Formal meromorphic functions on manifolds of finite type

Equilibrium measures of meromorphic self-maps on non-Kahler manifolds

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…

Kähler流形上的Meromorphic映射弱分享超平面问题

Meromorphic mappings on Kähler manifolds weakly sharing hyperplanes in $\mathbb P^n(\mathbb C)$

本文研究了Kähler流形到复射影空间的Meromorphic映射在共享超平面条件下的唯一性问题，并改进了相关代数依赖问题。

On the Removable Singularities for Meromorphic Mappings

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.

从凯勒流形到射影变分的非积分缺陷关系改进

Improvement of non-integrated defect relation for meromorphic maps from Kähler manifolds

本文改进了从凯勒流形到射影变分的非积分缺陷关系，通过独立估计总缺陷和截断水平，推广并优化了先前结果。

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 ver…

A non-integrated hypersurface defect relation for meromorphic maps over complete Kähler manifolds into projective algebraic varieties

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…

Algebraic dependence and finiteness problems of differentiably nondegenerate meromorphic mappings on Kähler manifolds

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 …

Meromorphic maps of Kahler manifolds with trivial canonical bundles

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.

Non-integrated defect relation for meromorphic mappings from a Kähler manifold with hypersurfaces of a projective variety in subgeneral position

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.

Non-integrated defect relation for meromorphic maps from a Kähler manifold intersecting hypersurfaces in subgeneral of $\mathbb P^n(\mathbb C)$

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…

Degeneracy theorems for meromorphic mappings of a complete Kähler manifold sharing hyperplanes in a projective space

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…

Compact singularities of meromorphic mappings between complex 3-dimensional manifolds

We prove that a meromorphic map defined on the complement of a compact subset of a three-dimensional Stein manifold M and with values in a compact complex three-fold X extends to the complement of a finite set of points. If X is simply connected, then the map extends to all of M.

Good formal structure for meromorphic flat connections on smooth projective surfaces

We prove the algebraic version of a conjecture of C. Sabbah on the existence of the good formal structure for meromorphic flat connections on surfaces after some blow up.

每个格罗莫夫椭圆域都是亚纯伪凸的

Every Gromov elliptic domain is meromorphically pseudoconvex

该研究证明每个格罗莫夫椭圆域都是亚纯伪凸的，明确了Stein流形中域的格罗莫夫椭圆性与Oka、Stein等性质的等价关系，还得出两类Oka流形非椭圆及射影空间中特定域非椭圆的结论。

Meromorphic almost rigid geometric structures

We study the local Killing Lie algebra of meromorphic almost rigid geometric structures on complex manifolds. This leads to classification results for compact complex manifolds bearing holomorphic rigid geometric structures.

Unfoldings of meromorphic connections and a construction of Frobenius manifolds

The existence of universal unfoldings of certain germs of meromorphic connections is established. This is used to prove a general construction theorem for Frobenius manifolds. A particular case is Dubrovin's theorem on semisimple Frobenius manifolds. Another special case starts with variations of Hodge structures. This case is used to compare two constructions of Frobenius manifolds, the one in singularity theory and the Barannikov-Kontsevich construction. For homogeneous polynomials which give Calabi-Yau hypersurfaces certain Frobenius submanifolds in both constructions are isomorphic.

完备凯勒流形上亚纯映射的唯一性问题

Unicity problem on meromorphic mappings of complete Kahler manifolds

本文将经典奈望林纳唯一性定理推广到具有非正截面曲率或非负里奇曲率的非紧完备凯勒流形上，拓展了值分布理论的适用范围。

On the local meromorphic extension of CR meromorphic mappings

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…

Meromorphic mappings of a complete connected Kähler manifold into a projective space sharing hyperplanes

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…

Green Currents for Meromorphic Maps of Compact Kähler Manifolds

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.

Thermodynamic Formalism and Geometric Applications for Transcendental Meromorphic and Entire Functions

This text provides an overview of the (geometric) thermodynamic formalism for transcendental meromorphic and entire functions with particular emphasis on geometric/fractal aspects such as Bowen's Formula expressing the hyperbolic dimension as a unique zero of a pressure function and the behavior of the latter when the transcendental functions vary in an analytic family.

一类双曲超越亚纯函数的热力学形式

Thermodynamic Formalism for a Class of Hyperbolic Transcendental Meromorphic Functions

针对极点重数有界、Nevanlinna阶有限且无穷远非渐近值的B类双曲超越亚纯函数，本文建立热力学形式，并证明共形测度及等价不变Gibbs测度的存在唯一性。

Comparison of dynamical degrees for semi-conjugate meromorphic maps

Let f be a dominant meromorphic self-map on a projective manifold X which preserves a meromorphic fibration pi: X --> Y of X over a projective manifold Y. We establish formulas relating the dynamical degrees of f, the dynamical degrees of f relative to the fibration and the dynamical degrees of the self-map g on Y induced by f. Applications are given.

Good formal structures for flat meromorphic connections, II: Excellent schemes

Given a flat meromorphic connection on an excellent scheme over a field of characteristic zero, we prove existence of good formal structures after blowing up; this extends a theorem of Mochizuki for algebraic varieties. The argument combines a numerical criterion for good formal structures from a previous paper, with an analysis based on the geometry of an associated valuation space (Riemann-Zariski space). We obtain a similar result over the formal completion of an excellent scheme along a closed subscheme. If we replace the excellent scheme by a complex analytic variety, we obtain a similar …

On the dynamical degrees of meromorphic maps preserving a fibration

Let f be a dominant meromorphic self-map on a compact Kaehler manifold X which preserves a fibration given by a meromorphic map from X to a compact Kaehler manifold Y. We compute the dynamical degrees of f in term of its dynamical degrees relative to the fibration and the dynamical degrees of the map g on Y which is induced by f. We derive from this result new properties of some fibrations intrinsically associated to X when this manifold admits an interesting dynamical system.

Décomposition en éléments simples des formes méromorphes formelles fermées

We show that any closed formal meromorphic 1-form admits a "partial fraction decomposition", which allows us in particular to define a notion of residue for closed formal meromorphic forms which extends the notion defined for usual forms.

Complex Curves in Almost-Complex Manifolds and Meromorphic Hulls

This are the notes of a course, given by the first author for the Graduiertenkollegs (=graduate students) at the Ruhr-University Bochum, in December 1997. These lectures pursued two main tasks: FIRST - to give a systematic and self-contained introduction to the Gromov theory of pseudoholomorphic curves. This is done in Chapters I,II,III. SECOND - to explain our join results on envelopes of meromorphy of real surfaces in complex two-dimensional manifolds. We do this in Chapter IV.

Elementary Pseudoconcavity and fields of CR meromorphic functions

Let M be a smooth CR manifold of CR dimension n and CR codimension k, which is not compact, but has the local extension property E. We introduce the notion of "elementary pseudoconcavity" for M, which extends to CR manifolds the concept of a "pseudoconcave" complex manifold. This notion is then used to obtain generalizations, to the noncompact case, of the results of our previous paper about algebraic dependence, transcendence degree and related matters for the field K(M) of CR meromorphic functions on M.

Meromorphic limits of automorphisms

Let $X$ be a compact complex manifold in the Fujiki class $\mathscr{C}$. We study the compactification of $\operatorname{Aut}^0(X)$ given by its closure in Barlet cycle space. The boundary points give rise to non-dominant meromorphic self-maps of $X$. Moreover convergence in cycle space yields convergence of the corresponding meromorphic maps. There are analogous compactifications for reductive subgroups acting trivially on $\operatorname{Alb} X$. If $X$ is Kähler, these compactifications are projective. Finally we give applications to the action of $\operatorname{Aut}(X)$ on the set of probab…

Good formal structures on flat meromorphic connections, I: Surfaces

We give a criterion under which one can obtain a good decomposition (in the sense of Malgrange) of a formal flat connection on a complex analytic or algebraic variety of arbitrary dimension. The criterion is stated in terms of the spectral behavior of differential operators, and generalizes Robba's construction of the Hukuhara-Levelt-Turrittin decomposition in the one-dimensional case. As an application, we prove the existence of good formal structures for flat meromorphic connections on surfaces after suitable blowing up; this verifies a conjecture of Sabbah, and extends a result of Mochizuki…

Rational connectedness and order of non-degenerate meromorphic maps from\ ${\bf C}^n$

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.

关于任意闭子方案的广义 $p$-抛物 manifold 上第二主定理的一般形式

General form of second main theorem on generalized $p$-Parabolic manifolds for arbitrary closed subschemes

本文提出了一种针对任意闭子方案的广义 $p$-抛物 manifold 上全纯映射的第二主定理的推广形式，并给出了与超曲面相交时的计数函数截断水平的显式结果。

Some applications of the Kohn-Rossi extension theorems

We prove extension results for meromorphic functions by combining the Kohn-Rossi extension theorems with Andreotti's theory on the algebraic and analytic dependence of meromorphic functions on pseudoconcave manifolds. Versions of Kohn-Rossi theorems for pseudoconvex domains are included.

Wedge extendability of CR-meromorphic functions: the minimal case

In this article, we consider metrically thin singularities E of the solutions of the tangential Cauchy-Riemann operators on a C^{2,a}-smooth embedded Cauchy-Riemann generic manifold M (CR functions on M - E) and more generally, we consider holomorphic functions defined in wedgelike domains attached to M - E. Our main result establishes the wedge- and the L^1-removability of E under the hypothesis that the (\dim M-2)-dimensional Hausdorff volume of E is zero and that M and M\backslash E are globally minimal. As an application, we deduce that there exists a wedgelike domain attached to an everyw…

Sequences of multivalued meromorphic maps and laminar currents

Let (F_n) be a sequence of (multivalued) meromorphic maps between compact Kaehler manifolds X1 and X2. We study the asymptotic distribution of preimages of points by F_n and the asymptotic distribution of fixed points for multivalued self-maps of a compact Riemann surface. Let (Z_n) be a sequence of holomorphic images of the projective space P^s in a projective manifold. We prove that the currents, defined by integration on Z_n, properly normalized, converge to weakly laminar currents. We also show that the Green currents, of suitable bidimensions, associated to a regular polynomial automorphi…

Regularity of the equilibrium measure for meromorphic correspondences

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.

通过尼文林纳理论解决皮卡问题

On Picard's Problem via Nevanlinna Theory

研究非抛物完全凯勒流形在非负曲率下的皮卡问题，利用全局格林函数方法，证明若满足体积增长条件，则任一全纯函数若避开三个不同值则为常数，且非多项式增长的全纯函数最多避开两个值。

Dynamics of Hyperbolic Meromorphic Functions

A definition of hyperbolic meromorphic functions is given and then we discuss the dynamical behavior and the thermodynamic formalism of hyperbolic functions on the Julia set. We prove the important expanding properties for hyperbolic functions on the complex plane and with respect to the euclidean metric. We establish the Bowen formula for hyperbolic functions on the complex plane, that is, the Poincare exponent equals to the Hausdorff dimension of the radial Julia set and furthermore, we prove that all the results in the Walters' theory hold for hyperbolic functions on the Riemann sphere.

Nevanlinna五值定理在非正曲率完备Kähler流形上

Nevanlinna's five-value theorem on non-positively curved complete Kähler manifolds

本文通过代数相关性理论，将Nevanlinna五值定理推广到非正截面曲率的非紧完备Kähler流形，在增长条件下证明两个非常数亚纯函数若忽略重数分担五个不同值则恒等。

Upper bound for topological entropy of a meromorphic correspondence

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.

具有非负Ricci曲率的完备Kähler流形上的Nevanlinna理论

Nevanlinna theory on complete Kähler manifolds with non-negative Ricci curvature

该研究在非负Ricci曲率的完备Kähler流形框架下发展亚纯映射的Nevanlinna等分布理论，得到带精细误差的第二基本定理、精确亏量关系，还推广了五值定理等多个唯一性结果。

Good formal structures for flat meromorphic connections, III: Irregularity and turning loci

Given a formal flat meromorphic connection over an excellent scheme over a field of characteristic zero, in a previous paper we established existence of good formal structures and a good Deligne-Malgrange lattice after suitably blowing up. In this paper, we reinterpret and refine these results by introducing some related structures. We consider the turning locus, which is the set of points at which one cannot achieve a good formal structure without blowing up. We show that when the polar divisor has normal crossings, the turning locus is of pure codimension 1 within the polar divisor, and henc…

带非抛物端的完备Kähler连通和上亚纯映射的值分布

Value distribution of meromorphic mappings on complete Kähler connected sums with non-parabolic ends

针对带非抛物端的完备Kähler连通和上调和函数Liouville性质的刚性问题，研究其上到复射影流形的亚纯映射值分布，建立几何条件下的Nevanlinna理论第二基本定理，证明体积增长条件下Cauchy-Riemann方程可保证该性质的刚性。

The ABC Theorem for Meromorphic Functions

Using a `height-to-radical' identity, we define the archimedean contribution to the radical, $r_\arch$, and we give a new proof of the abc theorem for the field of meromorphic functions. The first step of the proof is completely formal and yields that the height is bounded by the radical, $h\leq r$, where $r=r_\na+r_\arch$ is the radical completed with the archimedean contribution. The second step is analytic in nature and uses the lemma on the logarithmic derivative to derive a bound for $r_\arch$.

Densities of currents and complex dynamics

We extend the Dinh-Sibony notion of densities of currents to the setting where the ambient manifold is not necessarily Kähler and study the intersection of analytic sets from the point of view of densities of currents. As an application, we introduce the notion of exotic periodic points of a meromorphic self-map. We then establish the expected asymptotic for the sum of the number of isolated periodic points and the number of exotic periodic points for holomorphic self-maps with a simple action on the cohomology groups on a compact Kähler manifold. We also show that the algebraic entropy of mer…

The simplicity of the first spectral radius of a meromorphic map

Let $X$ be a compact Kähler manifold and let $f:X\rightarrow X$ be a dominant rational map which is 1-stable. Let $λ_1$ and $λ_2$ be the first and second dynamical degrees of $f$. If $λ_1^2>λ_2$, then we show that $λ_1$ is a simple eigenvalue of $f^*:H^{1,1}(X)\rightarrow H^{1,1}(X)$, and moreover the unique eigenvalue of modulus $>\sqrt{λ_2}$. A variant of the result, where we consider the first spectral radius in the case the map $f$ may not be 1-stable, is also given. An application is stated for bimeromorphic selfmaps of 3-folds. In the last section of the paper, we prove analogs of the ab…

Fields of CR meromorphic functions

Let $M$ be a smooth compact $CR$ manifold of $CR$ dimension $n$ and $CR$ codimension $k$, which has a certain local extension property $E$. In particular, if $M$ is pseudoconcave, it has property $E$. Then the field $\Cal K(M)$ of $CR$ meromorphic functions on $M$ has transcendence degree $d$, with $d\leq n+k$. If $f_1, f_2, \hdots , f_d$ is a maximal set of algebraically independent $CR$ meromorphic functions on $M$, then $\Cal K(M)$ is a simple finite algebraic extension of the field $\Bbb C(f_1, f_2, \hdots, f_d)$ of rational functions of the $f_1, f_2, \hdots , f_d$. When $M$ has a project…

关于超凯勒簇的反双曲性

On anti-hyperbolicity for hyperkähler varieties

本文研究超凯勒流形被同维数复仿射空间全纯或亚纯支配的反双曲性判据与例子，推广了 K3 曲面的相关已知结果。

幂幺群在紧致Kähler流形上的Hamilton作用

Hamiltonian actions of unipotent groups on compact Kähler manifolds

本文研究幂幺复李群在紧致Kähler流形上的亚纯作用，通过矩映射引入稳定性条件，证明半稳定点集Zariski开且存在带紧致化Kähler结构的几何商，并与Doran--Kirwan的GIT理论进行了详细比较。

Holomorphic Extension from Weakly Pseudoconcave CR Manifolds

Let M be a smooth locally embeddable CR manifold, having some CR dimension m and some CR codimension d. We find an improved local geometric condition on M which guarantees, at a point p on M, that germs of CR distributions are smooth functions, and have extensions to germs of holomorphic functions on a full ambient neighborhood of p. Our condition is a form of weak pseudoconcavity, closely related to essential pseudoconcavity as introduced in [HN1]. Applications are made to CR meromorphic functions and mappings. Explicit examples are given which satisfy our new condition,but which are not pseu…