$L^p$-estimates of extensions of holomorphic functions defined on a non-reduced subvariety

Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms

Let $X$ be a connected open Riemann surface. Let $Y$ be an Oka domain in the smooth locus of an analytic subvariety of $\mathbb C^n$, $n\geq 1$, such that the convex hull of $Y$ is all of $\mathbb C^n$. Let $\mathscr O_*(X, Y)$ be the space of nondegenerate holomorphic maps $X\to Y$. Take a holomorphic $1$-form $θ$ on $X$, not identically zero, and let $π:\mathscr O_*(X,Y) \to H^1(X,\mathbb C^n)$ send a map $g$ to the cohomology class of $gθ$. Our main theorem states that $π$ is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the div…

Extending holomorphic sections from complex subvarieties

Let X be a Stein manifold and let Y be a complex manifold which admits a spray in the sense of Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, pp. 851-897 (1989)). We prove that for every closed complex subvariety X_0 of X and for every continuous map f_0 from X to Y whose restriction to X_0 is holomorphic there exists a homotopy of maps f_t from X to Y whose restrictions to X_0 agree with f_0 and such that the map f_1 is holomorphic on X. We obtain analogous results for sections of holomorphic submersions with sprays over Stein manifolds or Stein s…

Extension of holomorphic functions defined on non reduced analytic subvarieties

The goal of this contribution is to investigate L${}^2$ extension properties for holomorphic sections of vector bundles satisfying weak semi-positivity properties. Using techniques borrowed from recent proofs of the Ohsawa-Takegoshi extension theorem, we obtain several new surjectivity results for the restriction morphism to a non necessarily reduced subvariety, provided the latter is defined as the zero variety of a multiplier ideal sheaf. These extension results come with precise L${}^2$ estimates and (probably) optimal curvature conditions.

Holomorphic Curves and Integral Points off Divisors

We deal with the distributions of holomorphic curves and integral points off divisors. We will simultaneouly prove an optimal dimension estimate from above of a subvariety W off a divisor D which contains a Zariski dense entire holomorphic curve, or a Zariski dense D-integral point set, provided that in the latter case everything is defined over a number field. Then, if the number of components of D is large, the estimate leads to the constancy of such a holomorphic curve or the finiteness of such an integral point set. At the begining, we extend logarithmic Bloch-Ochiai's Theorem to the Kaehl…

Quasiprojective varieties admitting Zariski dense entire holomorphic curves

Let $X$ be a complex quasiprojective variety. A result of Noguchi-Winkelmann-Yamanoi shows that if $X$ admits a Zariski dense entire curve, then its quasi-Albanese map is a fiber space. We show that the orbifold structure induced by a properly birationally equivalent map on the base is special in this case. As a consequence, if $X$ is of log-general type with $\bar q(X)\geq\dim X$, then any entire curve is contained in a proper subvariety in $X$.

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 $\C^n$.

Deformations of Stein structures and extensions of holomorphic mappings

Let X be a Stein manifold, A a closed complex subvariety of X, and f a continuous map from X to a complex manifold Y whose restriction to A is holomorphic. After a homotopic deformation of the Stein structure outside a neighborhood of A in X (and of its smooth structure when X is a Stein surface)we find a holomorphic map from X to Y which agrees with f on A and which is homotopic to f relative to A. The analogous results in the case when the variety A is empty have been obtained in the preprint math.CV/0507212.

Extending holomorphic maps from Stein manifolds into affine toric varieties

A complex manifold $Y$ is said to have the interpolation property if a holomorphic map to $Y$ from a subvariety $S$ of a reduced Stein space $X$ has a holomorphic extension to $X$ if it has a continuous extension. Taking $S$ to be a contractible submanifold of $X=\mathbb{C}^n$ gives an ostensibly much weaker property called the convex interpolation property. By a deep theorem of Forstnerič, the two properties are equivalent. They (and about a dozen other nontrivially equivalent properties) define the class of Oka manifolds. This paper is the first attempt to develop Oka theory for singular tar…

残差函数与延拓问题

Residue functions and Extension problems

针对Demailly定性延拓定理缺乏延拓估计的问题，研究利用残差函数获取一般情形下全纯延拓的$L^2$估计的可能性，并证明指标1残差函数定义的1-lc测度等于Ohsawa测度。

Index theorems for holomorphic self-maps

Let $M$ be a complex manifold and $S\subset M$ a (possibly singular) subvariety of $M$. Let $f\colon M\to M$ be a holomorphic map such that $f$ restricted to $S$ is the identity. We show that one can associate to $f$ a holomorphic section $X_f$ of a sheaf related to the embedding of $S$ in $M$ and that such a section reads the dynamical behavior of $f$ along $S$. In particular we prove that under generic hypotheses the canonical section $X_f$ induces a holomorphic action in the sense of Bott on the normal bundle of (the regular part of) $S$ in $M$ and this allows to obtain for holomorphic self…

全纯函数的能量渐近及其在复平面上的Calderón-Zygmund理论中的应用

Energy asymptotics of holomorphic functions with application to Calderón-Zygmund theory in $\mathbb{C}$

该研究针对复一维情形，推导了全纯函数水平集积分与次水平集能量的带显式常数的渐近公式，简化了p=1端点处Calderón-Zygmund理论普适反例的证明，还构造了p=∞端点处的新反例族，证明W^{2,∞}正则性失效是复一维情形的普适现象。

A global Brianç con-Skoda-Huneke-Sznajdman theorem

We prove a global effective membership result for polynomials on a non-reduced algebraic subvariety of $\C^N$. It can be seen as a global version of a recent local result of Sznajdman, generalizing the Briançon-Skoda-Huneke theorem for the local ring of holomorphic functions at a point on a reduced analytic space.

约化Bergman核的变换公式及其应用

Transformation formula for the Reduced Bergman kernel and its Application

本文证明了复平面有界域间真全纯对应及映射下约化Bergman核与加权约化Bergman核的变换公式，并给出了其应用。

Section Extension from Hyperbolic Geometry of Punctured Disk and Holomorphic Family of Flat Bundles

The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry. When the jet values are prescribed on a positive dimensional subvariety, it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive. We prove here an extension result when the curvature of the line bundle is only semipositive o…

Noncritical holomorphic functions on Stein spaces

We prove that every reduced Stein space admits a holomorphic function without critical points. Furthermore, any closed discrete subset of such a space is the critical locus of a holomorphic function. We also show that for every complex analytic stratification with nonsingular strata on a reduced Stein space there exists a holomorphic function whose restriction to every stratum is noncritical. These result also provide some information on critical loci of holomorphic functions on desingularizations of Stein spaces. In particular, every 1-convex manifold admits a holomorphic function that is non…

A pointwise norm on a non-reduced analytic space

Let $X$ be a possibly non-reduced space of pure dimension. We introduce an essentially intrinsic pointwise Hermitian norm on smooth $(0,*)$-forms, in particular on holomorphic functions, on $X$. We prove that the space of holomorphic functions is complete with respect to the natural topology induced by this norm.

全纯函数的 $L^2$ 延拓与对数典范位置

$L^2$ extension of holomorphic functions and log canonical places

本文指出在 Demailly 的 $L^2$ 延拓定理及 Zhou-Zhu 的推广中，存在许多非解析奇点定义函数（如环面奇点）使得仅有零函数具有有限的 Ohsawa 范数，导致 $L^2$ 延拓结论在这些情况下变得空泛。

用布朗运动方法研究亚纯映射的第二主定理及具有截断计数函数的超曲面

A Brownian-Motion Approach to the Second Main Theorem for Meromorphic Mappings and Hypersurfaces with Truncated Counting Functions

本文通过布朗运动和随机微积分方法，建立了亚纯映射进入复射影子流形V的第二主定理，并引入了截断计数函数以研究超曲面的分布常数。同时，给出了亚纯映射在复平面到V上共享任意超曲面的唯一性定理。

Pseudomeromorphic currents on subvarieties

Let $i\colon X \to Y$ be pure-dimensional reduced subvariety of a smooth manifold $Y$. We prove that the direct image of pseudomeromorphic currents on $X$ are pseudomeromorphic on $Y$. We also prove a partial converse: if $i_*τ$ is pseudomeromorphic and has the standard extension property, then $τ$ is pseudomermorphic on $X$.

On the number of factors in the unipotent factorization of holomorphic mappings into $\text{SL}_2(\mathbb{C})$

We estimate the number of unipotent elements needed to factor a null-homotopic holomorphic map from a finite dimensional reduced Stein spaces $X$ into $\text{SL}_2(\mathbb{C})$.

The $\bar{\partial}$-equation, duality, and holomorphic forms on a reduced complex space

We solve the $\bar{\partial}$-equation for $(p,q)$-forms locally on any reduced pure-dimensional complex space and we prove an explicit version of Serre duality by introducing suitable concrete fine sheaves of certain $(p,q)$-currents. In particular this gives a precise condition for the $\bar{\partial}$-equation to be globally solvable. Our results extend results for $(0,q)$-forms and give information about holomorphic $p$-forms on singular spaces.

Equidistribution of varieties for endomorphisms of projective spaces

Let f be a non-invertible holomorphic endomorphism of the complex projective space P^k and f^n its iterate of order n. Let V be an algebraic subvariety of P^k which is generic in the Zariski sense. We give here a survey on the asymptotic equidistribution of the sequence $f^{-n}(V)$ when n goes to infinity.

关于发散复几何积分的有限部分及其对 Hermitian 度量选择的依赖性

On Finite Parts of Divergent Complex Geometric Integrals and Their Dependence on a Choice of Hermitian Metric

本文研究了既约复空间上沿全纯向量丛截面零集奇异的发散积分，通过给定 Hermitian 度量定义其有限部分，并给出了该有限部分对度量选择依赖性的显式公式。

Stein neighborhood bases of embedded strongly pseudoconvex domains and approximation of mappings

In this paper we construct a Stein neighborhood basis for any compact subvariety $A$ with strongly pseudoconvex boundary $bA$ and Stein interior $A\backslash bA$ in a complex space $X$. This is an extension of a well known theorem of Siu. When $A$ is a complex curve, our result coincides with the result proved by Drinovec-Drnovšek and Forstnerič. We shall adapt their proof to the higher dimensional case, using also some ideas of Demailly's proof of Siu's theorem. For embedded strongly pseudoconvex domain in a complex manifold we also find a basis of tubular Stein neighborhoods. These results a…

Oka maps

Given a holomorphic submersion of reduced complex spaces, we prove that the basic Oka property of the submersion implies the parametric Oka property. This generalizes the corresponding result for complex manifolds (F. Forstneric, Oka Manifolds, C. R. Acad. Sci. Paris, Ser. I, 347 (2009) 1017-1020). It follows that a stratified elliptic (or subelliptic) holomorphic submersion, or a stratified holomorphic fiber bundle whose fibers are Oka manifolds, enjoys the parametric Oka property. As an application we give a parametric version of the factorization theorem due to Ivarsson and Kutzschebauch (A…

On the extension of holomorphic sections from reduced unions of strata of divisors

In this paper we study the problem of extension of holomorphic sections of line bundles/vector bundles from reduced unions of strata of divisors. An extension theorem of Ohsawa--Takegoshi type is proved. As consequences we deduce several qualitative results on extension from snc divisors and generic global generation of vector bundles.

局部可逼近 CR 函数、尖锐最大模原理与全纯延拓

Locally approximable CR functions, a sharp maximum modulus principle and holomorphic extension

本文提出约化复空间局部闭子集上的局部可逼近连续 CR 函数概念，并在弱伪凹条件下建立最优最大模原理，同时将 CR 子流形的全纯延拓结果推广到含 CR 奇点的实子流形。

Projective varieties invariant by one-dimensional foliations

This work concerns the problem of relating characteristic numbers of one-dimensional holomorphic foliations of P^n to those of algebraic varieties invariant by them. More precisely: if M is a connected complex manifold, a one-dimensional holomorphic foliation F of M is a morphism Φ:L -> TM where L is a holomorphic line bundle on M. The singular set of F is the analytic subvariety sing(F) = {p : Φ(p)=0} and the leaves of F are the leaves of the nonsingular foliation induced by F on M-sing(F). If M is P^n then, since line bundles over P^n are classified by the Chern class c_1(L) in H^2(P^n,Z) = …

开型的最优 $L^2$ 延拓

Optimal $L^2$ Extensions of Openness Type

本文研究复流形上开型最优 $L^2$ 延拓问题，证明弱伪凸 Kähler 流形上的最优延拓定理、极小延拓乘积性质，并应用于经典问题与 Suita 猜想。

Extension with log-canonical measures and an improvement to the plt extension of Demailly--Hacon--Paun

With a view to proving the conjecture of "dlt extension" related to the abundance conjecture, a sequence of potential candidates for replacing the Ohsawa measure in the Ohsawa-Takegoshi $L^2$ extension theorem, called the "lc-measures", which hopefully could provide the $L^2$ estimate of a holomorphic extension of any suitable holomorphic section on a subvariety with singular locus, are introduced in the first half of the paper. Based on the version of $L^2$ extension theorem proved by Demailly, a proof is provided to show that the lc-measure can replace the Ohsawa measure in the case where th…

A smoothness criterion for complex spaces in terms of differential forms

For a reduced pure dimensional complex space $X$, we show that if Barlet's recently introduced sheaf $α_X^1$ of holomorphic $1$-forms or the sheaf of germs of weakly holomorphic $1$-forms is locally free, then $X$ is smooth. Moreover, we discuss the connection to Barlet's well-known sheaf $ω_X^1$.

Antiholomorphic involutions of spherical complex spaces

Let X be a holomorphically separable irreducible reduced complex space, K a connected compact Lie group acting on X by holomorphic transformations, theta : K -> K a Weyl involution, and mu : X -> X an antiholomorphic involution map satisfying mu(kx) = theta(k) mu(x) for x in X and k in K. We show that if the holomorphic functions on X form a multiplicity free K-module then mu maps every K-orbit onto itself. For a spherical affine homogeneous space X=G/H of the reductive group G (the complexification of K) we construct an antiholomorphic map mu with these properties.

Segre 非退化全实子簇

Segre nondegenerate totally real subvarieties

本文研究复空间中不可约实解析簇的芽，在 Segre 非退化条件下定义推广 Moser-Webster 对合的平均算子，并利用其研究全纯函数限制及奇点平坦化条件。

从有界对称域穿出的局部全纯曲线的渐近全测地性及其在代数子集单值化问题中的应用

Asymptotic total geodesy of local holomorphic curves exiting a bounded symmetric domain and applications to a uniformization problem for algebraic subsets

本文研究全纯曲线与嵌入的渐近全测地性，证明从有界对称域穿出的全纯曲线渐近全测地，推导等变全纯嵌入的全测地性，并解决域内代数子集的单值化问题，给出余紧格下全测地子集的双代数性刻画。

从 $\mathbb{C}^p$ 到半阿贝尔簇的全纯映射

Holomorphic Maps from $\mathbb{C}^p$ into Semi-Abelian Varieties

本文证明半阿贝尔簇上具有 Zariski 稠密像的全纯映射与约化有效除子相交的第二基本定理（截断水平为一），给出计数函数与特征函数的不等式关系。

The zeros of Gaussian random holomorphic functions on $\C^n$, and hole probability

We consider a class of Gaussian random holomorphic functions, whose expected zero set is uniformly distributed over $\C^n $. This class is unique (up to multiplication by a non zero holomorphic function), and is closely related to a Gaussian field over a Hilbert space of holomorphic functions on the reduced Heisenberg group. For a fixed random function of this class, we show that the probability that there are no zeros in a ball of large radius, is less than $e^{-c_1 r^{2n+2}}$, and is also greater than $e^{-c_2 r^{2n+2}}$. Enroute to this result we also compute probability estimates for the e…

等单值变形、$\mathbb C^*$-作用以及Dolbeault模空间上非交换Noether-Lefschetz轨迹的表征

Isomonodromic deformations of Higgs bundles and characterization of the non-abelian Noether--Lefschetz locus

研究等单值变形与Dolbeault模空间上自然$\mathbb C^*$-作用的相互作用，证明在复解析子簇上缩放后的族仍是等单值的充要条件，并简化了Simpson非交换Noether-Lefschetz轨迹的局部表征证明。

全纯快慢系统

Holomorphic slow-fast systems

本文研究二维全纯快慢系统不变复流形的存在性，给出保证一维不变复流形存在的条件，并据此证明简化问题的中心、焦点和结点在奇异摄动下是持久的。

The Hartogs extension theorem on (n-1)-complete complex spaces

Employing Morse theory for the global control of monodromy and the method of analytic discs for local extension, we establish a version of the global Hartogs extension theorem in a singular setting: for every domain D of an (n-1)-complete normal complex space X of pure dimension n >= 2 and for every compact set K in D such that D - K is connected, holomorphic or meromorphic functions in D - K extend holomorphically or meromorphically to D. Normality is an unvavoidable assumption for holomorphic extension, but we show that meromorphic extension holds on a reduced globally irreducible (not neces…

Mixed tête-à-tête twists as monodromies associated with holomorphic function germs

Tête-à-tête graphs were introduced by N. A'Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed tête-à-tête graphs provide a generalization which define mixed tête-à-tête twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed tête-à-tête twists as those pseudo-periodic automorphisms that have a power which is a product of right-handed Dehn twists around disjoint simple closed curves, including all boundary components. It follows that the class of tête-à-tête twists coincides with that of monodromies associated with reduced functi…

约化凸预叶状结构在 $\mathbb{P}^{2}_{\mathbb{C}}$ 上的对偶织网是平坦的

Le tissu dual d'un pré-feuilletage convexe réduit sur $\mathbb{P}^{2}_{\mathbb{C}}$ est plat

本文证明了复射影平面 $\mathbb{P}^{2}_{\mathbb{C}}$ 上约化凸预叶状结构的对偶织网是平坦的，推广了此前仅针对不变直线情形的结果。

量化与可嵌入CR约化的约化可交换性

Quantization Commutes with Reduction for Embeddable CR Reductions

本文证明在严格伪凸且全局CR可嵌入的约化条件下，CR Guillemin-Sternberg映射为有界同构，通过消去Fredholm缺陷实现量化与约化的可交换性。