On holomorphic curves in algebraic torus

On the existence of holomorphic embeddings of strictly pseudoconvex algebraic hypersurfaces into spheres

We show that there are strictly pseudoconvex, real algebraic hypersurfaces in $\bC^{n+1}$ that cannot be locally embedded into a sphere in $\bC^{N+1}$ for any $N$. In fact, we show that there are strictly pseudoconvex, real algebraic hypersurfaces in $\bC^{n+1}$ that cannot be locally embedded into any compact, strictly pseudoconvex, real algebraic hypersurface.

Approximation of holomorphic maps from Runge domains to affine algebraic varieties

We present a geometric proof of the theorem saying that holomorphic maps from Runge domains to affine algebraic varieties admit approximation by Nash maps. Next we generalize this theorem.

Torus actions in the normalization problem

Let $f$ be a germ of biholomorphism of $\C^n$, fixing the origin. We show that if the germ commutes with a torus action, then we get information on the germs that can be conjugated to $f$, and furthermore on the existence of a holomorphic linearization or of a holomorphic normalization of $f$. We find out in a complete and computable manner what kind of structure a torus action must have in order to get a Poincaré-Dulac holomorphic normalization, studying the possible torsion phenomena. In particular, we link the eigenvalues of $df_O$ to the weight matrix of the action. The link and the struct…

Second main theorem for holomorphic curves into algebraic varieties intersecting moving hypersurfaces targets

Since the great work on holomorphic curves into algebraic varieties intersecting hypersurfaces in general position established by Ru in 2009, recently there has been some developments on the second main theorem into algebraic varieties intersecting moving hypersurfaces targets. The main purpose of this paper is to give some interesting improvements of Ru's second main theorem for moving hypersurfaces targets located in subgeneral position with index.

Holomorphic Curves into Algebraic Varieties Intersecting Divisors in Subgeneral Position

Recently, there are many developments on the second main theorem for holomorphic curves into algebraic varieties intersecting divisors in general position or subgeneral position. In this paper, we refine the concept of subgeneral position by introducing the notion of the index of subgeneral position. With this new notion we give some surprising improvement of the previous known second main theorem type results. Moreover, via the analogue between Nevanlinna theory and Diophantine approximation, the corresponding Schmidt's subspace type theorems are also established in the final section.

全纯映射到代数簇的非积分亏量关系

A non-integrated defect relation for holomorphic maps into algebraic varieties

本文推广Fujimoto关于全纯映射的非积分亏量理论，建立了全纯映射到射影簇的非积分亏量关系，将经典Nevanlinna-Cartan型结果从射影空间推广到一般射影簇。

Holomorphic curves into algebraic varieties intersecting moving hypersurface targets

In [Ann. of Math. 169 (2009)], Min Ru proved a second main theorem for algebraically nondegenerate holomorphic curves in complex projective varieties intersecting fixed hypersurface targets. In this paper, by introducing a new proof method for the case of projective varieties, we generalize this result to moving hypersurface targets.

满足代数关系的Hartogs级数的全纯性

On holomorphicity of Hartogs series satisfying algebraic relations

本文证明满足局部收敛性和代数性的Hartogs级数在直积空间上全纯，并给出条件(C2)必要性的例子。

Holomorphic line bundles over domains in Cousin groups and the algebraic dimension of OT-manifolds

In this paper we extend results due to Vogt on line bundles over Cousin groups to the case of domains stable by the maximal compact subgroup. This is used in the sequel to show that the algebraic dimension of OT-manifolds is zero. In the last part we establish that certain Cousin groups, in particular those arising from the construction of OT-manifolds, have finite-dimensional irregularity.

Holomorphic versus algebraic equivalence for deformations of real-algebraic CR manifolds

We consider (small) algebraic deformations of germs of real-algebraic CR submanifolds in complex space and study the biholomorphic equivalence problem for such deformations. We show that two algebraic deformations of minimal holomorphically nondegenerate real-algebraic CR submanifolds are holomorphically equivalent if and only if they are algebraically equivalent.

On the partial algebraicity of holomorphic mappings between real algebraic sets

In this paper, we consider local holomorphic mappings f: M\to M' between real algebraic CR generic manifolds (or more generally, real algebraic sets with singularities) in the complex euclidean spaces of different dimensions and we search necessary and sufficient conditions for f to be algebraic. These conditions appear to exclude two certain flatness of M and of M'. From the point of view of CR geometry, a real analytic CR manifold M can be flat in essentially to ways, being biholomorphic to a product M_1\times Δ^k, k\geq 1, by a polydisc (algebraic degeneracy), or to a product M_1\times I^l,…

Holomorphic extension of smooth CR-mappings between real-analytic and real-algebraic CR-manifolds

We establish results on holomorphic extension of CR-mappings of class $C^\infty$ between a real-analytic CR-submanifold of $\C^N$ and a real-algebraic CR-submanifold of $\C^{N'}$.

On the Nullstellensatz for c-holomorphic functions with algebraic graphs

C-holomorphic functions defined on algebraic sets and having algebraic graphs can be considered as a complex counterpart of regulous functions introduced recently in real geometry. This note is a part of our study on the subject; we prove herein some effective Nullstellensätze.

Degeneracy of Holomorphic Curves into Algebraic Varieties

Applying the Second Main Theorem we deal with the algebraic degeneracy of entire holomorphic curves from the complex plane into a complex algebraic normal variety of positive log Kodaira dimension that admits a finite proper morphism to a semi-abelian variety. We will also discuss applications to the Kobayashi hyperbolicity problem.

代数过剪密度性质

Algebraic Overshear Density Property

本文提出代数过剪密度性质，统一了代数灵活性与全纯密度性质，并证明具有该性质的复仿射曲面中带边黎曼面存在真全纯嵌入。

Holomorphic flexibility properties of complex manifolds

We obtain results on approximation of holomorphic maps by algebraic maps, jet transversality theorems for holomorphic and algebraic maps, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic manifolds.

Algebraic approximation of analytic sets and mappings

Let {X_n} be a sequence of analytic sets converging to some analytic set X in the sense of holomorphic chains. We introduce a condition which implies that every irreducible component of X is the limit of a sequence of irreducible components of the sets from {X_n}. Next we apply the condition to approximate a holomorphic solution y=f(x) of a system Q(x,y)=0 of Nash equations by Nash solutions. Presented methods allow to construct an algorithm of approximation of the holomorphic solutions.

关于环面叶的邻域与全纯叶状结构动力学

On the neighborhood of a torus leaf and dynamics of holomorphic foliations

本文研究复曲面中椭圆曲线邻域的 Ueda 分类与全纯叶状结构和乐动力学的关系，证明当存在沿某闭曲线的和乐为无理中性且不可线性化时该邻域为 ($\gamma$) 型，并基于 Pérez-Marco 的刺猬理论探讨相关线丛的度量半正性。

Basic cohomology of canonical holomorphic foliations on complex moment-angle manifolds

We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and Zaffran, who previously computed the basic Betti numbers for the canonical holomorphic foliation in the case of a shellable fan. Our proof uses an Eilenberg-Moore spectral sequence argument; the key ingredient is t…

Linearization of holomorphic families of algebraic automorphisms of the affine plane

Let $G$ be a reductive group. We prove that a family of polynomial actions of $G$ on $\mathbb{C}^2$, holomorphically parametrized by an open Riemann surface, is linearizable. As an application, we show that a particular class of reductive group actions on $\mathbb{C}^3$ is linearizable. The main step of our proof is to establish a certain restrictive Oka property for groups of equivariant algebraic automorphisms of $\mathbb{C}^2$.

Local approximation of the solutions of algebraic equations

A method of local approximation of holomorphic solutions of algebraic equations is discussed

On closed currents invariant by holomorphic foliations, I

We consider closed positive currents invariant by a singular holomorphic foliation on an algebraic surface. We show that under some conditions the foliation must leave invariant an algebraic curve.

Oka properties of groups of holomorphic and algebraic automorphisms of complex affine space

We show that the group of all holomorphic automorphisms of complex affine space $\mathbb C^n$, $n>1$, and several of its subgroups satisfy the parametric Oka property with approximation and with interpolation on discrete sets.

A characterization of real holomorphic chains and applications in representing homology classes by algebraic cycles

We show that a $2k$-current $T$ on a complex manifold is a real holomorphic $k$-chain if and only if $T$ is locally real rectifiable, $d$-closed and has $\mathcal{H}^{2k}$-locally finite support. This result is applied to study homology classes represented by algebraic cycles.

Holomorphic vector bundles on non-algebraic surfaces

The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is in general still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII. Our methods, which are based on Donaldson theory and deformation theory, can be used to solve the existence problem of holomorphic vector bundles on further classes of non-algebraic surfaces.

Existence Criteria for LCK Metrics

We investigate the relation between holomorphic torus actions on complex manifolds of LCK type and the existence of special LCK metrics. We show that if the group of biholomorphisms of such a manifold $(M,J)$ contains a non-real compact torus, then there exists a Vaisman metric on the manifold. Moreover, we show that if the group of biholomorphisms contains a compact torus whose dimension is half the real dimension of $M$, then $(M,J)$ admits an LCK metric with positive potential. As an application, we obtain a classification of manifolds of LCK type among all the manifolds having the structur…

代数锥乘积上的全纯张量

Holomorphic tensors on products of algebraic cones

研究两个代数锥乘积上的全纯张量，证明在特定条件下张量在收缩的Zariski闭包下不变，并应用于Sasaki流形乘积，得到Reeb场流下的不变性。

On the growth exponent of c-holomorphic functions with algebraic graphs

This paper is the first of a series dealing with c-holomorphic functions defined on algebraic sets and having algebraic graphs. These functions may be seen as the complex counterpart of the recently introduced \textit{regulous} functions. Herein we study their growth exponent at infinity. A general result on injectivity on fibres of an analytic set together with a theorem of Tworzewski and Winiarski gives a bound for the growth exponent of a c-holomorphic function with algebraic graph in terms of the projective degrees of the sets involved. We prove also that algebricity of the graph is equiva…

Completely integrable torus actions on complex manifolds with fixed points

We show that if a holomorphic $n$ dimensional compact torus action on a compact connected complex manifold of complex dimension $n$ has a fixed point then the manifold is equivariantly biholomorphic to a smooth toric variety.

Transcendental holomorphic maps between real algebraic manifolds in a complex space

We give an example of a real algebraic manifold embedded in a complex space that does not satisfy the Nash-Artin approximation Property. This Nash-Artin approximation Property is closely related to the problem of determining when the biholomorphic equivalence for germs of real algebraic manifolds coincides with the algebraic equivalence. This example is an elliptic Bishop surface, and its construction is based on the functional equation satisfied by the generating series of some walks restricted to the quarter plane.

A universal metric for the canonical bundle of a holomorphic family of projective algebraic manifolds

We prove that the canonical bundle of any holomorphic family of compact complex algebraic manifolds carries a singular Hermitian metric having non-negative curvature current and such that every holomorphic section of the canonical bundle of the central fiber is L^2 with respect to this metric. This result was proved by Siu when the members of the family are general type, and used to establish the deformation invariance of plurigenera in that case. In fact, we prove our result in a more general setting of a smooth hypersurface with not-necessarily normal trivial bundle in a manifold that is Ste…

Configuration spaces of tori

The configuration space C^n of unordered n-tuples of distinct points on a torus T^2 is a non-singular complex algebraic variety. We study holomorphic self-maps of C^n and prove that for n>4 any such map F either carries the whole of C^n into an orbit of the diagonal Aut(T^2) action in C^n or is of the form F(x)=T(x)x for some holomorphic map T:C^n-->Aut(T^2). We also prove that for n>4 any endomorphism of the torus braid group B_n(T^2) with a non-abelian image preserves the pure torus braid group PB_n(T^2).

关于全纯叶层代数重数的拓扑不变性

On the topological invariance of the algebraic multiplicity of holomorphic foliations

本文研究全纯叶层代数重数的拓扑不变性，证明二维下重数为1是拓扑不变量并推广至高维，验证多类叶层的不变性，还修正了Rosas-Bazan的相关结论。

Entire holomorphic curves into projective plane intersecting few generic algebraic curves

For $q\leq 3$ smooth plane algebraic curves $\mathcal{C}_i$ having simple normal crossings, if the invariant logarithmic $2$-jet differential bundle associated to $(\mathbb{P}^2(\mathbb{C}), \sum_{i=1}^q \mathcal{C}_i)$ has a nonzero section vanishing on some ample divisor, then, for every algebraically nondegenerate entire holomorphic curve $f\colon\mathbb{C}\rightarrow\mathbb{P}^2(\mathbb{C})$, we have a Second Main Theorem type estimate: \[ T_f(r) \leq c\sum_{i=1}^q\,N_f^{[1]}(r,\mathcal{C}_i) + o\big(T_f(r) \big)\parallel, \] where $T_f(r)$ and $N_f^{[1]}(r,C_i)$ stand for the order functi…

Rational approximation of holomorphic maps

Let X be a complex nonsingular affine algebraic variety, K a holomorphically convex subset of X, and Y a homogeneous variety for some complex linear algebraic group. We prove that a holomorphic map f:K-->Y can be uniformly approximated on K by regular maps K-->Y if and only if f is homotopic to a regular map K-->Y. However, it can happen that a null homotopic holomorphic map K-->Y does not admit uniform approximation on K by regular maps X-->Y. Here, a map g:K-->Y is called holomorphic (resp. regular) if there exist an open (resp. a Zariski open) neighborhood U of K in X and a holomorphic (res…

Rigidity for local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ up to conformal factors

In this article, we study local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic map between complex Euclidean spaces by the algebraic extension theorem derived along the lines of Mok and Mok-Ng. Applying holomorphic continuation and analyzing real analytic subvarieties carefully, we show that each component is either a constant map or a proper holomorphic map…

Approximation of holomorphic maps with a lower bound on the rank

Let $K$ be a closed polydisc or ball in $\C^n$, and let $Y$ be a quasi projective algebraic manifold which is Zariski locally equivalent to $\C^p$, or a complement of an algebraic subvariety of codimension $\ge 2$ in such manifold. If $r$ is an integer satisfying $(n-r+1) (p-r+1)\geq 2$ then every holomorphic map from a neighborhood of $K$ to $Y$ with rank $\ge r$ at every point of $K$ can be approximated uniformly on $K$ by entire maps $\C^n\to Y$ with rank $\ge r$ at every point of $\C^n$.

Algebraic dimension of complex nilmanifolds

Let M be a complex nilmanifold, that is, a compact quotient of a nilpotent Lie group endowed with an invariant complex structure by a discrete lattice. A holomorphic differential on M is a closed, holomorphic 1-form. We show that $a(M)\leq k$, where $a(M)$ is the algebraic dimension $a(M)$ (i.e. the transcendence degree of the field of meromorphic functions) and $k$ is the dimension of the space of holomorphic differentials. We prove a similar result about meromorphic maps to Kahler manifolds.

次高斯实代数极大曲线的一个指数稀疏性结果

An Exponential Rarefaction Result for Sub-Gaussian Real Algebraic Maximal Curves

本文证明了与次高斯随机实全纯截面相关的实代数极大曲线具有指数稀有性，推广了Gayet和Welschinger在高斯情形下的已有结果。