On the removable singularities for meromorphic mappings

If $E$ is a closed subset of locally finite Hausdorff $(2n-2)$-measure on an $n$-dimensional complex manifold $\Omega$ and all the points of $E$ are nonremovable for a meromorphic mapping of $\Omega \setminus E$ into a compact Kähler manifold, then $E$ is a pure $(n-1)$-dimensional complex analytic subset of $\Omega$

The role of Fourier modes in extension theorems of Hartogs-Chirka type

We generalize Chirka's theorem on the extension of functions holomorphic in a neighbourhood of graph(F)\cup(\partial D\times D) -- where D is the open unit disc and graph(F) denotes the graph of a continuous D-valued function F -- to the bidisc. We extend holomorphic functions by applying the Kontinuitaetssatz to certain continuous families of analytic annuli, which is a procedure suited to configurations not covered by Chirka's theorem.Comment: 17 page

On nonimbeddability of Hartogs figures into complex manifolds

5 pagesWe propose a method to construct examples of strange imbeddings of Hartogs figures into complex manifolds. It gives an imbedding of a "thin" Hartogs figure which does not have any neighborhood biholomorphic to an open set in a Stein manifold, thus unswering a question of E. Poletsky. Then we give an example of a foliated manifold which does not admit any nontrivial imbeddings of a "thick" (i.e. usual) Hartogs figure, giving thus a counterexample to some "selfevident" statements used in foliation theory

Functions holomorphic along holomorphic vector fields

The main result of the paper is the following generalization of Forelli's theorem: Suppose F is a holomorphic vector field with singular point at p, such that F is linearizable at p and the matrix is diagonalizable with the eigenvalues whose ratios are positive reals. Then any function $\phi$ that has an asymptotic Taylor expansion at p and is holomorphic along the complex integral curves of F is holomorphic in a neighborhood of p. We also present an example to show that the requirement for ratios of the eigenvalues to be positive reals is necessary

Tameness of complex dimension in a real analytic set

Given a real analytic set X in a complex manifold and a positive integer d, denote by A(d) the set of points p in X at which there exists a germ of a complex analytic set of dimension d contained in X. It is proved that A(d) is a closed semianalytic subset of X.Comment: Published versio

Enveloppe d'holomorphie locale des vari\'et\'es CR et \'elimination des singularit\'es pour les fonctions CR int\'egrables

Soient $M$ une vari\'et\'e CR localement plongeable et $\Phi\subset M$ un ferm\'e. On donne des conditions suffisantes pour que les fonctions $L_{loc}^1$ qui sont CR sur $M\backslash \Phi$ le soient aussi sur $M$ tout entier.Comment: 6 pages, LaTeX. To appear in C. R. Acad. Sci. Paris, 199

Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for H\"olderian almost complex structures

If $X$ is an almost complex manifold, with an almost complex structure $J$ of class \CC^\alpha, for some $\alpha >0$, for every point $p\in X$ and every tangent vector $V$ at $p$, there exists a germ of $J$-holomorphic disc through $p$ with this prescribed tangent vector. This existence result goes back to Nijenhuis-Woolf. All the $J$ holomorphic curves are of class \CC^{1,\alpha} in this case. Then, exactly as for complex manifolds one can define the Royden-Kobayashi pseudo-norm of tangent vectors. The question arises whether this pseudo-norm is an upper semi-continuous function on the tangent bundle. For complex manifolds it is the crucial point in Royden's proof of the equivalence of the two standard definitions of the Kobayashi pseudo-metric. The upper semi-continuity of the Royden-Kobayashi pseudo-norm has been established by Kruglikov for structures that are smooth enough. In [I-R], it is shown that \CC^{1,\alpha} regularity of $J$ is enough. Here we show the following: Theorem. There exists an almost complex structure $J$ of class \CC^{1\over 2} on the unit bidisc \D^2\subset \C^2, such that the Royden-Kobayashi seudo-norm is not an upper semi-continuous function on the tangent bundle.Comment: 5 page