937 research outputs found
Exhausting domains of the symmetrized bidisc
We show that the symmetrized bidisc may be exhausted by strongly linearly
convex domains. It shows in particular the existence of a strongly linearly
convex domain that cannot be exhausted by domains biholomorphic to convex ones.Comment: 6 page
Obstructions to embeddability into hyperquadrics and explicit examples
We give series of explicit examples of Levi-nondegenerate real-analytic
hypersurfaces in complex spaces that are not transversally holomorphically
embeddable into hyperquadrics of any dimension. For this, we construct
invariants attached to a given hypersurface that serve as obstructions to
embeddability. We further study the embeddability problem for real-analytic
submanifolds of higher codimension and answer a question by Forstneri\v{c}.Comment: Revised version, appendix and references adde
Powers of ideals and convergence of Green functions with colliding poles
Let us have a family of ideals of holomorphic functions vanishing at N
distinct points of a complex manifold, all tending to a single point. As is
known, convergence of the ideals does not guarantee the convergence of the
pluricomplex Green functions to the Green function of the limit ideal;
moreover, the existence of the limit of the Green functions was unclear.
Assuming that all the powers of the ideals converge to some ideals, we prove
that the Green functions converge, locally uniformly away from the limit pole,
to a function which is essentially the upper envelope of the scaled Green
functions of the limits of the powers. As examples, we consider ideals
generated by hyperplane sections of a holomorphic curve near its singular
point. In particular, our result explains recently obtained asymptotics for
3-point models.Comment: 15 pages; typesetting errors fixe
Life Sciences, Technology, and the Law - Symosium Transcript - March 7, 2003
Life sciences, Technology, and the Law Symposium held at the University of Michigan Law School Friday, March 7, 200
Extremal discs and the holomorphic extension from convex hypersurfaces
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
Pluricomplex Green and Lempert functions for equally weighted poles
For a domain in , the pluricomplex Green function with
poles is defined as .
When there is only one pole, or two poles in the unit ball, it turns out to be
equal to the Lempert function defined from analytic disks into by . It is known
that we always have . In the more general case where we
allow weighted poles, there is a counterexample to equality due to Carlehed and
Wiegerinck, with equal to the bidisk.
Here we exhibit a counterexample using only four distinct equally weighted
poles in the bidisk. In order to do so, we first define a more general notion
of Lempert function "with multiplicities", analogous to the generalized Green
functions of Lelong and Rashkovskii, then we show how in some examples this can
be realized as a limit of regular Lempert functions when the poles tend to each
other. Finally, from an example where in the case of
multiple poles, we deduce that distinct (but close enough) equally weighted
poles will provide an example of the same inequality. Open questions are
pointed out about the limits of Green and Lempert functions when poles tend to
each other.Comment: 25 page
Convergence and multiplicities for the Lempert function
Given a domain , the Lempert function is a
functional on the space Hol (\D,\Omega) of analytic disks with values in
, depending on a set of poles in . We generalize its definition
to the case where poles have multiplicities given by local indicators (in the
sense of Rashkovskii's work) to obtain a function which still dominates the
corresponding Green function, behaves relatively well under limits, and is
monotonic with respect to the indicators. In particular, this is an improvement
over the previous generalization used by the same authors to find an example of
a set of poles in the bidisk so that the (usual) Green and Lempert functions
differ.Comment: 24 pages; many typos corrected thanks to the referee of Arkiv for
Matemati
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