Laurent series of holomorphic functions smooth up to the boundary

Integrating holomorphic $L^1$-functions

Let $Ω\Subset\mathbb{C}^{n}$ be a domain with smooth boundary, $k\in\mathbb{N}$. It is shown that the integral of a holomorphic function in $L^1(Ω)$ may be represented as the integral of this function against a smooth function vanishing to order $k-1$ on $bΩ$. An application for a smoothing property of the Bergman projection for conjugate holomorphic functions is given.

Function theory and holomorphic maps on symmetric products of planar domains

We show that the $\bar{\partial}$-problem is globally regular on a domain in $\mathbb{C}^n$, which is the $n$-fold symmetric product of a smoothly bounded planar domain. Remmert-Stein type theorems are proved for proper holomorphic maps between equidimensional symmetric products and proper holomorphic maps from cartesian products to symmetric products. It is shown that proper holomorphic maps between equidimensional symmetric products of smooth planar domains are smooth up to the boundary

Hardy spaces of holomorphic functions for domains in $\mathbb C^n$ with minimal smoothness

We prove various representations and density results for Hardy spaces of holomorphic functions for two classes of bounded domains in $\mathbb C^n$, whose boundaries satisfy minimal regularity conditions (namely the classes $C^2$ and $C^{1,1}$ respectively) together with naturally occurring notions of convexity.

Duality of holomorphic functions spaces und smoothing properties of the Bergman projection

Let $Ω\subset\mathbb{C}^n$ be a bounded domain with smooth boundary, whose Bergman projection $B$ maps the Sobolev space $H^{k_{1}}(Ω)$ (continuously) into $H^{k_{2}}(Ω)$. We establish two smoothing results: (i) the full Sobolev norm $\|Bf\|_{k_{2}}$ is controlled by $L^2$ derivatives of $f$ taken along a single, distinguished direction (of order $\leq k_{1}$), and (ii) the projection of a conjugate holomorphic function in $L^{2}(Ω)$ is automatically in $H^{k_{2}}(Ω)$. There are obvious corollaries for when $B$ is globally regular.

A note on smoothing properties of the Bergman projection

Recently Herbig, McNeal, and Straube have showed that the Bergman projection of conjugate holomorphic functions is smooth up to the boundary on a class of pseudoconvex domains. We show that a further smoothing property holds on a family of Reinhardt domains; namely, the Bergman projection of conjugate holomorphic functions is holomorphic past the boundary.

Carleman approximation by holomorphic automorphisms of $\mathbb C^n$

We approximate smooth maps defined on non-compact totally real manifolds by holomorphic automorphisms of $\mathbb C^n$.

Analytic Continuation of Holomorphic Correspondences and Equivalence of Domains in ${\mathbb C}^n$

The following result is proved: Let $D$ and $D'$ be bounded domains in $\mathbb C^n$, $\partial D$ is smooth, real-analytic, simply connected, and $\partial D'$ is connected, smooth, real-algebraic. Then there exists a proper holomorphic correspondence $f:D\to D'$ if and only if $\partial D$ and $\partial D'$ a locally CR-equivalent. This leads to a characterization of the equivalence relationship between bounded domains in $\mathbb C^n$ modulo proper holomorphic correspondences in terms of CR-equivalence of their boundaries.

Condition R and holomorphic mappings of domains with generic corners

A piecewise smooth domain is said to have generic corners if the corners are generic CR manifolds. It is shown that a biholomorphic mapping from a piecewise smooth pseudoconvex domain with generic corners in complex Euclidean space that satisfies Condition R to another domain extends as a smooth diffeomorphism of the respective closures if and only if the target domain is also piecewise smooth with generic corners and satisfies Condition R. Further it is shown that a proper map from a domain with generic corners satisfying Condition R to a product domain of the same dimension extends continuou…

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 $\C^{N'}$.

关于全纯向量丛最优 $L^2$ 延拓定理的一个注记

A Remark on Optimal $L^2$ extension theorem for holomorphic vector bundles

本文在弱拟凸 Kähler 流形上建立了带光滑 Hermite 度量的全纯向量丛在连续增益条件下的最优 $L^2$ 延拓定理，统一了 Guan-Zhou 与 Zhou-Zhu 的已有结果。

Holomorphic extension of decomposable distributions from a CR submanifold of $\mathbb C^L$

Given $N$ a non generic smooth CR submanifold of $\C^L$, $N=\{(\n,h(\n))\}$ where $\n$ is generic in $\C^{L-n}$ and $h$ is a CR map from $\n$ into $\C^n$. We prove, using only elementary tools, that if $h$ is decomposable at $p'\in \n$ then any decomposable CR distribution on $N$ at $p=(p',h(p'))$ extends holomorphically to a complex transversal wedge. This gives an elementary proof of the well known equivalent for totally real non generic submanifolds, i.e if $N$ is a smooth totally real submanifold of $\C^L$ any continuous function on $N$ admits a holomorphic extension to a complex transvers…

Common boundary values of holomorphic functions for two-sided complex structures

Let $Ω_1,Ω_2$ be two disjoint open sets in $\mathbf C^n$ whose boundaries share a smooth real hypersurface $M$ as relatively open subsets. Assume that $Ω_i$ is equipped with a complex structure $J^i$ which is smooth up to $M$. Assume that the operator norm $\|J^2-J^1\|<2$ on $M$. Let $f$ be a continuous function on the union of $Ω_1,Ω_2, M$. If $f$ is holomorphic with respect to both structures in the open sets, then $f$ must be smooth on the union of $Ω_1$ with $M$. Although the result as stated is far more meaningful for integrable structures, our methods make it much more natural to deal wi…

On the tangential holomorphic vector fields vanishing at an infinite type point

Let (M,p) be a smooth non-Leviflat CR hypersurface germ in complex Euclidean space of dimension 2 where p is of infinite type. The purpose of this article is to investigate the holomorphic vector fields tangent to (M,p) vanishing at p.

On boundary properties of asymptotically holomorphic functions

We prove a Fatou type theorem for bounded functions with d_J -bar differential of a controled growth on smoothly bounded domains in an almost complex manifold.

Extremal discs and the holomorphic extension from convex hypersurfaces

Let D be a convex domain with smooth boundary in complex space and let f be a continuous function on the boundary of D. Suppose that f holomorphically extends to the extremal discs tangent to a convex subdomain of D. We prove that f holomorphically extends to D. The result partially answers a conjecture by Globevnik and Stout of 1991.

On the characteristic foliation on a smooth hypersurface in a holomorphic symplectic fourfold

Let $X$ be an irreducible holomorphic symplectic fourfold and $D$ a smooth hypersurface in $X$. It follows from a result by Amerik and Campana that the characteristic foliation (that is the foliation given by the kernel of the restriction of the symplectic form to $D$) is not algebraic unless $D$ is uniruled. Suppose now that the Zariski closure of its general leaf is a surface. We prove that $X$ has a lagrangian fibration and $D$ is the inverse image of a curve on its base.

Compactness of Hankel Operators with Conjugate Holomorphic Symbols on Complete Reinhardt Domains in $\mathbb{C}^2$

In this paper we characterize compact Hankel operators with conjugate holomorphic symbols on the Bergman space of bounded convex Reinhardt domains in $\mathbb{C}^2$. We also characterize compactness of Hankel operators with conjugate holomorphic symbols on smooth bounded pseudoconvex complete Reinhardt domains in $\mathbb{C}^2$.

一种用平面Lipschitz域上全纯函数表示边值的新方法

A new way to express boundary values in terms of holomorphic functions on planar Lipschitz domains

该研究提出一种将平面单连通Lipschitz域边界上的$p$可积函数分解为两个唯一确定的全纯函数边值之和的新方法，并给出相关正则性定理的新证明，还得到柯西变换核的新刻画，为经典问题提供新视角。

Tangential Lipschitz Gain for Holomorphic Functions on Domains of Finite Type

Functions that are holomorphic and Lipschitz in a smoothly bounded domain enjoy a gain in the order of Lipschitz regularity in the complex tangential directions near the boundary. We describe this gain explicitly in terms of the defining function near points of finite type in the boundary.

Moduli of holomorphic functions and logarithmically convex radial weights

Let $H(D)$ denote the space of holomorphic functions on the unit disk $D$. We characterize those radial weights $w$ on $D$, for which there exist functions $f, g \in H(D)$ such that the sum $|f| + |g|$ is equivalent to $w$. Also, we obtain similar results in several complex variables for circular, strictly convex domains with smooth boundary.

On sets of marked once-holed tori allowing holomorphic mappings into Riemann surfaces with marked handle

In our previous work, for a given Riemann surface $Y_0$ with marked handle, we investigated geometric properties of the set of marked once-holed tori $X$ allowing holomorphic mappings of $X$ into $Y_0$. It turned out that it is a closed domain with Lipschitz boundary. In the present paper we show that the boundary is never smooth. Also, we evaluate the critical extremal length for the existence of holomorphic mappings in terms of hyperbolic lengths.

恰当全纯映射族

Families of proper holomorphic maps

本文研究了在参数空间B中连续变化的复结构下，构造恰当全纯映射族的问题，并给出了具体的构造方法。

Simple zero property of some holomorphic functions on the moduli space of tori

We prove that some holomorphic functions on the moduli space of tori have only simple zeros. Instead of computing the derivative with respect to the moduli parameter $τ$, we introduce a conceptual proof by applying Painlevé VI\ equation. As an application of this simple zero property, we obtain the smoothness of all the degeneracy curves of trivial critical points for some multiple Green function.

The extension of holomorphic functions on a non-pluriharmonic locus

Let $n \geq 4$ and let $Ω$ be a bounded hyperconvex domain in $\mathbb{C}^{n}$. Let $φ$ be a negative exhaustive smooth plurisubharmonic function on $Ω$. We show that any holomorphic function defined on a connected open neighborhood of the support of $(i\partial \overline{\partial}φ)^{n-3}$ can be extended to the holomorphic function on $Ω$.

通过雅可比多项式对全纯函数的分数阶1/2拉普拉斯算子

Fractional $\frac{1}{2}$-Laplacian of holomorphic functions via Jacobi polynomials

研究原点处有幂型奇点的光滑函数及全纯函数的分数阶1/2拉普拉斯算子，基于特定测试函数空间定义，通过对洛朗级数应用构造得含雅可比多项式的新级数，还推导了相关应用及泛函演算。

Proper holomorphic curves attached to domains

Let D be a domain in C^n with smooth boundary, of finite 1-type at a point p in the boundary and such that the closure of D has a basis of Stein Runge neighborhoods. Assume that there exists an analytic disc which intersects the closure of D exactly at p. We construct proper holomorphic maps from any open Riemann surface S which are attached to the closure of D exactly at p.

Nonvanishing of the differential of holomorphic mappings at boundary points

In this paper we prove a general result of the ``Hopf lemma'' type for CR mappings, with nonidentically vanishing Jacobians, between real hypersurfaces in C^n with smooth or real analytic boundaries. Applications of this result to finiteness and holomorphic extendibility of such mappings are also given. The novelty here is that we make no assumption on the nonflatness of the mapping or its Jacobian, nor do we assume that the hypersurfaces are pseudoconvex or minimally convex.

Finite jet determination of holomorphic mappings at the boundary

Let M,M' be smooth real hypersurfaces in N-dimensional space and assume that M is k-nondegenerate at a point p in M. We prove that holomorphic mappings that extend smoothly to M, sending a neighborhood of p in M diffeomorphically into M' are completely determined by their 2k-jet at p. As an application of this result, we also give sufficient conditions on a smooth real hypersurface which guarantee that the space of infinitesimal CR automorphisms is finite dimensional.

On a neighborhood of a torus leaf of a certain class of holomorphic foliations on complex surfaces

Let $C$ be a smooth elliptic curve embedded in a smooth complex surface $X$ such that $C$ is a leaf of a suitable holomorphic foliation of $X$. We investigate complex analytic properties of a neighborhood of $C$ under some assumptions on complex dynamical properties of the holonomy function. As an application, we give an example of $(C, X)$ in which the line bundle $[C]$ is formally flat along $C$ however it does not admit a $C^\infty$ Hermitian metric with semi-positive curvature.

The Hele-Shaw flow and moduli of holomorphic discs

We present a new connection between the Hele-Shaw flow, also known as two-dimensional (2D) Laplacian growth, and the theory of holomorphic discs with boundary contained in a totally real submanifold. Using this we prove short time existence and uniqueness of the Hele-Shaw flow with varying permeability both when starting from a single point and also starting from a smooth Jordan domain. Applying the same ideas we prove that the moduli space of smooth quadrature domains is a smooth manifold whose dimension we also calculate, and we give a local existence theorem for the inverse potential proble…

Holomorphic one-forms without zeros on threefolds

We show that a smooth complex projective threefold admits a holomorphic one-form without zeros if and only if the underlying real 6-manifold fibres smoothly over the circle, and we give a complete classification of all threefolds with that property. Our results prove a conjecture of Kotschick in dimension three.

On Riesz type inequalities, Hardy-Littlewood type theorems and smooth moduli

The purpose of this paper is to develop some methods to study Riesz type inequalities, Hardy-Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions in high-dimensional cases. Initially, we prove some sharp Riesz type inequalities of pluriharmonic functions on bounded symmetric domains. The obtained results extend the main results in (\textit{Trans. Amer. Math. Soc.} {\bf 372} (2019)~ 4031--4051). Furthermore, some Hardy-Littlewood type theorems of holomorphic and pluriharmonic functions on John domains are established. Additionally, we also discuss the …

Holomorphic family of strongly pseudoconvex domains in a Kähler manifold

Let $p:X\rightarrow Y$ be a surjective holomorphic mapping between Kähler manifolds. Let $D$ be a bounded smooth domain in $X$ such that every generic fiber $D_y:=D\cap p^{-1}(y)$ for $y\in Y$ is a strongly pseudoconvex domain in $X_y:=p^{-1}(y)$, which admits the complete Kähler-Einstein metric. This family of Kähler-Einstein metrics induces a smooth $(1,1)$-form $ρ$ on $D$. In this paper, we prove that $ρ$ is positive-definite on $D$ if $D$ is strongly pseudoconvex. We also discuss the extensioin of $ρ$ as a positive current across singular fibers.

Proper pseudo-holomorphic maps between strictly pseudoconvex regions

We prove that proper pseudo-holomorphic maps between strictly pseudoconvex regions in almost complex manifolds extend to the boundary. The key point is that the Jacobian is far from zero near the boundary, and the proof is mainly based on an almost complex analogue of the scaling method. We also establish a link between the regularity of the extension and the regularity of the almost complex structures, and we determine explicit estimates for the Hölderian norms. As a corollary, we get in the smooth case a necessary and sufficient condition on the almost complex structure of the target's space…

On holomorphic domination, I

Let $X$ be a separable Banach space and $u{:} X\to\Bbb{R}$ locally upper bounded. We show that there are a Banach space $Z$ and a holomorphic function $h{:} X\to Z$ with $u(x)<\|h(x)\|$ for $x\in X$. As a consequence we find that the sheaf cohomology group $H^q(X,\Cal{O})$ vanishes if $X$ has the bounded approximation property (i.e., $X$ is a direct summand of a Banach space with a Schauder basis), $\Cal{O}$ is the sheaf of germs of holomorphic functions on $X$, and $q\ge1$. As another consequence we prove that if $f$ is a $C^1$-smooth $\overline\partial$-closed $(0,1)$-form on the space $X=L_…

Analyticity of smooth CR mappings of generic submanifolds

We consider a smooth CR mapping $f$ from a real-analytic generic submanifold $M$ in $\bC^N$ into $\bC^N$. For $M$ of finite type and essentially finite at a point $p\in M$, and $f$ formally finite at $p$, we give a necessary and sufficient condition for $f$ to extend as a holomorphic mapping in some neighborhood of $p$. In a similar vein, we consider a formal holomorphic mapping $H$ and give a necessary and sufficient condition for $H$ to be convergent.

On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four

We extend a result of Guan by showing that the second Betti number of a 4-dimensional primitively symplectic orbifold is at most 23 and there are at most 91 singular points. The maximal possibility 23 can only occur in the smooth case. In addition to the known smooth examples with second Betti numbers 7 and 23, we provide examples of such orbifolds with second Betti numbers 3, 5, 6, 8, 9, 10, 11, 14 and 16. In an appendix, we extend Salamon's relation among Betti/Hodge numbers of symplectic manifolds to symplectic orbifolds.

Holomorphic foliation associated with a semi-positive class of numerical dimension one

Let $X$ be a compact Kähler manifold and $α$ be a class in the Dolbeault cohomology class of bidegree $(1, 1)$ on $X$. When the numerical dimension of $α$ is one and $α$ admits at least two smooth semi-positive representatives, we show the existence of a family of real analytic Levi-flat hypersurfaces in $X$ and a holomorphic foliation on a suitable domain of $X$ along whose leaves any semi-positive representative of $α$ is zero. As an application, we give the affirmative answer to \cite[Conjecture 2.1]{K2019} on the relation between the semi-positivity of the line bundle $[Y]$ and the analyti…

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…

Integration of vector fields on smooth and holomorphic supermanifolds

We give a new and self-contained proof of the existence and unicity of the flow for an arbitrary (not necessarily homogeneous) smooth vector field on a real supermanifold, and extend these results to the case of holomorphic vector fields on complex supermanifolds. Furthermore we discuss local actions associated to super vector fields, and give several examples and applications, as, e.g., the construction of an exponential morphism for an arbitrary finite dimensional Lie supergroup. In the 2nd version we corrected typos, presentation of figures and other small points. We added references in Sec…

On the singular factor of a linear combination of holomorphic functions

We prove that the linear combinations of functions $f_0,...,f_n$ in $H^\infty$ have "few" singular inner factors, provided that the $f_j$'s are suitably smooth up to the boundary, while in general this is no longer true.

Holomorphic curves in Base Spaces of Families of Polarized Manifolds

For a smooth family $V \to U$ of polarized manifolds with semi-ample canonical sheaves, we show the following result: any entire curve must be contained in the fibers of the classifying map from the base space $U$ to the moduli space. This settles the Relative Isotriviality Conjecture, \cite[Conjecture 1.5]{DLSZ}.

Asymptotic behavior of discrete holomorphic maps z^c, log(z) and discrete Painleve transcedents

It is shown that discrete analogs of z^c and log(z) have the same asymptotic behavior as their smooth counterparts. These discrete maps are described in terms of special solutions of discrete Painleve-II equations, asymptotics of these solutions providing the behaviour of discrete z^c and log(z) at infinity.

Common boundary regular fixed points for holomorphic semigroups in strongly convex domains

Let $D$ be a bounded strongly convex domain with smooth boundary in $\mathbb C^N$. Let $(ϕ_t)$ be a continuous semigroup of holomorphic self-maps of $D$. We prove that if $p\in \partial D$ is an isolated boundary regular fixed point for $ϕ_{t_0}$ for some $t_0>0$, then $p$ is a boundary regular fixed point for $ϕ_t$ for all $t\geq 0$. Along the way we also study backward iteration sequences for elliptic holomorphic self-maps of $D$.

闭合3维流形嵌入到C³中的显式全纯结构

Explicit Holomorphic Structures for embeddings of closed 3-manifolds into $\mathbb{C}^3$

本文基于前人工作，证明了对于给定的无向同调链环L和任意的2平面结构η，存在一个全纯嵌入使得复切线集恰好为L，并且在每个点x∈L处的全纯切空间恰好为η_x。

On holomorphic distributions on Fano threefolds

This paper is devoted to the study of holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano manifolds threedimensional, with Picard number equal to one. We study the relations between algebro-geometric properties of the singular set of singular holomorphic distributions and their associated sheaves. We characterize either distributions whose tangent sheaf or conormal sheaf are arithmetically Cohen Macaulay (aCM) on smooth weighted projective complete intersection Fano manifolds threefold. We also prove that a codimension one locally…

非紧复流形上随机全纯截面因子的渐近正态性

Asymptotic Normality of Divisors of Random Holomorphic Sections on Non-compact Complex Manifolds

本文研究非紧 Hermite 流形上随机全纯截面零因子的统计行为，证明相关光滑线性统计量满足中心极限定理。

Complete complex hypersurfaces in the ball come in foliations

In this paper we prove that every smooth complete closed complex hypersurface in the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ is a level set of a noncritical holomorphic function on $\mathbb{B}_n$ all of whose level sets are complete. This shows that $\mathbb{B}_n$ admits a nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces and, what is the main point, that every hypersurface in $\mathbb{B}_n$ of this type can be embedded into such a foliation. We establish a more general result in which neither completeness nor smoothness of the given hypersurf…

Hilbert series of nearly holomorphic sections on generalized flag manifolds

Let X=G/P be a complex flag manifold and E->X be a G-homogeneous holomorphic vector bundle. Fix a U-invariant Kaehler metric on X with U in G maximal compact. We study the sheaf of nearly holomorphic sections and show that the space of global nearly holomorphic sections in E coincides with the space of U-finite smooth sections in E. The degree of nearly holomorphic sections defines a U-invariant filtration on this space. Using sheaf cohomology, we determine in suitable cases the corresponding Hilbert series. It turns out to be given in terms of Lusztig's q-analog of Kostant's weight multiplici…

Holomorphic correspondences between CR manifolds

It is proved that a germ of a real analytic CR map from a smooth real-analytic minimal CR manifold M to an essentially finite real-algebraic generic submanifold M' of P^N of the same CR-dimension extends as a holomorphic correspondence along M. Applications are given for pseudoconcave submanifolds of P^N.

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。

Kähler 三流形上无零点的全纯 1-形式

Holomorphic 1-forms without zeros on Kähler threefolds

本文分类了所有容许以圆为底的 $C^\infty$-纤维丛结构的连通紧致光滑 Kähler 三流形，证明其存在有限平展覆盖双亚满射于特定曲面丛，从而证明了 Kotschick 在 3 维情形下的猜想。