Extension Properties of Meromorphic Mappings with Values in Non-Kahler Manifolds
Sergey Ivashkovich
备注：
This version replaces the MSRI-1997-033 e-print and math.CV/9704219 e-print. Since the part of the old manuscript concerning the complex Plateau problem appeared in Ann. Pol. Math. it is removed from the new version. Also removed the part about separate meromorphicity which appeared in Americam J. Math. To appear in Annals of Mathematics
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.
The purpose of this paper has twofold. The first is to prove a unicity theorem for meromorphic mappings of a complete Kähler manifold M in P^n(C) sharing few hypersurfaces. The second is to give a unicity theorem for the case of differential nondegenerate meromorphic mappings sharing 2N-n+3+2ρ(2N-n+1) hyperplanes in N subgeneral position.
The purpose of this paper is to prove the finiteness theorems for meromorphic mappings of a complete connected Kähler manifold into projective space sharing few hyperplanes in subgeneral position without counting multiplicity, where all zeros with multiplicities more than a certain number are omitted. Our results are extensions and generalizations of some recent ones.
Let $X$ be a compact complex non-Kähler manifold and $f$ a dominant meromorphic self-map of $X$. Examples of such maps are self-maps of Hopf manifolds, Calabi-Eckmann manifolds, non-tori nilmanifolds and their blowups. We prove that if $f$ has a dominant topological degree, then $f$ possesses an equilibrium measure $μ$ satisfying well-known properties as in the Kähler case. The key ingredients are the notion of weakly d.s.h. functions substituting d.s.h. functions in the Kähler case and the use of suitable test functions in Sobolev spaces. A large enough class of holomorphic self-maps with dom…
Let M be a (bounded or not) domain of C^n which is complete with respect to a Kähler metric, or more generally, a complete Kähler manifold with trivial canonical bundle. Let f be a linearly nondegenerate meromorphic map from M to the complex projective space P^m. Under an assumption on the positivity of the pull-back by f of the Fubini-Study form on P^m, we prove that f can not omit a certain number of hyperplanes in subgeneral position in P^m. This is deduced directly from a non-integrated defect relation for such f which generalizes that obtained by Fujimoto in the case where M is a ball.
In this paper, a non-integrated defect relation for meromorphic maps from complete Kähler manifolds $M$ into smooth projective algebraic varieties $V$ intersecting hypersurfaces located in $k$-subgeneral position is proved. The novelty of this result lies in that both the upper bound and the truncation level of our defect relation depend only on $k$, $\dim_{\,\mathbb{C}}(V)$ and the degrees of the hypersurfaces considered. In addition, this defect relation recovers Hirotaka Fujimoto [Theorem 1.1; MR0884636 (88m:32049); Japan. J. Math. (N.S.) 11 (1985), no. 2, 233-264.] when subjected to the sa…
Let $M$ be a complete Kähler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0<R_0\le +\infty$). Our first aim in this paper is to study the algebraic dependence problem of differentiably meromorphic mappings. We will show that if $k$ differentibility nondegenerate meromorphic mappings $f^1,\ldots,f^k$ of $M$ into $\mathbb P^n(\mathbb C)\ (n\ge 2)$ satisfying the condition $(C_ρ)$ and sharing few hyperplanes in subgeneral position regardless of multiplicity then $f^1\wedge\cdots\wedge f^k\equiv 0$. For the second aim, we will show that there …
In this paper, we establish a truncated non-integrated defect relation for meromorphic mappings from a complete Kähler manifold into a projective variety intersecting a family of hypersurfaces located in subgeneral position, where the truncation level of the defect is explicitly estimated. Our result generalizes and improves previous ones. In particular, when the family of hypersurfaces located in general position, our theorem will implies the previous result of Min Ru-Sogome. In the last part of this paper we will apply ours to study the distribution of the Gauss map of minimal surfaces.
In this article, we establish a truncated non-integrated defect relation for meromorphic mappings from an $m$-dimensional complete Kähler manifold into $\mathbb P^n(\mathbb C)$ intersecting $q$ hypersurfaces $Q_1,...,Q_q$ in $k$-subgeneral position of degree $d_i$, i.e., the intersection of any $k+1$ hypersurfaces is emptyset. We will prove that $$ \sum_{i=1}^qδ_f^{[u-1]}(Q_i)\le (k-n+1)(n+1)+ε+\frac{ρu(u-1)}{d}, $$ where $u$ is explicitly estimated and $d$ is the least common multiple of $d_i'$s. Our result generalizes and improves previous results. In the last part of this paper we will appl…
Let $M$ be a complete Kähler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0<R_0\le +\infty$). In this article, we will show that if three meromorphic mappings $f^1,f^2,f^3$ of $M$ into $\mathbb P^n(\mathbb C)\ (n\ge 2)$ satisfying the condition $(C_ρ)$ and sharing $q\ (q> C+ρK)$ hyperplanes in general position regardless of multiplicity with certain positive constants $K$ and $C <2n$ (explicitly estimated), then there are some algebraic relation between them. A degeneracy theorem for $k\ (2\le k\le n+1)$ meromorphic mappings sharing hyperp…
We consider the dynamics of meromorphic maps of compact Kähler manifolds. In this work, our goal is to locate the non-nef locus of invariant classes and provide necessary and sufficient conditions for existence of Green currents in codimension one.
Let $M$ be a complete Kähler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0<R_0\le +\infty$). In this article, we will show that if three meromorphic mappings $f^1,f^2,f^3$ of $M$ into $\mathbb P^n(\mathbb C)\ (n\ge 2)$ satisfying the condition $(C_ρ)$ and sharing $q\ (q> 2n+1+α+ρK)$ hyperplanes in general position regardless of multiplicity with certain positive constants $K$ and $α<1$ (explicitly estimated), then $f^1=f^2$ or $f^2=f^3$ or $f^3=f^1$. Moreover, if the above three mappings share the hyperplanes with mutiplicity counted to l…
In this paper, we extend the uniqueness theorem for meromorphic mappings to the case where the family of hyperplanes depends on the meromorphic mapping and where the meromorphic mappings may be degenerate.
If E is a nonempty closed subset of the locally finite Hausdorff (2n-2)-measure on an n-dimensional complex manifold M and all points of E are nonremovable for a meromorphic mapping of M \ E into a compact Kähler manifold, then E is a pure (n-1)-dimensional complex analytic subset of M.
This article deals with the multiple values and algebraic dependences problem of meromorphic mappings sharing moving hyperplanes in projective space. We give some algebraic dependences theorems for meromorphic mappings sharing moving hyperplanes without counting multiplicity, where all zeros with multiplicities more than a certain number are omitted. Basing on these results, some unicity theorems regardless of multiplicity for meromorphic mappings in several complex variables are given. These results are extensions and strong improvements of some recent results.
We prove that a family of meromorphic mappings from a bidisc to a compact complex surface, which are equicontinuous in a neighborhood of the boundary of the bidisc, has the volumes of its graphs locally uniformly bounded.
In this article we prove a sufficient condition of quasi-normality in higher dimension for a family of meromorphic mappings in which each pair of functions of family shares some moving hypersurfaces. We also prove a normality criterion concerning shared moving hypersurfaces.
The purpose of this article is to study the uniqueness problem for meromorphic mappings from $\mathbb{C}^{n}$ into the complex projective space $\mathbb{P}^{N}(\mathbb{C}).$ By making using of the method of dealing with multiple values due to L. Yang and the technique of Dethloff-Quang-Tan respectively, we obtain two general uniqueness theorems which improve and extend some known results of meromorphic mappings sharing hyperplanes in general position.
In this article, we establish some new second main theorems for meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ and moving hypersurfaces with truncated counting functions. A uniqueness theorem for these mappings sharing few moving hypersurfaces without counting multiplicity is also given. This result is an improvement of the recent result of Dethloff - Tan [3]. Moreover the meromorphic mappings in our result may be algebraically degenerate. The last purpose of this article is to study uniqueness problem in the case where the meromorphic mappings agree on small identical set…
In this article, we show some uniqueness theorems for meromorphic mappings of $\C^n$ into the complex projective space $\pnc$ sharing different families of moving hyperplanes regardless of multiplicites, where all intersecting points between these mappings and moving hyperplanes with multiplicities more than a certain number do not need to be counted.
In this article, we show some new second main theorems for the mappings and moving hyperplanes of $¶^n(\C)$ with truncated counting functions. Our results are improvements of recent previous second main theorems for moving hyperplanes with the truncated (to level $n$) counting functions. As their application, we prove a unicity theorem for meromorphic mappings sharing moving hyperplanes.
In this article, by introducing a new method in estimating the counting function of the auxiliary function, we prove a new generalization of uniqueness theorems for meromorphic mappings into $¶^n(\C )$ which share few hyperplanes regardless of multiplicities. Our result improves the previous result in this topic. Moreover our proof is simpler than the previous proofs.
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.
In this paper, we prove a general second main theorem for meromorphic mappings into a subvariety $V$ of $\mathbb P^N(\mathbb C)$ with an arbitrary family of moving hypersurfaces. Our second main theorem generalizes and improves all previous results for meromorphic mappings with moving hypersurfaces, in particular for meromorphic mappings and families of moving hypersurfaces in subgeneral position. The method of our proof is different from that of previous authors used for the case of moving hypersurfaces.
In this paper, we prove some difference analogue of second main theorems of meromorphic mapping from Cm into an algebraic variety V intersecting a finite set of fixed hypersurfaces in subgeneral position. As an application, we prove a result on algebraically degenerate of holomorphic curves intersecting hypersurfaces and difference analogue of Picard's theorem on holomorphic curves. Furthermore, we obtain a second main theorem of meromorphic mappings intersecting hypersurfaces in N-subgeneral position for Veronese embedding in Pn(C) and a uniqueness theorem sharing hypersurfaces.
The purpose of this article is to deal with the multiple values and uniqueness problem of meromorphic mappings from $\mathbb{C}^{m}$ into the complex projective space $\mathbb{P}^{n}(\mathbb{C})$ sharing fixed and moving hypersurfaces. We obtain several uniqueness theorems which improve and extend some known results.
Let $c\in \mathbb{C}^{m},$ $f:\mathbb{C}^{m}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly nondegenerate meromorphic mapping over the field $\mathcal{P}_{c}$ of $c$-periodic meromorphic functions in $\mathbb{C}^{m}$, and let $H_{j}$ $(1\leq j\leq q)$ be $q(>2N-n+1)$ hyperplanes in $N$-subgeneral position of $\mathbb{P}^{n}(\mathbb{C}).$ We prove a new version of the second main theorem for meromorphic mappings of hyperorder strictly less than one without truncated multiplicity by considering the Casorati determinant of $f$ instead of its Wronskian determinant. As its applications, we obt…
Let $ι: \C^2 \hookrightarrow S$ be a compactification of the two dimensional complex space $\C^2$. By making use of Nevanlinna theoretic methods and the classification of compact complex surfaces K. Kodaira proved in 1971 (\cite{ko71}) that $S$ is a rational surface. Here we deal with a more general meromorphic map $f: \C^n \to X$ into a compact complex manifold $X$ of dimension $n$, whose differential $df$ has generically rank $n$. Let $ρ_f$ denote the order of $f$. We will prove that if $ρ_f<2$, then every global symmetric holomorphic tensor must vanish; in particular, {\it if $\dim X=2$ and…
Let $M$ be a generic CR submanifold in $\C^{m+n}$, $m= CRdim M \geq 1$,$n=codim M \geq 1$, $d=dim M = 2m+n$. A CR meromorphic mapping (in the sense of Harvey-Lawson) is a triple $(f,{\cal D}_f, [Γ_f])$, where: 1. $f: {\cal D}_f \to Y$ is a ${\cal C}^1$-smooth mapping defined over a dense open subset ${\cal D}_f$ of $M$ with values in a projective manifold $Y$; 2. The closure $Γ_f$ of its graph in $\C^{m+n} \times Y$ defines a oriented scarred ${\cal C}^1$-smooth CR manifold of CR dimension $m$ (i.e. CR outside a closed thin set) and 3. Such that $d[Γ_f]=0$ in the sense of currents. We prove in…
A proper or singular abelian mapping from $C^n$ to $\bar\C^n$ is parametrized by $n$ meromorphic functions with at most $2n$ periods. We develop the existence and structure theorems of the classical theory of an abelian mapping purely on the basis of its defining functional equation, the so-called algebraic addition theorem (AAT), with no appeal to any representation as quotients of theta functions. We offer two new proofs of the periodicity of a nonrational mapping admitting an AAT. We also prove by new arguments the existence of a rational group law on an associated algebraic variety, and th…
In this paper, we prove some results in normal family of meromorphic mappings intersecting with moving hypersurfaces. As some applications, we establish some results for normal mapping and extension of holomorphic mappings. A our result is strongly extended the Montel's normal criterion in the case several variables which is due to Tu in [Proc. Amer. Math. Soc. 127,1039-1049, 1999]. Our results are also strongly extended the results of Tu-Li in [Sci. China Ser. A. 48, 355-364, 2005] and [J. Math. Anal. Appl. 342, 629-638, 2006].
Let $M\subset \mathbb C^n$ be a real analytic hypersurface, $M'\subset \mathbb C^N$ $(N\geq n)$ be a strongly pseudoconvex real algebraic hypersurface of the special form and $F$ be a meromorphic mapping in a neighborhood of a point $p\in M$ which is holomorphic in one side of $M$. Assuming some additional conditions for the mapping $F$ on the hypersurface $M$, we proved that $F$ has a holomorphic extension to $p$. This result may be used to show the regularity of CR mappings between real hypersurfaces of different dimensions.
Nevanlinna showed that two non-constant meromorphic functions on $\mathbb C$ must be linked by a Möbius transformation if they have the same inverse images counted with multiplicities for four distinct values. After that this results is generalized by Gundersen to the case where two meromorphic functions share two values ignoring multiplicity and share other two values with multiplicities trucated by 2. Previously, the first author proved that for $n\ge 2,$ there are at most two linearly nondegenerate meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ sharing $2n+2$ hyperplane…
In this article, we prove that there are at most two meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)\ (n\geqslant 2)$ sharing $2n+2$ hyperplanes in general position regardless of multiplicity, where all zeros with multiplicities more than certain values do not need to be counted. We also show that if three meromorphic mappings $f^1,f^2,f^3$ of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)\ (n\geqslant 5)$ share $2n+1$ hyperplanes in general position with truncated multiplicity then the map $f^1\times f^2\times f^3$ is linearly degenerate.
In this article, we establish some new second main theorems for meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ and moving hyperplanes with truncated counting functions. Our results are improvements of the previous second main theorems for moving hyperplanes with truncated (to level $n$) counting functions.
The main aim of this article is to give some sufficient conditions for a family of meromorphic mappings on a domain D in C^n into P^N(C) to be meromorphically normal if they satisfy only some very weak conditions with respect to moving hypersurfaces in P^N(C), namely that their intersections with these moving hypersurfaces, which may moreover depend on the meromorphic maps, are in some sense uniform. Our results generalise and complete previous results in this area, especially the works of Fujimoto, Tu, Tu-Li, Mai-Thai-Trang and the recent work of Quang-Tan.
We show that an $n$-dimensional compact Kähler manifold $X$ admitting a non-degenerate meromorphic map $f:{\bf C}^n\to X$ of order $ρ_f<2$ is rationally connected.
Let $V$ be a projective subvariety of $\mathbb P^n(\mathbb C)$. A family of hypersurfaces $\{Q_i\}_{i=1}^q$ in $\mathbb P^n(\mathbb C)$ is said to be in $N$-subgeneral position with respect to $V$ if for any $1\le i_1<\cdots <i_{N+1}$, $ V\cap (\bigcap_{j=1}^{N+1}Q_{i_j})=\emptyset$. In this paper, we will prove a second main theorem for meromorphic mappings of $\mathbb C^m$ into $V$ intersecting hypersurfaces in subgeneral position with truncated counting functions. As an application of the above theorem, we give a uniqueness theorem for meromorphic mappings of $\mathbb C^m$ into $V$ sharing …
Let $f$ be an algebraically nondegenerate meromorphic mapping from $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ and let $Q_1,...,Q_q$ be $q$ hypersurfaces in $\mathbb P^n(\mathbb C)$ of degree $d_i$, in $N-$subgeneral position. In this paper, we will prove that, for every $ε>0$, there exists a positive integer $M$ such that $$||\ (q-(N-n+1)(n+1)-ε) T_f(r)\le\sum_{i=1}^q\frac{1}{d_i}N^{[M]}(r,f^*Q_i)+o(T_f(r)).$$ Moreover, an explicit estimate for $M$ is given. Our result is an extension of the previous second main theorem for the mappings and moving hyperplanes or moving hypersurfaces.
Let $Σ_k(p)$ be the class of univalent meromorphic functions defined on $\mathbb{D}$ with $k$-quasiconformal extension to the extended complex plane $\widehat{\mathbb{C}}$, where $0\leq k < 1$. Let $Σ_k^0(p)$ be the class of functions $f \in Σ_k(p)$ having expansion of the form $f(z)= 1/(z-p) + \sum_{n=1}^{\infty}b_n z^{n}$ on $\mathbb{D}$. In this article, we obtain sharp area distortion and weighted area distortion inequalities for functions in $Σ_k^0(p)$. As a consequence of the obtained results, we present a sharp estimate for the bounds of the Hilbert transform.
Let $f$ be a meromorphic correspondence on a compact Kähler manifold $X$ of dimension $k$. Assume that its topological degree is larger than the dynamical degree of order $k-1$. We obtain a quantitative regularity of the equilibrium measure of $f$ in terms of its super-potentials.
We give several unequivalent notions of convergency of meromorphic functions and more generally meromorphic mappings (strong, weak, $Γ$-convergency and some others). Relations between them are investigated. A version of Rouche theorem for the strongly converging sequences of meromorphic mappings is given. Further we investigate the sets of normality of families of meromorphic mappings, connections with extension properties of those mappings, some apriori estimates for the volume. Finally we apply the technique developped to the study of Fatou sets of meromorphic endomorphisms of Kähler surface…
We follow our study of non-Kähler complex structures on $R^4$ that we defined in a previous paper. We prove that these complex surfaces do not admit any smooth complex compactification. Moreover, we give an explicit description of their meromorphic functions. We also prove that the Picard groups of these complex surfaces are uncountable, and give an explicit description of the canonical bundle. Finally, we show that any connected non-compact oriented 4-manifold admits complex structures without Kähler metrics.
Let f be a meromorphic correspondence on a compact Kahler manifold. We show that the topological entropy of f is bounded from above by the logarithm of its maximal dynamical degree. An analogous estimate for the entropy on subvarieties is given. We also discuss a notion of Julia and Fatou sets.
The purpose of this paper has twofold. The first is to establish a second main theorem with truncated counting functions for algebraically nondegenerate meromorphic mappings into an arbitrary projective variety intersecting a family of hypersurfaces in subgeneral position. In our result, the truncation level of the counting functions is estimated explicitly. Our result is an extension of the classical second main theorem of H. Cartan, also is a generalization of the recent second main theorem of M. Ru and improves some recent results. The second purpose of this paper is to give another proof f…
In this paper, using techniques of value distribution theory, we give a uniqueness theorem for meromorphic mappings of C^m into P^n with (3n+1) moving targets and truncated multiplicities.
In this paper, we will show that if two meromorphic mappings $f$ and $g$ of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ have the same inverse images for $(2n+2)$ moving hyperplanes $\{a_i\}_{i=1}^{2n+2}$ with multiplicities counted to level $l_0$ then the map $f\times g$ must be algebraically degenerated over the field $\mathcal R\{a_i\}_{i=1}^{2n+2}$, where $l_0=3n^3(n+1)q(q-2)$ with $q=\binom{2n+2}{n+2}$. Our result generalizes the previous result for fixed hyperplanes case of Fujimoto and also improves his result by giving an explicit estimate for the number $l_0$.
We consider the dynamics of a meromorphic map on a compact kahler surface whose topological degree is smaller than its first dynamical degree. The latter quantity is the exponential rate at which its iterates expand the cohomology class of a kahler form. Our goal in this article and its sequels is to carry out a conjectural program for constructing and analyzing a natural measure of maximal entropy for each such map. Here we take the first step, converting information about the linear action of the map on cohomology to invariant currents with special geometric structure. We also give some exam…
