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Rasul Shafikov

Publications and source records attributed to Rasul Shafikov.

At least 19 recordsLinked to original sources
arXiv2026

Meromorphic Convexity on Complex Manifolds

The notion of meromorphic convexity is defined and studied on complex manifolds. Using this notion, in analogy with Stein manifolds, a new class of complex manifolds, called {\calligra M }-manifolds, is introduced. This is a class of complex manifolds with a good supply of global meromorphic functions, in particular, it includes all Stein manifolds and projective manifolds. It is also shown that there exist noncompact complex manifolds, known as long $\mathbb C^2$, that are {\calligra M }-manifolds but do not contain any nonconstant holomorphic functions.

math.CV↗
arXiv2026

Arc-analytic subanalytic functions on complex manifolds

We show that an arc-analytic subanalytic function on a complex manifold M, which is holomorphic near one point, is a holomorphic function on M. More generally, an arc-analytic subanalytic function on a real analytic CR-manifold M, which is CR on a nonempty open subset of M, is a CR-function on the whole M.

math.CV↗
arXiv2025

Levi-flats in $\mathbb{CP}^n$: a survey for nonexperts

This survey paper, aimed at nonexperts in the field, explores various proofs of nonexistence of real analytic Levi-flat hypersurfaces in $\mathbb CP^n$, $n>2$. Some generalizations and other related results are also discussed.

math.CV↗
arXiv2025

Polynomially convex embeddings and CR singularities of real manifolds

It is proved that any smooth manifold $\mathcal M$ of dimension $m$ admits a smooth polynomially convex embedding into $\mathbb C^n$ when $n\geq \lfloor 5m/4\rfloor$. Further, such embeddings are dense in the space of smooth maps from $\mathcal M$ into $\mathbb C^n$ in the $\mathcal C^3$-topology. The components of any such embedding give smooth generators of the algebra of complex-valued continuous functions on $\mathcal M$. A key ingredient of the proof is a coordinate-free description of certain notions of (non)degeneracy, as defined by Webster and Coffman, for CR-singularities of order one of an embedded real manifold in $\mathbb C^n$. The main result is obtained by inductively perturbing each stratum of degeneracy to produce a global polynomially convex embedding.

math.CV↗
arXiv2024

Hypersurface Convexity and Extension of Kähler Forms

The following generalization of a result of S. Nemirovski is proved: if $X$ is either a projective or a Stein manifold and $K\subset X$ is a compact sublevel set of a strictly plurisubharmonic function $φ$ defined in a neighborhood of $K$, then $X\setminus K$ is a union of positive divisors if and only if $dd^cφ$ extends to a Hodge form on $X$. For an arbitrary compact subset $K\subsetneq X$, this gives that $X\setminus K$ is a union of positive divisors if and only if $K$ admits a neighbourhood basis of sublevel sets of strictly plurisubharmonic functions with the $dd^c$-extension property.

math.CV↗
arXiv2023

On Rational Convexity of Totally Real Sets

Under a mild technical assumption, we prove a necessary and sufficient condition for a totally real compacdt set in $\mathbb{C}^n$ to be rationally convex. This generalizes a classical result of Duval-Sibony

math.CV↗
arXiv2020

Polynomially convex embeddings of odd-dimensional closed manifolds

It is shown that any smooth closed orientable manifold of dimension $2k + 1$, $k \geq 2$, admits a smooth polynomially convex embedding into $\mathbb C^{3k}$. This improves by $1$ the previously known lower bound of $3k+1$ on the possible ambient complex dimension for such embeddings (which is sharp when $k=1$). It is further shown that the embeddings produced have the property that all continuous functions on the image can be uniformly approximated by holomorphic polynomials. Lastly, the same technique is modified to construct embeddings whose images have nontrivial hulls containing no nontrivial analytic disks. The distinguishing feature of this dimensional setting is the appearance of nonisolated CR-singularities, which cannot be tackled using only local analytic methods (as done in earlier results of this kind), and a topological approach is required.

math.CV↗
arXiv2018

Polynomially Convex Embeddings of Even-Dimensional Compact Manifolds

The totally-real embeddability of any $2k$-dimensional compact manifold $M$ into $\mathbb C^n$, $n\geq 3k$, has several consequences: the genericity of polynomially convex embeddings of $M$ into $\mathbb C^n$, the existence of $n$ smooth generators for the Banach algebra $\mathcal C(M)$, the existence of nonpolynomially convex embeddings with no analytic disks in their hulls, and the existence of special plurisubharmonic defining functions. We show that these results can be recovered even when $n=3k-1$, $k>1$, despite the presence of complex tangencies, thus lowering the known bound for the optimal $n$ in these (related but inequivalent) questions.

math.CV↗
arXiv2017

Tameness of complex dimension in a real analytic set

Given a real analytic set X in a complex manifold and a positive integer d, denote by A(d) the set of points p in X at which there exists a germ of a complex analytic set of dimension d contained in X. It is proved that A(d) is a closed semianalytic subset of X.

math.CV↗
arXiv2017

Uniformization and Steinness

It is shown that the unit ball in ${\mathbb C}^n$ is the only complex manifold that can universally cover both Stein and non-Stein strictly pseudoconvex domains.

math.CV↗
arXiv2017

Some aspects of holomorphic mappings: a survey

This expository paper is concerned with the properties of proper holomorphic mappings between domains in complex affine spaces. We discuss some of the main geometric methods of this theory, such as the Reflection Principle, the scaling method, and the Kobayashi-Royden metric. We sketch the proofs of certain principal results and discuss some recent achievements. Several open problems are also stated.

math.CV↗
arXiv2016

Rational approximation and Lagrangian inclusions

We show that a Lagrangian inclusion in $\mathbb C^2$ with double transverse self-intersection points and standard open Whitney umbrellas is rationally convex. As an application we show that any compact surface $S$, except $S^2$ and $\mathbb RP_2$, admits a pair of smooth complex-valued functions $f_1$, $f_2$ with the property that any continuous complex valued function on $S$ is a uniform limit of a sequence of $R_j(f_1,f_2)$, where $R_j(z_1,z_2)$ are rational functions in $\mathbb C^2$.

math.CV↗
arXiv2016

Rational and Polynomial Density on Compact Real Manifolds

We establish a characterization for an $m$-manifold $M$ to admit $n$ functions $f_1$,...,$f_n$ and $n'$ functions $g_1,...,g_{n'}$ in $\mathcal{C}^\infty(M)$ so that every element of $\mathcal{C}^k(M)$ can be approximated by rational combinations of $f_1,...,f_n$ and polynomial combinations of $g_1,...,g_{n'}$. As an application, we show that the optimal value of $n$ and $n'$ for all manifolds of dimension $m$ is [3m/2], when $k\geq 1$ and $m\geq 2$.

math.CV↗