E. M. Chirka
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.
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We give a sufficient condition for a meromorphic correspondence to be a holomorphic correspondence in a neighbourhood of a smooth real hypersurface
There is proved the sufficiency of several conditions for the removability of singularities of complex-analytic sets in domains of $\mathbb C^n$.
We endeavour a systematic approach for the removal of singularities for CR functions on an arbitrary embeddable CR manifold.
A paper is devoted to study of local behavior of so-called $Q$-mappings including qua\-si\-con\-for\-mal mappings and mappings with bounded distortion. It is showed that, such mappings have removable isolated singularities whenever the grow of the mappings is note more than some function of a radius of a ball.
Complete pluripolar sets and removable singularities of plurisubharmonic functions
本研究受已有成果启发，证明了多重次调和函数穿过紧完全多重极集的Hartogs型延拓定理，补充了Shiffman的经典定理。
Removable singularities of mappings with inverse Poletksy inequality on Rie\-man\-nian manifolds
本文研究Riemannian流形上满足逆Poletsky型模不等式的开离散映射，证明在遗漏至少两点且控制函数几乎处处球面可积时，边界孤立点为可去奇点。
In this paper, we study meromorphic functions on a domain $Ω\subset \mathbb{C}$ whose image has finite spherical area, counted with multiplicity. The paper is composed of two parts. In the first part, we show that the limit of a sequence of meromorphic functions is naturally defined on $Ω$ union a tree of spheres. In the second part, we show that a set $E \subset Ω$ is removable if and only if it is negligible for extremal distance.
We establish the holomorphic wedge extendability of CR functions, defined on an everywhere locally minimal generic submanifold M of C^n and having singularities contained in a submanifold N of codimension 1, 2 or 3, assuming some transversality conditions about the relative disposition of N with respect to the complex tangent bundle to M. The statements hold in arbitrary codimension and are obtained by applying the theory of normal deformations of analytic discs, due to A.E. Tumanov in 1994 and also the FBI propagation of singularities phenomenon enjoyed by CR functions, due to J.-M. Trepreau …
This paper surveys some recent results on existence, uniqueness and removable singularities for fully nonlinear differential equations on manifolds. The discussion also treats restriction theorems and the strong Bellman principle.
In this note it is shown that two key results on transcendental singularities for meromorphic functions of finite lower order have refinements which hold under the weaker hypothesis that the logarithmic derivative has finite lower order.
This is an extensive (published) survey on CR geometry, whose major themes are: formal analytic reflection principle; generic properties of Systems of (CR) vector fields; pairs of foliations and conjugate reflection identities; Sussmann's orbit theorem; local and global aspects of holomorphic extension of CR functions; Tumanov's solution of Bishop's equation in Hoelder classes with optimal loss of smoothness; wedge-extendability on C^2,a generic submanifolds of C^n consisting of a single CR orbit; propagation of CR extendability and edge-of-the-wedge theorem; Painlevé problem; metrically thin …
Let $n\ge 2$ be an integer, and $B^{n}\subset \mathbb{C}^{n}$ the unit ball. Let $K\subset B^{n}$ be a compact subset such that $B^n\setminus K$ is connected, or $K=\{z=(z_1,\cdots, z_n)|z_1=z_2=0\}\subset \mathbb{C}^{n}$. By the theory of developing maps, we prove that a Kähler metric on $B^{n}\setminus K$ with constant holomorphic sectional curvature uniquely extends to $B^{n}$.
A two-parameter characteristic of functions meromorphic on annuli is introduced and an extension of the Nevanlinna value distribution theory for such functions is proposed.
We present several aspects of the "topology of meromorphic functions", which we conceive as a general theory which includes the topology of holomorphic functions, the topology of pencils on quasi-projective spaces and the topology of polynomial functions.
The affine geometry of meromorphic connections with irregular singularities
本文研究黎曼曲面切丛上亚纯联络的仿射几何，引入局部不变量和局部模型，推广Veech构造定义Delaunay分解，并给出复杂度上界。
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.
On the Ohsawa-Takegoshi $L^2$ extension theorem and removable singularities of plurisubharmonic functions
本文推广了Ohsawa-Takegoshi $L^2$延拓定理至Stein流形上扭曲典范丛的全纯截面，并建立了多次调和函数在紧完备多极集上的Hartogs型延拓定理，填补了文献空白。
Let $D_j\subset\mathbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N. $$ Let $M\subset X$ be relatively closed. For any $j\in\{1,...,N\}$ let $Σ_j$ be the set of all $(z',z'')\in(A_1\times...\times A_{j-1})\times(A_{j+1}\times...\times A_N)$ such that the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\mathbb C^{n_j}: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $Σ_1,...,Σ_N$ are pluripolar. Put ${multline*} X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1…
Deformations of singularities of meromorphic $\mathfrak{sl}_2(\mathbb{C})$-connections and meromorphic quadratic differentials
本文研究无迹 $2\times2$ 亚纯线性 ODE 奇点的形变与汇合，建立形式不变量与二次微分射流的对应，并刻画等单值形变及其退化。
We show that if a meromorphic function has a direct singularity over infinity, then the escaping set has an unbounded component and the intersection of the escaping set with the Julia set contains continua. This intersection has an unbounded component if and only if the function has no Baker wandering domains. We also give estimates of the Hausdorff dimension and the upper box dimension of the Julia set of a meromorphic function with a logarithmic singularity over infinity. The above theorems are deduced from more general results concerning functions which have "direct or logarithmic tracts", …
A survey of general results on the singularities of inverses to meromorphic functions is given, with applications to holomorphic dynamics. This is a lecture delivered at the workshop "The role of complex analysis in complex dynamics" in Edinburgh on May 22 2013, with corrected mistakes and updated references.
Meromorphic continuation and non-polar singularities of local zeta functions in some smooth cases
本文研究了特定非实解析光滑函数的局部 zeta 函数，证明了其在某些奇点处具有非极点奇点，并验证了特定光滑函数类下亚纯延拓下界估计的最优性。
On meromorphic pencils, cusp singularities and holomorphic foliations in the complex plane
研究复平面\(\mathbb{C}^2\)中沿特定亚纯铅笔纤维同调平凡的多项式全纯\(1\)-形式，通过建立同调特征和积分原理，应用于尖点叶状结构，证明全局化定理，给出标准形式和第一积分，在多方面架起桥梁。
We consider a continuous function $f$ on a domain in $\mathbf C^n$ satisfying the inequality that $|\bar \partial f|\leq |f|$ off its zero set. The main conclusion is that the zero set of $f$ is a complex variety. We also obtain removable singularity theorem of Rado type for J-holomorphic maps. Let $Ω$ be an open subset in $\mathbf C$ and let $E$ be a closed polar subset of $Ω$. Let $u$ be a continuous map from $Ω$ into an almost complex manifold $(M,J)$ with $J$ of class $C^1$. We show that if $u$ is J-holomorphic on $Ω\setminus E$ then it is J-holomorphic on $Ω$.
We study vanishing cycles naturally attached to a meromorphic function with isolated singularities, in both local and global settings.
We show that for each $d\in (0,2]$ there exists a meromorphic function $f$ such that the inverse function of $f$ has three singularities and the Julia set of $f$ has Hausdorff dimension $d$.
In the main part of the paper, on the basis of contour integration of complex meromorphic functions whose singularities lie onto an integration contour, in the first step, a concept of improper integrals absolute existence of meromorphic functions, as more general one with respect to the concept of improper integrals convergence (existence), is introduced into analysis. In the second step, in the case when a modulus of complex parameter tends to infinity, an interval of improper integrals convergence of parametric meromorphic functions is defined. In accordance with this, it is shown that the …
We study some type of mappings with finite distortion $f:D\rightarrow D^{\prime},$ $D, D^{\prime}\subset{\Bbb R}^n,$ $n\ge 2,$ which admit branch points. It is proved some inequality playing essential role at investigation of some problems of plane and space mappings. As application, it is investigated a problem of removable singularities for open discrete mappings with finite length distortion.
Relative Cohomology and Deformations of Logarithmic Foliations on Complex Spaces
该论文研究复空间上对数叶状结构的相对上同调，在合适假设下得到其分解式，建立亚纯延拓定理，处理奇异对数叶状结构，并将结果应用于推导解析可积形变的形式正规形。
Given a real- or complex-analytic singular foliation $θ$ with $n$ first integrals of meromorphic or Darboux type $(f_1,\dots,f_n)$, we prove that there exists a local monomialization of the first integrals. In particular, if $θ$ is generated by the $n$ first integrals, we prove the existence of a local reduction of singularities of $θ$ to monomial singularities.
Closed meromorphic 1-forms
本文从闭亚纯 1-形式与平亚纯联络视角综述叶状结构经典理论，并应用于研究半整体分界线代数性及紧 Kähler 流形上数值平凡法丛超曲面的几何。
On the classification of Meromorphic Affine Connections on Complex Compact Surfaces
本研究分类了代数维数为1的紧复曲面上的亚纯仿射联络，拓展了前人在全纯情形下的相关工作，旨在探索复紧曲线单值化原理在高维带奇点几何结构中的拓展可能。
We construct a (non Kähler) compact complex 3-dimensional manifold $X$ having two following properties: 1) for any domain $D$ in $C^2$ every meromorphic map $f$ from this domain into $X$ extends to a meromorphic map from the envelope of meromorphy $\hat D$ of $D$ into $X$; 2) but there exist a meromorphic map $F$ from a punctured ball $B_*$ in $C^3$ into $X$ which doens't extend meromorphically to the origin. In other words, one can allways remove the singularities of complex codimesion two for the meromorphic maps into this $X$, but only up to some subset of complex codimension three. A descr…
Meromorphic Projective Structures: Signed Spaces, Grafting and Monodromy
本文研究了可微复结构的带符号空间，证明了Grafting定理的类比，并展示了单值性映射的局部双全纯性质。
We study transcendental singularities of a Schröder map arising from a rational function $f$, using results from complex dynamics and Nevanlinna theory. These maps are transcendental meromorphic functions of finite order in the complex plane. We show that their transcendental singularities lie over the set where $f$ is not semihyperbolic (unhyperbolic). In addition, if they are direct, then they lie over only attracting periodic points of $f$, and moreover, if $f$ is a polynomial, then both direct and indirect singularities lie over attracting, parabolic and Cremer periodic points of $f$. We a…
Logarithmic models and meromorphic functions in dimension two
本文研究二维空间中对数模型的构造及其在亚纯函数和向量场中的应用，通过分离子和Camacho-Sad指数构建具有特定奇点结构的亚纯函数。
We study the removable singularities for solutions to the Beltrami equation $\bar\partial f=μ\partial f$, assuming that the coefficient $μ$ lies on some Sobolev space $W^{1,p}$, $p\leq 2$. Our results are based on an extended version of the well known Weyl's lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly supported Beltrami coefficient $μ\in W^{1,2}$ preserve compact sets of $σ$-finite length and vanishing analytic capacity, even though they need not be bilipschitz.
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