M. S. Baouendi, Peter Ebenfelt, Linda P. Rothschild
In this paper, we consider holomorphic mappings between real hypersurfaces in different dimensional complex spaces. We give a number of conditions that imply that such mappings are transversal to the target hypersurface at most points.
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We prove analogs of Thom's transversality theorem and Whitney's theorem on immersions for pseudo-holomorphic discs. We also prove that pseudo-holomorphic discs form a manifold.
We show that a holomorphic mapping sending one generic submanifold into another of the same dimension is CR transversal to the target submanifold provided that the source manifold is of finite type and the map is of generic full rank. This result and its corollaries completely resolve two questions posed by Linda P. Rothschild and the first author in a paper from 2006.
Let $M_\ell$ be a smooth Levi-nondegenerate hypersurface of signature $\ell$ in $\mathbf C^n$ with $ n\ge 3$, and write $H_\ell^N$ for the standard hyperquadric of the same signature in $\mathbf C^N$ with $N-n< \frac{n-1}{2}$. Let $F$ be a holomorphic map sending $M_\ell$ into $H_\ell^N$. Assume $F$ does not send a neighborhood of $M_\ell$ in $\mathbf C^n$ into $H_\ell^N$. We show that $F$ is necessarily CR transversal to $M_\ell$ at any point. Equivalently, we show that $F$ is a local CR embedding from $M_\ell$ into $H_\ell^N$.
In this paper, we establish general stratawise higher jet evaluation transversality of $J$-holomorphic curves for a generic choice of almost complex structures $J$ tame to a given symplectic manifold $(M,ω)$. Using this transversality result, we prove that there exists a subset $\CJ_ω^{ram} \subset \CJ_ω$ of second category such that for every $J \in \CJ_ω^{ram}$, the dimension of the moduli space of (somewhere injective) $J$-holomorphic curves with a given ramification profile goes down by $2n$ or $2(n-1)$ depending on whether the ramification degree goes up by one or a new ramification point…
We prove here new results about transversality and related geometric properties of a holomorphic, formal, or CR mapping, sending one generic submanifold of $\bC^N$ into another. One of our main results is that a finite mapping is transversal to the target manifold provided this manifold is of finite type. For the case of hypersurfaces, transversality in this context was proved by Baouendi and the second author in 1990. The general case of generic manifolds of higher codimension, which we treat in this paper, had remained an open problem since then. Applications of this result include a suffici…
In this paper, we examine holomorphic Segre preserving maps between the complexifications of real hypersurfaces in $\mathbb{C}^{n+1}$. In particular, we find several sufficient conditions ensuring that Segre transversality and total Segre nondegeneracy of the maps must hold.
We study the classification of singularities of holomorphic foliations and non-integrable one-forms under the hypothesis of transversality with real hypersurfaces.
We study a natural generalization of transversally intersecting smooth hypersurfaces in a complex manifold: hypersurfaces, whose components intersect in a transversal way but may be themselves singular. Such hypersurfaces will be called splayed divisors. A splayed divisor is characterized by a property of its Jacobian ideal. This yields an effective test for splayedness. Two further characterizations of a splayed divisor are shown, one reflecting the geometry of the intersection of its components, the other one using K. Saito's logarithmic derivations. As an application we prove that a union o…
We consider holomorphic mappings $H$ between a smooth real hypersurface $M\subset \bC^{n+1}$ and another $M'\subset \bC^{N+1}$ with $N\geq n$. We provide conditions guaranteeing that $H$ is transversal to $M'$ along all of $M$. In the strictly pseudoconvex case, this is well known and follows from the classical Hopf boundary lemma. In the equidimensional case ($N=n$), transversality holds for maps of full generic rank provided that the source is of finite type in view of recent results by the authors (see also a previous paper by the first author and L. Rothschild). In the positive codimension…
We obtain an explicit parametrization of stationary discs glued to some Levi non-degenerate hypersurfaces. These discs form a family which is invariant under the action of biholomorphisms. We use this parametrization to construct a local circular representation of these hypersurfaces. As a corollary, we get the uniqueness of biholomorphisms with given 1-jet at some convenient point.
We prove a sharp lower bound for the Tanaka-Webster holomorphic sectional curvature of strictly pseudoconvex real hypersurfaces that are "semi-isometrically" immersed in a Kähler manifold of nonnegative holomorphic sectional curvature under an appropriate convexity condition. This gives a partial answer to a question posed by Chanillo, Chiu, and Yang regarding the positivity of the Tanaka-Webster scalar curvature of the boundary of a strictly convex domain in $\mathbb{C}^2$ from 2012. In fact, the main result proves a stronger positivity property, namely the $\frac12$-positivity in the sense o…
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 give a complete description and classification of locally homogeneous real hypersurfaces in $\mathbb C^3$. Various groups of mathematicians have been studying this problem in the last 25 years, and several significant classes of hypersurfaces under consideration have been studied and classified. The final results in the classification problem presented in this paper are obtained by using the classification of abstract 5-dimensional real Lie algebras, and by studying their representations by algebras of holomorphic vector fields in complex 3-space. The complete list of pairwise inequivalent …
Inspired by an article of R. Bryant on holomorphic immersions of unit disks into Lorentzian CR manifolds, we discuss the application of Cartan's method to the question of the existence of bi-disk $\mathbb{D}^{2}$ in a smooth $9$-dimensional real analytic real hypersurface $M^{9}\subset\mathbb{C}^{5}$ with Levi signature $(2,2)$ passing through a fixed point. The result is that the lift to $M^{9}\times U(2)$ of the image of the bi-disk in $M^{9}$ must lie in the zero set of two complex-valued functions in $M^{9}\times U(2)$. We then provide an example where one of the functions does not identic…
It is proved that the germ of a holomorphic map from a real analytic hypersurface M in C^n into a strictly pseudoconvex compact real algebraic hypersurface M' in C^N, 1 < n < N extends holomorphically along any path on M.
In this paper, we establish a second main theorem for holomorphic curve intersecting hypersurfaces in general position in projective space with level of truncation. As an application, we reduce the number hypersurfaces in uniqueness problem for holomorphic curve of authors before.
We investigate germs of real analytic Levi-flat hypersurfaces tangent to germs of codimension one holomorphic webs. We introduce the notion of first integrals for local webs. In particular, we prove that a $k$-web with finitely many invariant analytic subvarieties through the origin tangent to a Levi-flat hypersurface has a holomorphic first integral.
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.
In this paper, we prove a Second Main Theorem for holomorphic mappings in a disk whose image intersects some families of nonlinear hypersurfaces (totally geodesic hypersurfaces with respect to a meromorphic connection) in the complex projective space $\mathbb{P}^k$. This is a generalization of Cartan's Second Main Theorem. As a consequence, we establish a uniqueness theorem for holomorphic mappings which intersects $O(k^3)$ many totally geodesic hypersurfaces.
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 $\mathbb{P}^2$.
Replaced by a shorter version which appeared in press.
It is shown that the Levi foliation of a real analytic Levi-flat hypersurface extends to a $d$-web near a nondicritical singular point and admits a multiple-valued meromorphic first integral.
We discuss a general result of holomorphic extension of a real analytic function $f$ defined on the boundary $\partial D$ of a real analytic strictly convex subset $D\subset\subset \C^n$. We show that this follows from the hypothesis of separate holomorphic extension along stationary/extremal discs.
Yan and Chen proved a weak Cartan-type second main theorem for holomorphic curves meeting hypersurfaces in projective space that included truncated counting functions. Here we give an explicit estimate for the level of truncation.
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…
On real-analytic Levi-flat hypersurfaces and holomorphic webs
本文研究紧复曲面上与实解析Levi平坦超曲面相切的全纯网，证明其在特定条件下具有多值亚纯首次积分，并证明由Grassmannian中不可约实解析曲线诱导的Levi叶状结构可延拓为复射影空间上的代数网。
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.
Let $M\subset \mathbb C^n$ be a real analytic hypersurface, $M'\subset \mathbb C^N$ $(N\geq n)$ be a strongly pseudoconvex real algebraic hypersurface of the special form and $F$ be a meromorphic mapping in a neighborhood of a point $p\in M$ which is holomorphic in one side of $M$. Assuming some additional conditions for the mapping $F$ on the hypersurface $M$, we proved that $F$ has a holomorphic extension to $p$. This result may be used to show the regularity of CR mappings between real hypersurfaces of different dimensions.
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.
The floating body approach to affine surface area is adapted to a holomorphic context providing an alternate approach to Fefferman's invariant hypersurface measure.
Defect relation for holomorphic maps from complex discs into projective varieties and hypersurfaces
本文针对复圆盘上有限增长指数的全纯映射，建立了与射影簇中任意超曲面族相交的第二主要定理，给出截断亏量之和的上界，并推广改进了先前相关结果。
We prove several new transversality results for formal CR maps between formal real hypersurfaces in complex space. Both cases of finite and infinite type hypersurfaces are tackled in this note.
We give in \mathbb{R}^6 a real analytic almost complex structure J, a real analytic hypersurface M and a vector v in the Levi null set at 0 of M, such that there is no germ of J-holomorphic disc f included in M with f(0)=0 and \frac{\partial f}{\partial x}(0)=v, although the Levi form of M has constant rank. Then for any hypersurface M and any complex structure J, we give necessary conditions under which there exists such a germ of disc.
We study the non-embddability property for a class of real hypersurfaces, called real hypersurfaces of involution type, into the sphere in the low codimensional case, by making use of property of a naturally related Gauss curvature. We also study rigidity problems for conformal maps between a class of Kähler manifolds with pseud-conformally flat metrics.
Holomorphic maps from complex discs intersecting families of hypersurfaces
本文针对射影簇中与超曲面族相交的复圆盘上具有有限增长指数的全纯映射建立了第二主定理，并详细估计了小项，推广并扩展了以往关于从$\mathbb C$出发的全纯映射的第二主定理。
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 bundles framed along a real hypersurface and the Riemann-Hilbert problem
本文研究了沿实超曲面标架的全纯丛的模空间，证明了其到带边形式全纯丛模空间纤维积的自然映射在特定 Hölder 正则性下为同胚，并基于粘合原理推广至任意定向超曲面与复 Lie 群主丛，给出 Riemann-Hilbert 问题解空间的复几何解释。
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.
Let $D$ be a strictly pseudoconvex domain and $X$ be a singular analytic set of pure dimension $n-1$ in $C^n$ such that $X\cap D\neq \emptyset$ and $X\cap bD$ is transverse. We give sufficient conditions for a function holomorphic on $D\cap X$ to admit a holomorphic extension which belongs to $L^q(D),$ $q\in [1,+\infty[$, or to $BMO(D)$. The extension is given by mean of integral representation formulas and residue currents.
We study the analytic continuation problem for a germ of a biholomorphic mapping from a non-minimal real hypersurface $M\subset\CC{n}$ into a real hyperquadric $\mathcal Q\subset\CP{n}$ and prove that under certain non-degeneracy conditions any such germ extends locally biholomorphically along any path lying in the complement $U\setminus X$ of the complex hypersurface $X$ contained in $M$ for an appropriate neighborhood $U\supset X$. Using the monodromy representation for the multiple-valued mapping obtained by the analytic continuation we establish a connection between nonminimal real hypersu…
The purpose of this paper is to investigate the geometric properties of real hypersurfaces of D'Angelo infinite type in ${\mathbb C}^n$. In order to understand the situation of flatness of these hypersurfaces, it is natural to ask whether there exists a nonconstant holomorphic curve tangent to a given hypersurface to infinite order. A sufficient condition for this existence is given by using Newton polyhedra,which is an important concept in singularity theory. More precisely,equivalence conditions are given in the case of some model hypersurfaces.
We study regularity properties of CR maps in positive codimension valued in pseudoconvex manifolds which carry a nontrivial Levi foliation. We introduce an invariant which can be used to deduce that any sufficiently regular CR map from a minimal manifold into such a foliated target is either generically smooth or geometrically highly constrained, and to show generic smoothness of sufficiently regular CR transversal CR maps between pseudoconvex hypersurfaces. As an application, we discuss boundary regularity of proper holomorphic maps into bounded symmetric domains.
Holomorphic foliations tangent to Rolle-pfaffian hypersurfaces
本文研究复平面原点处满足Rolle-Khovanskii条件的与Pfaffian超曲面相切的全纯叶状结构芽，证明其由闭亚纯1-形式定义，并完成奇点消解中简单模型的分类。
In this paper, we prove some fundamental theorems for holomorphic curves on angular domain intersecting a hypersurface, finite set of fixed hyperplanes in general position and finite set of fixed hypersurfaces in general position on complex projective variety with the level of truncation. As applications of the second main theorems for an angle, we will discuss the uniqueness problem of holomorphic curves in an angle instead of the whole complex plane. Detail, we establish a result for uniqueness problem of holomorphic curve by inverse image of a hypersurface. In my knowledge, this is the firs…
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.
Almost holomorphic curves in real analytic hypersurfaces
本文利用外微分系统理论研究实解析近复流形的超曲面中伪全纯圆盘芽的存在性，并指出不可积情形因缺少高维 J 不变对象而与可积情形本质不同。
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.
A general class of singular real hypersurfaces, called subanalytic, is defined. For a subanalytic hypersurface M in C^n, Cauchy-Riemann (or simply CR) functions on M are defined, and certain properties of CR functions discussed. In particular, sufficient geometric conditions are given for a point p on a subanalytic hypersurface M to admit a germ at p of a smooth CR function f that cannot be holomorphically extended to either side of M. As a consequence it is shown that a well-known condition of the absence of complex hypersurfaces contained in a smooth real hypersurface M, which guarantees one…
The Lewy extension theorem asserts the holomorphic extendability of CR functions defined in a neighborhood of a point on a hypersurface in C^{n+1}. The edge-of-the-wedge theorem asserts the extendability of holomorphic functions defined in wedges in C^{n+1} with edge a maximally real submanifold. In this article we prove under suitable hypotheses the holomorphic extendability to an open set in C^{n+1} of CR functions defined in the intersection of a hypersurface with a wedge whose edge is contained in the hypersurface. Unlike the situation for the classical edge-of-the-wedge theorem, for this …
We prove that if $M$ and $M'$ are algebraic hypersurfaces in $ C^ N$, i.e. both defined by the vanishing of real polynomials, then any sufficiently smooth CR mapping with Jacobian not identically zero extends holomorphically provided the hypersurfaces are holomorphically nondegenerate . Conversely, we prove that holomorphic nondegeneracy is necessary for this property of CR mappings to hold. For the case of unequal dimensions, we also prove that if $M$ is an algebraic hypersurface in $ C^N$ which does not contain any complex variety of positive codimension and $M'$ is the sphere in $ C^{N+1 }$…
It is shown that if a proper holomorphic map $f: \mathbb C^n \to \mathbb C^N$, $1<n\le N$, sends a pseudoconvex real analytic hypersurface of finite type into another such hypersurface, then any $n-1$ dimensional component of the critical locus of $f$ intersects both sides of $M$. We apply this result to the problem of boundary regularity of proper holomorphic mappings between bounded domains in $\mathbb C^n$.
Let $Q^N_l\subset \bC\bP^{N+1}$ denote the standard real, nondegenerate hyperquadric of signature $l$ and $M\subset \bC^{n+1}$ a real, Levi nondegenerate hypersurface of the same signature $l$. We shall assume that there is a holomorphic mapping $H_0\colon U\to \bC\bP^{N_0+1}$, where $U$ is some neighborhood of $M$ in $\bC^{n+1}$, such that $H_0(M)\subset Q^{N_0}_l$ but $H(U)\not\subset Q^{N_0}_l$. We show that if $N_0-n<l$ then, for any $N\geq N_0$, any holomorphic mapping $H\colon U\to \bC\bP^{N+1}$ with $H(M)\subset Q^{N}_l$ and $H(U)\not\subset Q^{N_0}_l$ must be the standard linear embedd…
We explore connections between geometric properties of the Levi foliation of a Levi-flat hypersurface M and holomorphic convexity of compact sets in M, or bounded in part by M. Applications include extendability of Cauchy-Riemann functions, solvability of the dibar_b-equation, approximability of holomorphic and CR functions, and global regularity of the dibar-Neumann operator.
A formal invertible equivalence between two minimal real analytic hypersurfaces converges if and only if the hypersurfaces are holomorphically nondegenerate
We study the holomorphic embedding problem from a compact strongly pseudoconvex real algebraic hypersurface into a sphere of higher dimension. We construct a family of compact strongly pseudoconvex hypersurfaces $M_ε$ in $\mathbb{C}^2,$ and prove that for any integer $N$, there is a number $ε(N)$ with $0<ε(N)<1$ such that for any $ε$ with $0<ε<ε(N)$, $M_ε$ can not be locally holomorphically embedded into the unit sphere $\mathbb{S}^{2N-1}$ in $\mathbb{C}^N.$
A unimodular complex surface is a complex 2-manifold X endowed with a holomorphic volume form. A strictly pseudoconvex real hypersurface M in X inherits not only a CR-structure but a canonical coframing as well. In this article, this canonical coframing on M is defined, its invariants are discussed and interpreted geometrically, and its basic properties are studied. A natural evolution equation for strictly pseudoconvex real hypersurfaces in unimodular complex surfaces is defined, some of its properties are discussed, and several examples are computed. The locally homogeneous examples are dete…
Degeneracy of holomorphic mappings into or avoiding Fermat type hypersurfaces
本文证明了进入 Fermat 型超曲面的全纯映射的退化性，强化了 Green 的经典结论，并为 Etesse 的近期结果提供了 Nevanlinna 理论证明。
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.
We construct a family of small analytic discs attached to Levi non-degenerate hypersurfaces in $\mathbb{C}^{n+1}$, which is globally biholomorphically invariant. We then apply this technique to study unique determination problems along Levi non-degenerate hypersurfaces that are merely of class $\mathcal{C}^4$. This method gives 2-jet determination results for germs of biholomorphisms, CR diffeomorphisms, as well as in the almost complex setting.
In this note, we establish the following Second Main Theorem type estimate for every entire non-algebraically degenerate holomorphic curve $f\colon\mathbb{C}\rightarrow\mathbb{P}^n(\mathbb{C})$, in present of a {\sl generic} hypersuface $D\subset\mathbb{P}^n(\mathbb{C})$ of sufficiently high degree $d\geq 15(5n+1)n^n$: \[ T_f(r) \leq \,N_f^{[1]}(r,D) + O\big(\log T_f(r) + \log r \big)\parallel, \] where $T_f(r)$ and $N_f^{[1]}(r,D)$ stand for the order function and the $1$-truncated counting function in Nevanlinna theory. This inequality quantifies recent results on the logarithmic Green--Grif…
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