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S. Ivashkovich

Publications and source records attributed to S. Ivashkovich.

16 recordsLinked to original sources
arXiv2024

Non-existence of a holomorphic embedding of the Sobolev loop space into the projective Hilbert space

The goal of this paper is to understand the properties of meromorphic mappings with values in two model complex Hibert manifolds: projective Hilbert space $\pp(l^2)$ and Sobolev loop space of the Riemann sphere $L\pp^1$. It occurs that these properties are quite different. Based on our study we obtain as a corollary that $L\pp^1$ does not admit a closed holomorphic embedding to $\pp(l^2)$. In other words $L\pp^1$ is {\slsf not} a projective Hilbert variety despite of the fact that it is K\"ahler and meromorphic functions separate points on it. Moreover, we prove that $L\pp^1$ doesn't admit even a non-degenerate meromorphic map to $\pp (l^2)$.

math.CV↗
arXiv2022

Riemann surface of the Riemann zeta function

In this paper we treat the classical Riemann zeta function as a function of three variables: one is the usual complex $\adyn$-dimensional, customly denoted as $s$, another two are complex infinite dimensional, we denote it as $\b = \{b_n\}_{n=1}^{\infty}$ and $\z =\{z_n\}_{n=1}^{\infty}$. When $\b = \{1\}_{n=1}^{\infty}$ and $\z = \{\frac{1}{n}\}_{n=1}^{\infty}$ one gets the usual Riemann zeta function. Our goal in this paper is to study the meromorphic continuation of $ζ(\b , \z ,s)$ as a function of the triple $(\a , \z , s)$. Minor corrections, to appear in the Journal of Mathematical Analysis and Applications.

math.CV↗
arXiv2020

Loop Spaces as Hilbert-Hartogs Manifolds

We prove that generalized loop spaces of Hartogs manifolds are Hilbert-Hartogs. We prove also that Hilbert-Hartogs manifolds possess a better extension properties that it is postulated in their definition. Finally, we give a list of examples of Hilbert-Hartogs manifolds.

math.CV↗
arXiv2018

One side continuity of meromorphic mappings between real analytic hypersurfaces

We prove that a meromorphic mapping, which sends a peace of a real analytic strictly pseudoconvex hypersurface in $\cc^2$ to a compact subset of $\cc^N$ which doesn't contain germs of non-constant complex curves is continuous from the concave side of the hypersurface. This implies the analytic continuability along CR-paths of germs of holomorphic mappings from real analytic hypersurfaces with non-vanishing Levi form to the locally spherical ones in all dimensions.

math.CV↗
arXiv2010

Vanishing Cycles in Holomorphic Foliations by Curves and Foliated Shells

The purpose of this paper is the study of vanishing cycles in holomorphic foliations by complex curves on compact complex manifolds. The main result consists in showing that a vanishing cycle comes together with a much richer complex geometric object - we call this object a foliated shell.

math.CV↗
arXiv2010

Local properties of J-complex curves in Lipschitz-continuous structures

We prove the existence of primitive curves and positivity of intersections of $J$-complex curves for Lipschitz-continuous almost complex structures. These results are deduced from the Comparison Theorem for $J$-holomorphic maps in Lipschitz structures, previously known for $J$ of class $C^{1, Lip}$. We also give the optimal regularity of curves in Lipschitz structures. It occurs to be $C^{1,LnLip}$, i.e. the first derivatives of a $J$-complex curve for Lipschitz $J$ are Log-Lipschitz-continuous. A simple example that nothing better can be achieved is given. Further we prove the Genus Formula for $J$-complex curves and determine their principal Puisieux exponents (all this for Lipschitz-continuous $J$-s).

math.CV↗
arXiv2009

On convex to pseudoconvex mappings

In the works of Darboux and Walsh it was remarked that a one to one self mapping of $\rr^3$ which sends convex sets to convex ones is affine. It can be remarked also that a $\calc^2$-diffeomorphism $F:U\to U^{'}$ between two domains in $\cc^n$, $n\ge 2$, which sends pseudoconvex hypersurfaces to pseudoconvex ones is either holomorphic or antiholomorphic. \smallskip In this note we are interested in the self mappings of $\cc^n$ which send convex hypersurfaces to pseudoconvex ones. Their characterization is the following: {\it A $\calc^2$ - diffeomorphism $F:U'\to U$ (where $U', U\subset \cc^n$ are domains) sends convex hypersurfaces to pseudoconvex ones if and only if the inverse map $Φ\deff F^{-1}$ is weakly pluriharmonic, i.e. it satisfies some nice second order PDE very close to $\d\bar\d Φ= 0$.} In fact all pluriharmonic $Φ$-s do satisfy this equation, but there are also other solutions.

math.CV↗
arXiv2008

Extra extension properties of equidimensional holomorphic mappings: results and open questions

Holomorphic (nondegenerate) mappings between complex manifolds of the same dimension are of special interest. For example, they appear as coverings of complex manifolds. At the same time they have very strong "extra" extension properties in compare with mappings in different dimensions. The aim of this paper is to put together the known results on this subject, give some perspective on the general strategy for future progress, prove some new results and formulate open questions.

math.CV↗
arXiv2004

On nonimbeddability of Hartogs figures into complex manifolds

We propose a method to construct examples of strange imbeddings of Hartogs figures into complex manifolds. It gives an imbedding of a "thin" Hartogs figure which does not have any neighborhood biholomorphic to an open set in a Stein manifold, thus unswering a question of E. Poletsky. Then we give an example of a foliated manifold which does not admit any nontrivial imbeddings of a "thick" (i.e. usual) Hartogs figure, giving thus a counterexample to some "selfevident" statements used in foliation theory.

math.CV↗
arXiv2004

Hyperbolic distance to submanifolds in an almost-complex manifold

Complete hyperbolicity of small Euclidean balls with respect to a C^1-smooth almost complex structure standard at origin is improved to give a complete hyperbolicity of strictly pseudoconvex domains. More precise (and lower) regularity assumptions on almost complex structure are made also for another results.

math.CV↗
arXiv2000

Complete hyperbolic neighborhoods in almost-complex surfaces

We prove that each point in an almost-complex surface has a basis of complete hyperbolic neighborhoods. The problem is local, and therefore we can consider the case when our surface is ${\bf R^4}$ with an arbitrary almost-complex structure $J$ of class $C^{1.α}$. Let $C$ be a non-singular $J$-complex curve passing through the origin. Our result cah be stated as follows: There exists a basis $\{U_j\}$ of neighborhoods of zero in ${\bf R^4}$, such that $(U_j,J)$ are complete hyperbolic in the sence of Kobayashi, moreover $(U_j\setminus C,J)$ are complete hyperbolic as well. The fact that this result remains true for any almost-complex structure is somewhat suprising. Really, given any germ of a non-singular real surface $C\ni 0$ in ${\bf R^4}$, one can easily construct an almost-complex structure $J$ in a neighborhood of zero, such that $C$ becomes a $J$-complex curve. Typical corollary is the following: Let ${\cal M}_{ω, 5l}$ be the Banach manifold consisting of pairs $(J,\{D_j\}_{j=1}^5)$, where $J$ is any almost-complex structure on ${\bf CP^2}$ tamed by the Fubini-Studi form $ω$ and $\{D_j\}_{j=1}^5$ the union of five $J$-complex lines in ${\bf CP^2}$ in general position. The set ${\cal H}_{ω, 5l}$ consisting of $(J, \{D_j\}_{j=1}^5)$ with $Y=({\bf CP^2}\setminus \bigcup_{j=1}^5 D_j,J)$ hyperbolically imbedded into $({\bf CP^2}, J)$ is an open nonempty subset of ${\cal M}{ω, 5l}$.

math.CV↗
arXiv1999

Gromov compactness theorem for stable curves

We give a proof of the Gromov compactness theorem using the language of stable curves (i.e. cusp-curve of Gromov, or stable maps of Kontsevich and Manin) in general setting: An almost complex structure on a target manifold is only continuous and can vary; the curves are only assumed to have fixed ``topological type'', in particular they can be non-closed and the complex structures on them can vary arbitrarily. In connection with this, we study in §2 moduli spaces of nodal curves with boundary and define a natural complex structure for such moduli spaces. We obtain an apriori estimate for pseudoholomorphic maps of ``long cylinders'' (see §3), which gives a uniform description for degeneration of complex structure on the curves and for the ``bubbling'' phenomenon. It also implies the Hausdorff convergence of the curves. We also prove in §5 the compactness theorem for curves with boundary on totally real submanifolds. For this ``boundary'' case we give appropriate generalizations of all ``inner'' constructions and estimates.

math.DG↗