Mark Elin, David Shoikhet, Nikolai Tarkhanov
We prove a theorem on separation of boundary null points for generators of continuous semigroups of holomorphic self-mappings of the unit disk in the complex plane. Our construction demonstrates the existence and importance of a particular role of the binary operation $\circ$ given by $1 / f \circ g = 1/f + 1/g$ on generators.
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We study holomorphic foliations tangent to singular real-analytic Levi-flat hypersurfaces in compact complex manifolds of complex dimension two. We give some hypotheses to guarantee the existence of dicritical singularities of these objects. As consequence, we give some applications to holomorphic foliations tangent to real-analytic Levi-flat hypersurfaces with singularities in $\mathbb{P}^2$.
This paper has been withdrawn by the author(s) and included into the new version of "An extension theorem for separately holomorphic functions with singularities", math.CV/0104089.
We study the pluripolar hull of the graph of a holomorphic function f, defined on a domain D in the complex plane outside a polar set A of D. This leads to a theorem that describes under what conditions f is nowhere extendable over A, while the graph of f over D \ A is NOT complete pluripolar.
These are the notes of a series of lectures delivered by the author at the Graduate School of Mathematics of the University of Tokyo, during the month of October 2015. They were meant to be as self-contained as possible, taking into account time and space. The basic idea was to introduce the concepts and some of the basic results in the theory of holomorphic foliations with singularities. Another goal is to guide the reader to some of the recent questions and problems in the field, providing in this way a motivating introduction to those who are interested in studying a new subject.
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 present a rigidity property of holomorphic generators on the open unit ball $\mathbb{B}$ of a Hilbert space $H$. Namely, if $f\in\Hol (\mathbb{B},H)$ is the generator of a one-parameter continuous semigroup ${F_t}_{t\geq 0}$ on $\mathbb{B}$ such that for some boundary point $τ\in \partial\mathbb{B}$, the admissible limit $K$-$\lim\limits_{z\toτ}\frac{f(x)}{\|x-τ\|^{3}}=0$, then $f$ vanishes identically on $\mathbb{B}$.
Let $f$ be the infinitesimal generator of a one-parameter semigroup $\left\{ F_{t}\right\} _{t\ge0}$ of holomorphic self-mappings of the open unit disk $Δ$. In this paper we study properties of the family $R$ of resolvents $(I+rf)^{-1}:Δ\toΔ~ (r\ge0)$ in the spirit of geometric function theory. We discovered, in particular, that $R$ forms an inverse Löwner chain of hyperbolically convex functions. Moreover, each element of $R$ satisfies the Noshiro-Warschawski condition and is a starlike function of order at least $\frac12$,. This, in turn, implies that each element of $R$ is also a holomorphi…
Heat diffusions on holomorphic foliations with non-degenerate singularities
本文研究紧复流形上带非退化奇点的 Brody 双曲全纯叶状结构，证明叶状热扩散与抽象热扩散一致，并由此得出抽象热扩散的唯一性。
Poincaré metric of holomorphic foliations with non-degenerate singularities
本文研究紧复流形上具有非退化奇点的布罗迪双曲叶层，证明其叶向庞加莱度量在奇点集附近具有带对数斜率的横向赫尔德连续性。
We define the Milnor number -- as the intersection number of two holomorphic sections -- of a one-dimensional holomorphic foliation $\mathscr{F}$ with respect to a compact connected component $C$ of its singular set. Under certain conditions, we prove that the Milnor number of $\mathscr{F}$ on a three-dimensional manifold with respect to $C$ is invariant by $C^1$ topological equivalences.
Let $X$ be a Hermitian complex space of pure dimension $n$ with isolated singularities. In the present paper, we give a natural resolution for the canonical sheaf of square-integrable holomorphic $n$-forms with Dirichlet boundary condition on $X$. As application, we obtain an explicit smooth model for the $L^2$-$\bar{\partial}$-cohomology, including natural resolutions for sheaves of $\bar{\partial}$-closed (holomorphic) $L^2$-functions.
We study codimension one holomorphic foliations on complex projective spaces and compact manifolds under the assumption that the foliation has a projective transverse structure in the complement of some invariant codimension one analytic subset. The basic motivation is the characterization of pull-backs of Riccati foliations on projective spaces. Our techniques apply to give a description of the generic models of codimension one foliations on compact manifolds of dimension $\ge 3$.
Let $D_j\subset\Bbb C^{k_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluripolar set, $j=1,...,N$. Put$$X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N\subset\Bbb C^{k_1+...+k_N}.$$Let $U$ be an open connected neighborhood of $X$ and let $M\varsubsetneq U$ be an analytic subset. Then there exists an analytic subset $\hat M$ of the `envelope of holomorphy' $\hat X$ of $X$ with $\hat M\cap X\subset M$ such that for every function $f$ separately holomorphic on $X\setminus M$ there exists an $\hat f$ holomorphic on $\hat X\setminus\hat…
Finiteness of hyperbolic entropy for holomorphic foliations with non-degenerate singularities
本文研究了具有非退化奇点的全纯叶系的超几何熵有限性问题，证明其熵值为有限。
We prove the existence of non-positively curved Kähler-Einstein metrics with cone singularities along a given simple normal crossing divisor on a compact Kähler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved Kähler-Einstein metrics with cone singularities. As an application we extend to this setting classical results of Lichnerowicz and Kobayashi on the parallelism and vanishing of appropriate holomorphic tensor fields.
In this paper I give a condition, in terms of boundary behaviour, that forces the infinitesimal generator of a one-parameter semigroup of holomorphic maps to vanish.
In this paper we establish a rigidity property of holomorphic generators by using their local behavior at a boundary point $τ$ of the open unit disk $Δ$. Namely, if $f\in\mathrm{Hol}(Δ,\mathbb{C})$ is the generator of a one-parameter continuous semigroup $\{F_{t}\}_{t\geq0}$, we state that the equality $f(z)=o(|z-τ|^{3})$ when $z\toτ$ in each non-tangential approach region at $τ$ implies that $f$ vanishes identically on $Δ$. Note, that if $F$ is a self-mapping of $Δ$ then $f=I-F$ is a generator, so our result extends the boundary version of the Schwarz Lemma obtained by D. Burns and S. Krantz.…
We prove that rational and 1-rational singularities of complex spaces are stable under taking quotients by holomorphic actions of reductive and compact Lie groups. This extends a result of Boutot to the analytic category and yields a refinement of his result in the algebraic category. As one of the main technical tools vanishing theorems for cohomology groups with support on fibres of resolutions are proven.
We prove that in the extension theorem for separately holomorphic functions on an $N$-fold cross with singularities the case of analytic singularities follows from the case of pluripolar singularities.
We study boundary singularities which can appear for infinitesimal generators of one-parameter semigroups of holomorphic self-maps in the unit disc. We introduce "regular" fractional singularities and characterize them in terms of the behavior of the associated semigroups and Koenigs functions. We also provide necessary and sufficient geometric criteria on the shape of the image of the Koenigs function for having such singularities. In order to do this, we study contact points of semigroups and prove that any contact (not fixed) point of a one-parameter semigroup corresponds to a maximal arc o…
The aim of this paper is to present an extension theorem for the functions separately holomorphic on generalized (N,k)-crosses with pluripolar singularities.
The two-dimensional Matkowski--Sutô equation with holomorphic and strictly increasing generators
本文研究二维Matkowski--Sutô方程，在全纯与单调算子两种设定下刻画解，全纯情形仅有仿射与指数对，单调情形存在无限维非仿射解。
We prove that a transversely product component of the singular set of a holomorphic foliation on $\mathbb P^n$ is necessarily a Kupka component.
In dimensions greater than or equal to 3, the local structure of a singular holomorphic foliation conceals a globally defined foliation on the projective space of dimension one less. In this paper, we will discuss how the global dynamics of the latter foliation exerts influence on several problems that apparently have a purely local nature. In the course of the discussion, a few recent results and open problems in the area will be reviewed as well.
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 G be a finite group acting on a symplectic complex vector space V. Assume that the quotient V/G has a holomorphic symplectic resolution. We prove that G is generated by "symplectic reflectionsd"', i.e. symplectomorphisms with fixed space of codimension 2 in V. Symplectic resolutions are always semismall. A crepant resolution of V/G is always symplectic. We give a symplectic version of Nakamura conjectures.
We investigate the minimal singularities of metrics on a big line bundle $L$ over a projective manifold when the stable base locus $Y$ of $L$ is a submanifold of codimension $r\geq 1$. Under some assumptions on the normal bundle and a neighborhood of $Y$, we give a explicit description of the minimal singularity of metrics on $L$. We apply this result to study a higher (co-)dimensional analogue of Zariski's example, in which the line bundle $L$ is not semi-ample, however it is nef and big.
We give a sufficient condition for a meromorphic correspondence to be a holomorphic correspondence in a neighbourhood of a smooth real hypersurface
Let \Fc be a holomorphic foliation by curves defined in a neighborhood of 0 in \C^2 having 0 as a hyperbolic singularity. Let T be a harmonic current directed by \Fc which does not give mass to any of the two separatrices. Then we show that the Lelong number of T at 0 vanishes. Next, we apply this local result to investigate the global mass-distribution for directed harmonic currents on singular holomorphic foliations living on compact complex surfaces. Finally, we apply this global result to study the recurrence phenomenon of a generic leaf.
We present necessary and sufficient conditions on planar compacta $K$ and continuous functions $f$ on $K$ in order that $f$ generates the algebras $P(K), R(K), A(K)$ or $C(K)$. We also unveil quite surprisingly simple examples of non-polynomial convex compacta $K\subseteq\mathbb C$ and $f\in P(K)$ with the property that $f \in P(K)$ is a homeomorphism, but for which $f^{-1}\notin P(f(K))$. As a consequence, such functions do not admit injective holomorphic extensions to the interior of the polynomial convex hull $\widehat K$. On the other hand, it will be shown that the restriction $f^*|_G$ of…
Sharp Coefficient and Inverse Problems for Holomorphic Semigroup Generators
本文研究了与全纯半群生成元相关系数函数的极值问题，得出了关于初始对数系数、反系数及对数反系数的精确界，并探讨了连续系数问题及广义Fekete--Szegö函数的极值。
We prove that for any complex manifold X, the set of all holomorphic maps from the unit disc to X whose images are everywhere dense in X forms a dense subset in the space of all holomorphic maps from the disc to X. We show by an example that this need not hold for maps from the disc to complex spaces with singularities.
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。
In these introductory notes we give the basics of the theory of holomorphic foliations and laminations. The emphasis is on the theory of harmonic currents and unique ergodicity for laminations transversally Lipschitz in CP^2 and for generic holomorphic foliations in CP^2.
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 $B^{n}$.
Let $D_j\subset\Bbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times ...\times A_N\subset\Bbb C^{n_1}\times...\times\Bbb C^{n_N}=\Bbb C^n. $$ Let $U\subset\Bbb C^n$ be an open neighborhood of $X$ and let $M\subset U$ be a relatively closed subset of $U$. For $j\in\{1,...,N\}$ let $Σ_j$ be the set of all $(z',z'')\in(A_1\times...\times A_{j-1}) \times(A_{j+1}\times...\times A_N)$ for which the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\Bbb C^{n_j}\: (z',z_j,z'')\in M…
We study the classification of the pairs $(N, \,X)$ where $N$ is a Stein surface and $X$ is a complete holomorphic vector field with isolated singularities on $N$. We describe the role of transverse sections in the classification of $X$ and give necessary and sufficient conditions on $X$ in order to have $N$ biholomorphic to $\mathbb C^2$. As a sample of our results, we prove that $N$ is biholomorphic to $\mathbb C^2$ if $H^2(N,\mathbb Z)=0$, $X$ has a finite number of singularities and exhibits a non-nilpotent singularity with three separatrices or, equivalently, a singularity with first jet …
We present enumerative results for holomorphic foliations by curves on $\mathbb{P}^n$, $n\geq 3$, with singularities of positive dimension. Some of the results presented improve previous ones due to Corrêa--Fernández-Pérez--Nonato Costa--Vidal Martins and Nonato Costa. We also present an enumerative result bounding the number of isolated singularities in a projective subvariety invariant by a holomorphic foliation by curves on $\mathbb{P}^n$ with a singularity of positive dimension.
The aim of this article is twofold: First we study holomorphic germs of parabolic diffeomorphisms of $(\mathbb{C}^2,0)$ that are reversed by a holomorphic reflection and posses an analytic first integral with non-degenerate critical point at the origin. We find a canonical formal normal form and provide a complete analytic classification (in formal generic cases) in terms of a collection of functional invariants. Their restriction to an irreductible component of the zero locus of the first integral reduces to the Birkhoff--Écalle--Voronin modulus of the 1-dimensional restricted parabolic germ.…
Formulae for indices of holomorphic foliations via reduction of singularities
本文通过奇点约简过程，推导出全纯叶系的多种指标公式，包括歧变向量、Milnor数、经典指数等，并统一了相关理论框架。
This article is concerned with the geometry of germs of real analytic surfaces in $(\mathbb{C}^2,0)$ having an isolated Cauchy-Riemann (CR) singularity at the origin. These are perturbations of {\it Bishop quadrics}. There are two kinds of CR singularities stable under perturbation~: {\it elliptic} and {\it hyperbolic}. Elliptic case was studied by Moser-Webster \cite{moser-webster} who showed that such a surface is locally, near the CR singularity, holomorphically equivalent to {\it normal form} from which lots of geometric features can be read off. In this article we focus on perturbations o…
Starting from some remarkable singularities of holomorphic vector fields, we construct (open) complex surfaces over which the singularities in question are realized by complete vector fields. Our constructions lead to manifolds and vector fields beyond the algebraic setting and provide examples of complete vector fields with some new dynamical phenomena.
Generic singularities of holomorphic foliations by curves on $\mathbb{P}^n$
本文研究射影空间上一般奇异曲线全纯叶状结构，证明满测度集合中叶状结构的奇点均可线性化且双曲，并在次数至少为2时无不变代数曲线。
We establish extension theorems for separately holomorphic mappings defined on sets of the form W\setminus M with values in a complex analytic space which possesses the Hartogs extension property. Here W is a 2-fold cross of arbitrary complex manifolds and M is a set of singularities which is locally pluripolar (resp. thin) in fibers.
We present a list of open questions in the theory of holomorphic foliations, possibly with singularities. Some problems have been around for a while, others are very accessible.
We study analytic integrable deformations of the germ of a holomorphic foliation given by $df=0$ at the origin $0 \in \mathbb C^n, n \geq 3$. We consider the case where $f$ is a germ of an irreducible and reduced holomorphic function. Our central hypotheses is that, {\em outside of a dimension $\leq n-3$ analytic subset $Y\subset X$, the analytic hypersurface $X_f : (f=0)$ has only normal crossings singularities}. We then prove that, as germs, such deformations also exhibit a holomorphic first integral, depending analytically on the parameter of the deformation. This applies to the study of in…
One of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations (\cite{moussu}). In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with "many" periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in di…
Let a real-analytic manifold $M$ formally (holomorphically) equivalent to the following model \begin{equation*}w=z_{1}\overline{z}_{1}+\dots+z_{N}\overline{z}_{N}+λ_{1}\left(z_{1}^{2}+\overline{z}_{1}^{2}\right)+\dots+λ_{N}\left(z_{N}^{2}+\overline{z}_{N}^{2}\right),{equation*} assuming that $$ λ_{1},\dots, λ_{N}\in \left[0,\frac{1}{2}\right).$$ It is proven that $M$ is holomorphically equivalent to this model by developing a partial normal form for such real-analytic submanifold using generalized Fischer Decompositions. In particular, there are defined certain Spaces of Normalizations used al…
We study the relationship between singular holomorphic foliations in $(\mathbb{C}^{2},0)$ and their separatrices. Under mild conditions we describe a complete set of analytic invariants characterizing foliations with quasi-homogeneous separatrices. Further, we give the full moduli space of quasi-homogeneous plane curves. This paper has an expository character in order to make it accessible also to non-specialists.
We prove that given a holomorphic family of holomorphic functions with isolated singularities at zero and constant Milnor number, it is possible to obtain the gradient inequality with a uniform exponent.
We introduce the notion of regular (boundary) poles for infinitesimal generators of semigroups of holomorphic self-maps of the unit disc. We characterize such regular poles in terms of $β$-points (i.e. pre-images of values with positive Carleson-Makarov $β$-numbers) of the associated semigroup and of the associated Königs intertwining function. We also define a natural duality operation in the cone of infinitesimal generators and show that the regular poles of an infinitesimal generator correspond to the regular null poles of the dual generator. Finally we apply such a construction to study ra…
Let $D$ and $G$ be copies of the open unit disc in $\C,$ let $A$ (resp. $B$) be a measurable subset of $\partial D$ (resp. $\partial G$), let $W$ be the 2-fold cross $\big((D\cup A)\times B\big)\cup \big(A\times(B\cup G)\big),$ and let $M$ be a relatively closed subset of $W.$ Suppose in addition that $A$ and $B$ are of positive one-dimensional Lebesgue measure and that $M$ is fiberwise polar (resp. fiberwise discrete) and that $M\cap (A\times B)=\varnothing.$ We determine the "envelope of holomorphy" $\hat{W\setminus M}$ of $W\setminus M$ in the sense that any function locally bounded on $W\s…
Let X be a complex analytic space. A short analytic arc is a holomorphic map of the closed unit disc to X such that only the origin is mapped to a singular point. In contrast with the space of formal arcs studied by Nash, the moduli space of short analytic arcs usually has infinitely many connected components. We describe these for surface singularities, in terms of certain conjugacy classes of the fundamental group of the link. For quotient singularities (in any dimension), this gives a concrete realization of the McKay correspondence. Our results also give new connections between a surface c…
We study the sheaf of locally square integrable holomorphic section of vector bundle with semi-positive curved singular Hermitian metric. We confirm the coherence when its induced determinant metric has analytic singularities.
Singular holomorphic Morse inequalities on non-compact manifolds
本文研究了非紧复流形上带代数奇异性度量的线丛同调维数渐近估计，提出统一方法建立多种流形的奇异全纯Morse不等式，并推广了Berndtsson估计及更一般奇异性情形。
We study properties of a Green function G_A with singularities along a complex subspace A of a complex manifold X. It is defined as the largest negative plurisubharmonic function u satisfying locally u\leq \log|ψ|+C, where ψ=(ψ_1, ...,ψ_m), ψ_1, ...,ψ_m are local generators for the ideal sheaf I_A of A, and C is a constant depending on the function u and the generators. A motivation for this study is to estimate global bounded functions from the sheaf I_A and thus proving a ``Schwarz Lemma'' for I_A.
We are interested in characterizing the holonomy maps associated to integral curves of non-degenerate singularities of holomorphic vector fields. Such a description is well-known in dimension 2 where is a key ingredient in the study of reduced singularities. The most intricate case in the 2 dimensional setting corresponds to (Siegel) saddle singularities. This work treats the analogous problem for saddles in higher dimension. We show that any germ of holomorphic biholomorphism, in any dimension, can be obtained as the holonomy map associated to an integral curve of a saddle singularity. A natu…
The Schwartz index and the residue of logarithmic foliations along a hypersurface with isolated singularities
本文证明了对数叶状结构沿孤立奇点超曲面的Baum-Bott型公式，用Schwartz指数表达留数，并推广了Poincaré问题的障碍结果至任意偶数维射影空间。
We prove a result on removing singularities of almost complex structures pulled back by a non-diffeomorphic map. As an application we prove the existence of global J-holomorphic discs with boundaries attached to real tori.
Let f and g be holomorphic function-germs vanishing at the origin of a complex analytic germ of dimension three. Suppose that they have no common irreducible component and that the real analytic map-germ given by the multiplication of f by the conjugate of g has an isolated critical value at 0. We give necessary and sufficient conditions for the real analytic map-germ to have a Milnor fibration and we prove that in this case the boundary of its Milnor fibre is a Waldhausen manifold. As an intermediate milestone we describe geometrically the Milnor fibre of map-germs given by the multiplication…
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