We consider CR submersive mappings between generic submanifolds in complex space. We show that, under suitable conditions on the manifolds, there is an integer k such that any jet of the CR mapping at a given point is a rational function of its k-jet at that point. As a consequence, it is shown that the stability group of a (suitably nondegenerate) real-analytic generic submanifold is a real algebraic Lie group. Moreover, it is shown that a formal equivalence between two such real-analytic submanifolds is necessarily convergent.
Constructing Rational Curves via Jets on Projective Varieties with Non-Nef Canonical Bundle
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学国家重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
The topic of the paper is the study of germs of local holomorphisms $f$ between $C^n$ and $C^{n'}$ such that $f(M)\subset M'$ and $df(T^cM)=T^cM'$ for $M\subset C^n$ and $M'\subset C^{n'}$ generic real-analytic CR submanifolds of arbitrary codimensions. It is proved that for $M$ minimal and $M'$ finitely nondegenerate, such germs depend analytically on their jets. As a corollary, an analytic structure on the set of all germs of this type is obtained.
We show that germs of local real-analytic CR automorphisms of a real-analytic hypersurface $M$ in $\C^2$ at a point $p\in M$ are uniquely determined by their jets of some finite order at $p$ if and only if $M$ is not Levi-flat near $p$. This seems to be the first necessary and sufficient result on finite jet determination and the first result of this kind in the infinite type case. If $M$ is of finite type at $p$, we prove a stronger assertion: the local real-analytic CR automorphisms of $M$ fixing $p$ are analytically parametrized (and hence uniquely determined) by their 2-jets at $p$. This r…
For a smooth strictly pseudoconvex hypersurface in a complex manifold, we give a necessary and sufficient condition for being CR-diffeomorphic to a real-analytic CR manifold. Our condition amounts to a holomorphic extension property for the canonically associated function expressing $2$-jets of the formal Segre varieties in terms of their $1$-jets. We also express this condition in equivalent terms for a Fefferman type determinant
Gonchar-Stahl's $ρ^2$-theorem characterizes the rate of convergence of best uniform (Chebyshev) rational approximations (with free poles) for one basic class of analytic functions. The theorem itself, its modifications and generalizations, methods involved in the proof and other related details constitute an important subfield in the theory of rational approximations of analytic functions and complex analysis. The paper briefly outlines essentials of the subfield. Fundamental contributions by A. A. Gonchar and H. Stahl are in the center of the exposition.
For any real-analytic hypersurface M in complex euclidean space of dimension >= 2 which does not contain any complex-analytic subvariety of positive dimension, we show that for every point p in M the local real-analytic CR automorphisms of M fixing p can be parametrized real-analytically by their l(p)-jets at p. As a direct application, we derive a Lie group structure for the topological group Aut(M,p). Furthermore, we also show that the order l(p) of the jet space in which the group Aut(M,p) embeds can be chosen to depend upper-semicontinuously on p. As a first consequence, it follows that th…
We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having self-intersection +1. We study the analytic classification, existence of normal forms, pencil/fibration decomposition, infinitesimal symmetries. We deduce some transcendental result about Painlev'e equations.
We introduce a family of discrete analytic functions, called expandable discrete analytic functions, which includes discrete analytic polynomials, and define two products in this family. The first one is defined in a way similar to the Cauchy-Kovalevskaya product of hyperholomorphic functions, and allows us to define rational discrete analytic functions. To define the second product we need a new space of entire functions which is contractively included in the Fock space. We study in this space some counterparts of Schur analysis.
Let (M,p) and (M',p') be the germs of real-analytic 1-infinite type hypersurfaces in \C^2. We prove that any formal equivalence sending (M,p) into (M',p') is formally parametrized (and hence uniquely determined by) its jet at p of a predetermined order depending only on (M,p). As an application, we use this to examine the local formal transformation groups of such hypersurfaces.
The hyperbolicity statements for subvarieties and complements of hypersurfaces in abelian varieties admit arithmetic analogues, due to Faltings (and Vojta for the semi-abelian case). In Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018) by the second author, an analogy between the analytic and arithmetic theories was shown to hold also at proof level, namely in a proof of Raynaud's theorem (Manin-Mumford Conjecture). The first aim of this paper is to extend to the relative setting the above mentioned hyperbolicity results. We shall be concerned with analytic sections of a relative (semi…
For an interval $E=[a,b]$ on the real line, let $μ$ be either the equilibrium measure, or the normalized Lebesgue measure of $E$, and let $V^μ$ denote the associated logarithmic potential. In the present paper, we construct a function $f$ which is analytic on $E$ and possesses four branch points of second order outside of $E$ such that the family of the admissible compacta of $f$ has no minimizing elements with regard to the extremal theoretic-potential problem, in the external field equals $V^{-μ}$.
For a rational matrix function $Φ$ with poles outside the unit circle, we estimate the degree of the unique superoptimal approximation $\AΦ$ by matrix functions analytic in the unit disk. We obtain sharp estimates in the case of $2\times2$ matrix functions. It turns out that ``generically'' $°\AΦ\le\degΦ-2$. We prove that for an arbitrary $2\times2$ rational function $Φ$, $°\AΦ\le2\degΦ-3$ whenever $\degΦ\ge2$. On the other hand, for $k\ge2$, we construct a $2\times2$ matrix function $Φ$, for which $\degΦ=k$, while $°\AΦ=2k-3$. Moreover, we conduct a detailed analysis of the situation when the…
We consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic. Combining this with a result obtained by the first author, we show that if for each semigroup of such an analytic family of expanding semigroups satisfies the open set condition, then the function of the Hausdorff dimension of the Julia set is real-analytic and plurisubharmonic. Moreover, we provide an extensive collection of classes of examples of analytic fam…
Using Zeilberger generating function formula for the values of a discrete analytic function in a quadrant we make connections with the theory of structured reproducing kernel spaces, structured matrices and a generalized moment problem. An important role is played by a Krein space realization result of Dijksma, Langer and de Snoo for functions analytic in a neighborhood of the origin. A key observation is that, using a simple Moebius transform, one can reduce the study of discrete analytic functions in the upper right quadrant to problems of function theory in the open unit disk. As an example…
It is shown that no purely topological condition implies the equality of the polynomial and rational hulls of a set: For any compact subset $K$ of a Euclidean space, there exists a set $X$, in some ${\mathbb C}^N$, that is homeomorphic to $K$ and is rationally convex but not polynomially convex. In addition, it is shown that for the surfaces in ${\mathbb C}^3$ constructed by Izzo and Stout, whose polynomial hulls are nontrivial but contain no analytic discs, the polynomial and rational hulls coincide, thereby answering a question of Gupta. Equality of polynomial and rational hulls is shown als…
Analytic and Algebraic Oka-1 Approximation for Smooth Projective Morphisms with Rationally Connected Fibers
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
We investigate the asymptotic behavior for type II Hermite-Pade approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite-Pade approximants are described by an algebraic function of order 3 and genus 0. This situation gives rise to a vector-potential equilibrium problem for three measures and the poles of the common denominator are asymptotically distributed like one of these measures. We also work out the strong asymptotics f…
