73 research outputs found
On the removable singularities for meromorphic mappings
If is a closed subset of locally finite Hausdorff -measure on an -dimensional complex manifold and all the points of are nonremovable for a meromorphic mapping of into a compact Kähler manifold, then is a pure -dimensional complex analytic subset of
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
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
Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for H\"olderian almost complex structures
If is an almost complex manifold, with an almost complex structure of
class \CC^\alpha, for some , for every point and every
tangent vector at , there exists a germ of -holomorphic disc through
with this prescribed tangent vector. This existence result goes back to
Nijenhuis-Woolf. All the 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 is enough.
Here we show the following:
Theorem. There exists an almost complex structure 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
Remarks on the rank properties of formal CR maps
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
Residue currents associated with weakly holomorphic functions
We construct Coleff-Herrera products and Bochner-Martinelli type residue
currents associated with a tuple of weakly holomorphic functions, and show
that these currents satisfy basic properties from the (strongly) holomorphic
case, as the transformation law, the Poincar\'e-Lelong formula and the
equivalence of the Coleff-Herrera product and the Bochner-Martinelli type
residue current associated with when defines a complete intersection.Comment: 28 pages. Updated with some corrections from the revision process. In
particular, corrected and clarified some things in Section 5 and 6 regarding
products of weakly holomorphic functions and currents, and the definition of
the Bochner-Martinelli type current
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