Approximation of holomorphic maps by algebraic morphisms

J. Bochnak; W. Kucharz

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The rigidity properties of the local invariants of real algebraic Cauchy-Riemann structures imposes upon holomorphic mappings some global rational properties (Poincaré 1907) or more generally algebraic ones (Webster 1977). Our principal goal will be to unify the classical or recent results in the subject, building on a study of the transcendence degree, to discuss also the usual assumption of minimality in the sense of Tumanov, in arbitrary dimension, without rank assumption and for...

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In this paper, we show that if $M_{1}$ and $M_{2}$ are algebraic real hypersurfaces in (possibly different) complex spaces of dimension at least two and if $f$ is a holomorphic mapping defined near a neighborhood of $M_{1}$ so that $f (M_{1}) \subset M_{2}$ , then $f$ is also algebraic. Our proof is based on a careful analysis on the invariant varieties and reduces to the consideration of many cases. After a slight modification, the argument is also used to prove a reflection principle, which allows our main result to be stated for...

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Germs of holomorphic mappings between real algebraic hypersurfaces

Nordine Mir (1998)

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We study germs of holomorphic mappings between general algebraic hypersurfaces. Our main result is the following. If $(M, p_{0})$ and $(M^{'}, p_{0}^{'})$ are two germs of real algebraic hypersurfaces in $ℂ^{N + 1}$ , $N \geq 1$ , $M$ is not Levi-flat and $H$ is a germ at $p_{0}$ of a holomorphic mapping such that $H (M) \subseteq M^{'}$ and $Jac (H) ≢ 0$ then the so-called reflection function associated to $H$ is always holomorphic algebraic. As a consequence, we obtain that if $M^{'}$ is given in the so-called normal form, the transversal component of $H$ is always algebraic. Another...

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Investigated are continuous rational maps of nonsingular real algebraic varieties into spheres. In some cases, necessary and sufficient conditions are given for a continuous map to be approximable by continuous rational maps. In particular, each continuous map between unit spheres can be approximated by continuous rational maps.

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It is shown that there exists a twistor space on the n-fold connected sum of complex projective planes nCP2, whose algebraic dimension is one and whose general fiber of the algebraic reduction is birational to an elliptic ruled surface or a K3 surface. The former kind of twistor spaces are constructed over nCP2 for any n ≥ 5, while the latter kind of example is constructed over 5CP2. Both of these seem to be the first such example on nCP2. The algebraic reduction in these examples is...

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