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Oka properties of some hypersurface complements. (English) Zbl 1297.32015

This paper in several complex variables studies the complement \(X\) of a complex curve in a projective or affine space from the point of view of the paucity or abundance of entire holomomorphic curves in \(X\). In the case of paucity, the author proves instances of Kobayashi hyperbolicity, and in the case of abundance, instances of the Oka property of Forstnerič. There are two main results to be quoted below.
Theorem 3.1. Let \({\mathbb P}^n\) be the complex projective space of dimension \(n\), and \(H_1,\dots,H_N\) distinct hyperplanes in it. The complement \(X={\mathbb P}^n\setminus\bigcup_{j=1}^N H_j\) is Oka if and only if the hyperplanes \(H_1,\dots,H_N\) are in general position and \(N\leq n+1\). Furthermore, if \(X\) is not Oka, then it is neither dominable by \({\mathbb C}^n\) nor \({\mathbb C}\)-connected.
Here a complex manifold \(X\) is Oka if it satisfies Forstnerič’s equivalent properties of convex approximation and convex interpolation. The proof of Theorem 3.1 involves mainly work with the Picard-Borel theorem on the range of entire holomorphic functions, and some theorems from the theories of Kobayashi hyperbolic manifolds and of Oka manifolds.
Theorem 4.6. Let \(X\) be a complex manifold and \(m: X\to{\mathbb P}^1\) a holomorphic map regarded as a meromorphic function on \(X\) with no points of indeterminacy. If \(m\) can be written as \(m=f+\frac1g\) for holomorphic functions \(f\) and \(g\) on \(X\), then both or neither \(X\) and the graph complement \(\{(x,y)\in X\times{\mathbb C}: m(x)\not=\infty,y\not=m(x)\}\) enjoy the Oka property of Forstnerič.
The proof of Theorem 4.6 is done by reduction to the case when the manifold is \(X={\mathbb C}^n\). The reduction is achieved by considering the case \(f=0\) and the sets \(\{(x,y)\in{\mathbb C}^n\times{\mathbb C}: 1-g(x)y\not=0\}\), \(\{(x,y,z)\in{\mathbb C}^n\times{\mathbb C}\times{\mathbb C}: 1-g(x)y=e^z\}\). The case of \(X={\mathbb C}^n\) is dealt with the help of fiber bundles and a Lemma 4.8 that writes a directional derivative \(g'(x_0)s\) at a zero \(x_0\) of \(g(x_0)=0\) as the limit of \(\frac{1}{g(x)}-\frac{1}{g(x+g(x)^2s)}\) as \(x\to x_0\) in the complement \(g\not=0\). There is also Lemma 4.2 that states, in effect, that a meromorphic function \(m=h/k\) on \({\mathbb C}^n\) without indeterminate points can be written as \(m=f+\frac1g\) for holomorphic functions on \({\mathbb C}^n\) if and only the numerator \(h\) has a branch of logarithm along the pole sheet \(k=0\) (or on a neighborhood of it).

MSC:

32Q28 Stein manifolds
32E10 Stein spaces
14J70 Hypersurfaces and algebraic geometry
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32H04 Meromorphic mappings in several complex variables
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References:

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