We investigate the limits of the ideals of holomorphic functions vanishing on three points in C^2 when all three points tend to the origin, and what happens to the associated pluricomplex Green functions. This is a continuation of the work of Magnusson, Rashkovskii, Sigurdsson and Thomas, where those questions were settled in a generic case. 2010 Mathematics Subject Classification: 32U35, 32A27.
En reprenant le travail de Weinstock concernant l’union de deux sous-espaces, nous montrons que peut être obtenu comme l’union d’un nombre fini de sous-espaces vectoriels totalement réels maximaux, pour tout supérieur à un. Ceci contraste avec le cas des droites complexes de , dont il faut un ensemble de capacité positive pour que l’enveloppe soit tout l’espace. On étudie aussi le cas des trois plans réels de : si les trois unions deux à deux ne sont pas polynomialement convexes, alors l’enveloppe...
We consider the problem of whether the union of complex hyperplanes can be a subset of a zero variety for the Hardy classes of the ball. A sufficient condition is found, consisting in a strong geometric separatedness requirement, together with a quantitative requirement slightly stronger than the necessary condition for Nevanlinna class zero varieties.
Let a be a sequence of points in the unit ball of Cn. Eric Amar and the author have introduced the nonnegative quantity ρ(a) = infα infk Πj:j≠k dG(αj, αk), where dG is the Gleason distance in the unit disk and the first infimum is taken over all sequences α in the unit disk which map to a by a map from the disk to the ball. The...
A sufficient condition is given to make a sequence of hyperplanes in the complex unit ball an for , i.e. bounded holomorphic functions on the hyperplanes can be boundedly extended.
In the space A(D) of functions of polynomial growth, weakly sufficient sets are those such that the topology induced by restriction to the set coincides with the topology of the original space. Horowitz, Korenblum and Pinchuk defined sampling sets for A(D) as those such that the restriction of a function to the set determines the type of growth of the function. We show that sampling sets are always weakly sufficient, that weakly sufficient sets are always of uniqueness, and provide examples of discrete...
A complex analytic space is said to have the D*-extension property if and only if any holomorphic map from the punctured disk to the given space extends to a holomorphic map from the whole disk to the same space. A Hartogs domain H over the base X (a complex space) is a subset of X x C where all the fibers over X are disks centered at the origin, possibly of infinite radius. Denote by φ the function giving the logarithm of the reciprocal of the radius of the fibers, so that, when X is pseudoconvex,...
Given A∈ Ωₙ, the n²-dimensional spectral unit ball, we show that if B is an n×n complex matrix, then B is a “generalized” tangent vector at A to an entire curve in Ωₙ if and only if B is in the tangent cone to the isospectral variety at A. In the case of Ω₃, the zero set of the Kobayashi-Royden pseudometric is completely described.
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