If Σ is a compact subset of a domain Ω ⊂ ℂ and the cluster values on ∂Σ of a holomorphic function f in Ω∖Σ, f' ≢ 0, are contained in a compact null-set for the holomorphic Dirichlet class, then f extends holomorphically onto the whole domain Ω.
We begin this article with a graph theorem and a kind of Nullstellensatz for weakly holomorphic functions. This yields a general Nullstellensatz for c-holomorphic functions on locally irreducible sets. In Section 2 some methods of Płoski-Tworzewski permit us to prove an effective Nullstellensatz for c-holomorphic functions in the case of a proper intersection with the degree of the intersection cycle as exponent. We also extend this result to the case of isolated improper intersection, generalizing...
A version of the Schwarz lemma for correspondences is studied. Two applications are obtained namely, the 'non-increasing' property of the Kobayashi metric under correspondences and a weak version of the Wong-Rosay theorem for convex, finite type domains admitting a 'non-compact' family of proper correspondences.
Let be a two dimensional totally real submanifold of class in . A continuous map of the closed unit disk into that is holomorphic on the open disk and maps its boundary into is called an analytic disk with boundary in . Given an initial immersed analytic disk with boundary in , we describe the existence and behavior of analytic disks near with boundaries in small perturbations of in terms of the homology class of the closed curve in . We also prove a regularity theorem...
We study the singularities of plurisubharmonic functions using methods from convexity theory. Analyticity theorems for a refined Lelong number are proved.
Let V be an analytic variety in a domain Ω ⊂ ℂⁿ and let K ⊂ ⊂ V be a closed subset. By studying Jensen measures for certain classes of plurisubharmonic functions on V, we prove that the relative extremal function is continuous on V if Ω is hyperconvex and K is regular.
We endow the module of analytic p-chains with the structure of a second-countable metrizable topological space.
Let be polynomials in variables without a common zero. Hilbert’s Nullstellensatz says that there are polynomials such that . The effective versions of this result bound the degrees of the in terms of the degrees of the . The aim of this paper is to generalize this to the case when the are replaced by arbitrary ideals. Applications to the Bézout theorem, to Łojasiewicz–type inequalities and to deformation theory are also discussed.