Behnke for holomorphic/meromorphic functions in terms of the lift to the envelope of holomorphy. We consider two non-equivalent formulations: "discrete" and "continuous" ones. Giving a proof of the "discrete" version we, somehow unexpectedly, construct a counterexample to the "continuous" one when convergence/continuity of analytic sets is considered in Hausdorff topology or, even in the stronger topology of currents. But we prove the "continuous" version of the Kontinuitätssatz if continuity is understood with respect to the Gromov topology. Our formulations seem to be not yet existing in the literature. A number of relevant examples and open questions is given as well.

math.CV↗
arXiv2012

Banach Analytic Sets and a Non-Linear Version of the Levi Extension Theorem

We prove a certain non-linear version of the Levi extension theorem for meromorphic functions. This means that the meromorphic function in question is supposed to be extendable along a sequence of complex curves, which are arbitrary, not necessarily straight lines. Moreover, these curves are not supposed to belong to any finite dimensional analytic family. The conclusion of our theorem is that nevertheless the function in question meromorphically extends along an (infinite dimensional) analytic family of complex curves and its domain of existence is a pinched domain filled in by this analytic family.

math.CV↗
arXiv2010

Limiting behavior of trajectories of complex polynomial vector fields

We prove that every trajectory of a polynomial vector field on the complex projective plane accumulates to the singular locus of the vector field. This statement represents a holomorphic version of the Poincare-Bendixson theorem and solves the complex analytic counterpart of Hilbert's 16th problem. The main result can be also reformulated as the nonexistence of "exceptional minimals" of holomorphic foliations on $\pp^2$ and, in particular, implies the nonexistence of real analytic Levi flat hypersurfaces in the complex projective plane. Finally, we describe (in the first approximation) the way a minimal complex trajectory approaches the singular locus of the vector field.

math.CV↗
arXiv2010

On fixed points of rational self-maps of complex projective plane

For any given natural $d\ge 1$ we provide examples of rational self-maps of complex projective plane $\pp^2$ of degree $d$ without (holomorphic) fixed points. This makes a contrast with the situation in one dimension. We also prove that the set of fixed point free rational self-maps of $\pp^2$ is closed (modulo "degenerate" maps) in some natural topology on the space of rational self-maps of $\pp^2$.

math.CV↗
arXiv2004

Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for Hölderian almost complex structures

If $X$ is an almost complex manifold, with an almost complex structure $J$ of class $\CC^α$, for some $α>0$, for every point $p\in X$ and every tangent vector $V$ at $p$, there exists a germ of $J$-holomorphic disc through $p$ with this prescribed tangent vector. This existence result goes back to Nijenhuis-Woolf. All the $J$ holomorphic curves are of class $\CC^{1,α}$ in this case. Then, exactly as for complex manifolds one can define the Royden-Kobayashi pseudo-norm of tangent vectors. The question arises whether this pseudo-norm is an upper semi-continuous function on the tangent bundle. For complex manifolds it is the crucial point in Royden's proof of the equivalence of the two standard definitions of the Kobayashi pseudo-metric. The upper semi-continuity of the Royden-Kobayashi pseudo-norm has been established by Kruglikov for structures that are smooth enough. In [I-R], it is shown that $\CC^{1,α}$ regularity of $J$ is enough. Here we show the following: Theorem. There exists an almost complex structure $J$ of class $\CC^{1\over 2}$ on the unit bidisc $\D^2\subset \C^2$, such that the Royden-Kobayashi seudo-norm is not an upper semi-continuous function on the tangent bundle.

math.CV↗
arXiv2004

Extension Properties of Meromorphic Mappings with Values in Non-Kahler Manifolds

We prove an analogue of E. Levi's Continuity Principle for meromorphic mappings with values in arbitrary compact complex manifolds in place of the Riemann sphere $\cc\pp^1$. The result is achieved by introducing a new extension method for meromorphic mappings. One of the corollaries reads as follows: If a compact complex surface $X$ is not "among the known ones" then for every domain $Ω$ in a Stein surface every meromorphic mapping $f:Ω\to X$ is in fact holomorphic and extends as a holomorphic mapping $\hat f:\hat D\to X$ of the envelope of holomorphy $\hat D$ of $D$ into $X$. In this last version also two examples of compact complex maniflds are described with meromoprhic mappings into these manifolds having thin but non-analytic singularity sets.

math.CV↗