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Some applications of metric currents to complex analysis
Get PDFThe aim of this paper is to extend the theory of metric currents, developed by Ambrosio and Kirchheim, to complex spaces. We define the bidimension of a metric current on a complex space and we discuss the Cauchy-Riemann equation on a particular class of singular spaces. As another application, we investigate the Cauchy-Riemann equation on complex Banach spaces, by means of a homotopy formula. © 2012 Springer-Verlag
Weakly complete domains in Grauert-type surfaces
Get PDFThe aim of this short note is to investigate the geometry of weakly complete subdomains of Grauert-type surfaces, i.e., open connected sets D, sitting inside a Grauert-type surface X, which admit a smooth plurisubharmonic exhaustion function. We prove that they are either modifications of Stein spaces or Grauert-type surfaces themselves, and we apply these results to the special case of Hopf surfaces
Oka principle for Levi flat manifolds
Get PDFThe name of Oka principle, or Oka–Grauert principle, is traditionally used to refer to the holomorphic incarnation of the homotopy principle: on a Stein space, every problem that can be solved in the continuous category, can be solved in the holomorphic category as well. In this note, we begin the study of the same kind of questions on a Levi-flat manifold; more precisely, we try to obtain a classification of CR-bundles on a semiholomorphic foliation of type (n, 1). Our investigation should only be considered a preliminary exploration, as it deals only with some particular cases, either in terms of regularity or bidegree of the bundle, and partial results
Generalized Levi Currents and Singular Loci for Families of Plurisubharmonic Functions
No full textWe show how the formalism of Levi currents on complex manifolds, as introduced by Sibony, can be used to study the analytic structure of singular sets associated to families of plurisubharmonic functions, in the sense of Slodkowski
Geometry - Transversally pseudoconvex semiholomorphic foliations
No full textA semiholomorphic foliations of type (n, d) is a differentiable real manifold X of dimension 2n + d, foliated by complex leaves of complex dimension n. The aim of the present note is to outline some results obtained in studying such spaces along the lines of the classical theory of complex spaces. Complete proofs will appear elsewhere
The Cauchy transform in the slice hyperholomorphic setting and related topics
Get PDFIn this paper we study the additive splitting associated to the quaternionic Cauchy transform defined by the Cauchy formula of slice hyperholomorphic functions. Moreover, we introduce and study the analogue of the fundamental solution of the global operator of slice hyperholomorphic functions. We state our results in the quaternionic setting but several results hold for Clifford algebra-valued function with minor changes in the proofs
Positive metric currents and holomorphic chains in Hilbert spaces
Get PDFWe present some results concerning currents of integration on finite-dimensional analytic spaces in Hilbert spaces, using the setting of metric currents. In particular, we obtain the characterization of such currents as positive closed (k, k)-integer rectifiable currents and solve the boundary problem for holomorphic chains
On weakly complete surfaces [Sur les surfaces faiblement complètes]
No full textA weakly complete space is a (connected) complex space endowed with a (smooth) plurisubharmonic exhaustion function. In this paper, we classify the weakly complete surfaces (i.e. weakly complete manifolds of dimension 2) for which such exhaustion function can be chosen to be real analytic: they can be modifications of Stein spaces or proper (i.e. endowed with a proper surjective holomorphic map onto) a non-compact (possibly singular) complex curve or surfaces of Grauert type i.e. foliated with real analytic Levi flat hypersurfaces whose Levi foliation has dense complex leaves. In the last case, we also show that such Levi flat hypersurfaces are in fact level sets of a global proper pluriharmonic function, up to passing to a holomorphic double covering.Un espace complexe est dit faiblement
complet s'il est muni d'une fonction d'exhaustion plurisousharmonique. Dans ce
papier on classie les surfaces complexes faiblement complètes qui admettent une
fonction d'exhaustion plurisousharmonique et analytique réelle. Elles sont des types
suivants : modications des espaces de Stein, surfaces complexes propres sur des
courbes complexes non compactes, ou bien surfaces complexes de type Grauert
i.e. feuilletées par des hypersurfaces Levi plates dont les feuilles du feuilletage de
Levi sont partout denses. Dans ce dernier cas on montre aussi que, sauf à passer à
un double revêtement, les hypersurfaces Levi plates sont en fait les niveaux d'une
fonction pluriharmonique globale
Non compact boundaries of complex analytic varieties in Hilbert spaces
Get PDFWe treat the boundary problem for complex varieties with isolated
singularities, of complex dimension greater than or equal to 3, non necessarily
compact, which are contained in strongly convex, open subsets of
a complex Hilbert space H. We deal with the problem by cutting with a
family of complex hyperplanes and applying the already known result for
the compact case
Weakly complete complex surfaces
Get PDFA weakly complete space is a complex space that admits a (smooth) plurisubharmonic exhaustion function. In this paper, we classify those weakly complete complex surfaces for which such an exhaustion function can be chosen to be real analytic: they can be modifications of Stein spaces or proper over a non-compact (possibly singular) complex curve, or foliated with real-analytic Levi flat hypersurfaces which in turn are foliated by dense complex leaves (these we call "surfaces of Grauert type"). In the last case, we also show that such Levi flat hypersurfaces are in fact level sets of a global proper pluriharmonic function, up to passing to a holomorphic double cover of the space. Our method of proof is based on the careful analysis of the level sets of the given exhaustion function and their intersections with the \emph{minimal singular set}, that is, the set where every plurisubharmonic exhaustion function has a degenerate Levi form
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