The density property for a Stein manifold X implies that the group of holomorphic diffeomorphisms of X is infinite-dimensional and, in a certain well-defined sense, as large as possible.
We prove that if G is a complex semisimple Lie group of adjoint type and K is a reductive subgroup, then G/K has the density property. This theorem is a non-trivial extension of an earlier result of ours, which handles the case of complex semi-simple Lie groups. We also establish the density property for some other complex homogeneous spaces by ad hoc methods. Finally, we introduce a lifting method that extends many results on complex manifolds with the density property to covering spaces of such manifolds.
We consider stable manifolds of a holomorphic diffeomorphism of a complex manifold. Using a conjugation of the dynamics to a (non-stationary) polynomial normal form, we show that typical stable manifolds are biholomorphic to complex Euclidean space.
We prove the existence (and give a characterization) of a germ of complex analytic set left invariant by an abelian group of germs of holomorphic diffeomorphisms at a common fixed point.We also give condition that ensure that such a group can be linearized holomorphically near the fixed point. It rests on a "small divisors condition" of the family of linear parts. The second part of this article is devoted to the study families of totally real intersecting n-submanifolds of (C n , 0). We give some conditions which allow to straighten holomorphically the family. If this is not possible to do it…
Let f_1,...,f_N be commuting germs of holomorphic diffeomorphisms in C fixing the origin with irrational rationally independent rotation numbers alpha_1,...,alpha_N. We adapt Yoccoz' renormalization of germs to this setting to show that a Brjuno-type condition on simultaneous Diophantine approximability of the rotation numbers is sufficient for simultaneous linearizability of f_1,...,f_N. This generalizes a result of Moser's. In the absence of periodic orbits we show that a weaker arithmetic condition analogous to that of Perez-Marco's for the case of a single germ is sufficent for linearizabi…
We investigate the local dynamics of antiholomorphic diffeomorphisms around a parabolic fixed point. We first give a normal form. Then we give a complete classification including a modulus space for antiholomorphic germs with a parabolic fixed point under analytic conjugacy. We then study some geometric applications: existence of real analytic invariant curve, existence of holomorphic and antiholomorphic roots of holomorphic and antiholomorphic parabolic germs, commuting holomorphic and antiholomorphic parabolic germs.
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.…
In this paper, we generalize the Kobayashi pseudo-distance to complex manifolds which admit holomorphic bracket generating distributions. The generalization is based on Chow's theorem in sub-Riemannian geometry. Let G be a linear semisimple Lie group. For a complex $G$-homogeneous manifold M with a G-invariant holomorphic bracket generating distribution D, we prove that (M,D) is Kobayashi hyperbolic if and only if the universal covering of M is a canonical flag domain and the induced distribution is the superhorizontal distribution.
The purpose of this paper is to organize some results on the local geometry of CR singular real-analytic manifolds that are images of CR manifolds via a CR map that is a diffeomorphism onto its image. We find a necessary (sufficient in dimension 2) condition for the diffeomorphism to extend to a finite holomorphic map. The multiplicity of this map is a biholomorphic invariant that is precisely the Moser invariant of the image when it is a Bishop surface with vanishing Bishop invariant. In higher dimensions, we study Levi-flat CR singular images and we prove that the set of CR singular points m…
We prove that a topological homeomorphism conjugating two generic 1-parameter unfoldings of 1-variable complex analytic resonant diffeomorphisms is holomorphic or anti-holomorphic by restriction to the unperturbed parameter. We provide examples that show that the genericity hypothesis is necessary. Moreover we characterize the possible behavior of conjugacies for the unperturbed parameter in the general case. In particular they are always real analytic outside of the origin. We describe the structure of the limits of orbits when we approach the unperturbed parameter. The proof of the rigidity …
We study groups of formal diffeomorphisms in several complex variables. For abelian, metabelian or nilpotent groups we investigate the existence of suitable formal vector fields and closed differential forms which exhibit an invariance property under the group action. Our results are applicable in the construction of suitable integrating factors for holomorphic foliations with singularities. We believe they are a starting point in the study of the connection between Liouvillian integration and transverse structures of holomorphic foliations with singularities in the case of arbitrary codimensi…
For a representation of a finite group $G$ on a complex vector space $V$ we determine when a holomorphic $\binom{p}{q}$-tensor field on the principle stratum of the orbit space $V/G$ can be lifted to a holomorphic $G$-invariant tensor field on $V$. This extends also to connections. As a consequence we determine those holomorphic diffeomorphisms on $V/G$ which can be lifted to orbit preserving holomorphic diffeomorphisms on $V$. This in turn is applied to characterize complex orbifolds.
Let $M$ be a two dimensional complex manifold, $p \in M $ and \Fl a germ of holomorphic foliation of \M at $p$. Let $S\subset M$ be a germ of an irreducible, possibly singular, curve at $p$ in $M$ which is a separatrix for \Fl. We prove that if the Camacho-Sad-Suwa index $\id(\F,S,p)\not \in \Q^+\cup \{0\} $ then there exists another separatrix for \Fl at $p$. A similar result is proved for the existence of parabolic curves for germs of holomorphic diffeomorphisms near a curve of fixed points.
We study groups of formal or germs of analytic diffeomorphisms in several complex variables. Such groups are related to the study of the transverse structure and dynamics of Holomorphic foliations, via the notion of holonomy group of a leaf of a foliation. For dimension one, there is a well-established dictionary relating analytic/formal classification of the group, with its algebraic properties (finiteness, commutativity, solvability, ...). Such system of equivalences also characterizes the existence of suitable {\it integrating factors}, i.e., invariant vector fields and one-forms associated…
We study germs of singular holomorphic vector fields at the origin of $\Bbb C^n$ of which the linear part is 1-resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some "sectorial domains" which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms. We show that the Gevrey ordrer of the latter is linked to the diophantine type of th…
The group $Diff$ of diffeomorphisms of the circle is an infinite dimensional analog of the real semisimple Lie groups $U(p,q)$, $Sp(2n,R)$, $SO^*(2n)$; the space $Ξ$ of univalent functions is an analog of the corresponding classical complex Cartan domains. We present explicit formulas for realizations of highest weight representations of $Diff$ in the space of holomorphic functionals on $Ξ$, reproducing kernels on $Ξ$ determining inner products, and expressions ('canonical cocycles') replacing spherical functions.
Let $f$ be a germ of holomorphic diffeomorphism of $\C^n$ fixing the origin $O$, with $df_O$ diagonalizable. We prove that, under certain arithmetic conditions on the eigenvalues of $df_O$ and some restrictions on the resonances, $f$ is locally holomorphically linearizable if and only if there exists a particular $f$-invariant complex manifold. Most of the classical linearization results can be obtained as corollaries of our result.
A manifold $M$ is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. Let $M$ be an LCK manifold admitting a holomorphic conformal flow of diffeomorphisms, lifted to a non-isometric homothetic flow on its covering. We show that $M$ admits an automorphic potential. This version was added 10 years after publication to correct some errors in the original.
We give a complete proof of a propagation theorem of multiplicity-free property from fibers to spaces of global sections for holomorphic vector bundles. The propagation theorem is formalised in three ways, aiming for producing various multiplicity-free theorems in representation theory for both finite and infinite dimensional cases in a systematic and synthetic manner. The key geometric condition in our theorem is an orbit-preserving anti-holomorphic diffeomorphism on the base space, which brings us to the concept of visible actions on complex manifolds.
We prove that planar homeomorphisms can be approximated by diffeomorphisms in the Sobolev space $W^{1,2}$ and in the Royden algebra. As an application, we show that every discrete and open planar mapping with a holomorphic Hopf differential is harmonic.
In this paper we study the dynamics of germs of holomorphic diffeomorphisms of $(\mathbb{C}^{n},0)$ with a fixed point at the origin with exactly one neutral eigenvalue. We prove that the map on any local center manifold of $0$ is quasiconformally conjugate to a holomorphic map and use this to transport results from one complex dimension to higher dimensions.
In this paper we introduce (weakly) root graded Banach--Lie algebras and corresponding Lie groups as natural generalizations of group like $\GL_n(A)$ for a Banach algebra $A$ or groups like $C(X,K)$ of continuous maps of a compact space $X$ into a complex semisimple Lie group $K$. We study holomorphic induction from holomorphic Banach representations of so-called parabolic subgroups $P$ to representations of $G$ on holomorphic sections of homogeneous vector bundles over $G/P$. One of our main results is an algebraic characterization of the space of sections which is used to show that this spac…
Let $G$ be a real semisimple Lie group, $K$ its maximal complex subgroup, and $G_C$ its complexification. It is known that all the $K$-finite matrix elements on $G$ admit holomorphic continuation to branching functions on $G_C$ having singularities at the a prescribed divisor. We propose a geometric explanation of this phenomenon. The note also contsins a general survey of holomorphic continuations of infinite-dimensional representations.
Let $f$ be a harmonic map from a Riemann surface to a Riemannian $n$-manifold. We prove that if there is a holomorphic diffeomorphism $h$ between open subsets of the surface such that $f\circ h = f$, then $f$ factors through a holomorphic map onto another Riemann surface. If such $h$ is anti-holomorphic, we obtain an analogous statement. For minimal maps, this result is well known and is a consequence of the theory of branched immersions of surfaces due to Gulliver-Osserman-Royden. Our proof relies on various geometric properties of the Hopf differential.
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 …
Let $F\in\mathrm{Diff}(\mathbb{C}^2,0)$ be a germ of a holomorphic diffeomorphism and let $Γ$ be an invariant formal curve of $F$. Assume that the restricted diffeomorphism $F|_Γ$ is either hyperbolic attracting or rationally neutral non-periodic (these are the conditions that the diffeomorphism $F|_Γ$ should satisfy, if $Γ$ were convergent, in order to have orbits converging to the origin). Then we prove that $F$ has finitely many stable manifolds, either open domains or parabolic curves, consisting of and containing all converging orbits asymptotic to $Γ$. Our results generalize to the case …
We consider a commutative family of holomorphic vector fields in an neighbourhood of a common singular point, say $0\in \Bbb C^n$. Let $\lie g$ be a commutative complex Lie algebra of dimension $l$. Let $λ_1,...,λ_n\in \lie g^*$ and let us set $S(g)=\sum_{i=1}^nλ_i(g)x_i\frac{\partial}{\partial x_i}$. We assume that this Lie morphism is {\bf diophantine} in the sense that a diophantine condition $(ω(S))$ is satisfied. Let $X_1$ be a holomorphic vector field in a neighbourhood of $0\in \Bbb C^n$. We assume that its linear part $s$ is regular relatively to $S$, that is belongs to $S(\lie g)$ and…
Let (M,J) be an almost complex manifold. We show that the infinite-dimensional space Tau of totally real submanifolds in M carries a natural connection. This induces a canonical notion of geodesics in Tau and a corresponding definition of when a functional, defined on Tau, is convex. Geodesics in Tau can be expressed in terms of families of J-holomorphic curves in M; we prove a uniqueness result and study their existence. When M is Kähler we define a canonical functional on Tau; it is convex if M has non-positive Ricci curvature. Our construction is formally analogous to the notion of geodesic…
If G_0 is a real form of a complex semisimple Lie group G and Z is compact G-homogeneous projective algebraic manifold, then G_0 has only finitely many orbits on Z. Complex analytic properties of open G_0-orbits D (flag domains) are studied. Schubert incidence-geometry is used to prove the Kobayashi hyperbolicity of certain cycle space components C_q(D). Using the hyperbolicity of C_q(D) and analyzing the action of Aut(D) on it, an exact description of Aut(D) is given. It is shown that, except in the easily understood case where D is holomorphically convex with a nontrivial Remmert reduction, …
The goal of this paper is to study the dynamics of holomorphic diffeomorphisms in C^n such that the resonances among the first 1<= r<= n eigenvalues of the differential are generated over N by a finite number of Q-linearly independent multi-indices (and more resonances are allowed for other eigenvalues). We give sharp conditions for the existence of basins of attraction where a Fatou coordinate can be defined. Furthermore, we obtain a generalization of the Leau-Fatou flower theorem, providing a complete description of the dynamics in a full neighborhood of the origin for 1-resonant parabolical…
The topology of Kähler manifolds and 1-forms without zeros
Università degli Studi di Bari Aldo Moro（巴里阿尔多·莫罗大学）；Princeton University（普林斯顿大学）；Leibniz Universität Hannover（汉诺威莱布尼茨大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
Neretin and Segal independently defined a semigroup of annuli with boundary parametrizations, which is viewed as a complexification of the group of diffeomorphisms of the circle. By extending the parametrizations to quasisymmetries, we show that this semigroup is a quotient of the Teichmueller space of doubly-connected Riemann surfaces by a Z action. Furthermore, the semigroup can be given a complex structure in two distinct, natural ways. We show that these two complex structures are equivalent, and furthermore that multiplication is holomorphic. Finally, we show that the class of quasiconfor…
A real semisimple Lie group G_0 embedded in its complexification G has only finitely many orbits in any G-fag manifold Z = G/Q. The complex geometry of its open orbits D (flag domains) is studied from the point of view of compact complex submanifolds C (cycles) which arise as orbits of certain distinguished subgroups. Normal bundles E of the cycles are analyzed in some detail. It is shown that E is trivial if and only if D is holomorphically convex, in fact a product of C and a Hermitian symmetric space, and otherwise D is pseudoconcave.
The basic setup consists of a complex flag manifold $Z=G/Q$ where $G$ is a complex semisimple Lie group and $Q$ is a parabolic subgroup, an open orbit $D = G_0(z) \subset Z$ where $G_0$ is a real form of $G$, and a $G_0$--homogeneous holomorphic vector bundle $\mathbb E \to D$. The topic here is the double fibration transform ${\cal P}: H^q(D;{\cal O}(\mathbb E)) \to H^0({\cal M}_D;{\cal O}(\mathbb E'))$ where $q$ is given by the geometry of $D$, ${\cal M}_D$ is the cycle space of $D$, and $\mathbb E' \to {\cal M}_D$ is a certain naturally derived holomorphic vector bundle. Schubert intersecti…
The first part of this article is devoted to the study families of totally real intersecting $n$-submanifolds of $(\Bbb C^n,0)$. We give some conditions which allow to straighten holomorphically the family. If this is not possible to do it formally, we construct a germ of complex analytic set at the origin which interesection with the family can be holomorphically staightened. The second part is devoted to the study real analytic $(n+r)$-submanifolds of $(\Bbb C^n,0)$ having a CR-singularity at the origin ($r$ is a nonnegative integer). We consider deformations of quadrics and we define genera…
