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The complement to the general hypersurface of degree 2n in \({\mathbb{C}}P^ n\) is not hyperbolic. (Russian) Zbl 0625.32021

Let \({\mathcal E}_{n,d}\) be the set of hypersurfaces of degree d in complex projective space \({\mathbb{P}}^ n(={\mathbb{C}}P^ n)\). A hypersurface \(D\in {\mathcal E}_{n,d}\) and a line 1 are said to have an intersection of type \(m=(m_ 1,...,m_ t)\), where \(m_ i\in N\); \(i=1,...,t\); \(\sum^{t}_{i=1}m_ i=d\), if the multiplicity of \(1\cap D\) at point \(a_ i\) is equal to \(m_ i\). Particularly, \({\mathcal E}_{n,d,m}\) is the set of such \(D\in {\mathcal E}_{n,d}\) which have an intersection of type m with some lines, and \({\mathcal E}_{n,d,\infty}\) denotes hypersurfaces containing complete lines. Let \(\bar k\) denote the type (k,2n-k) for every \(k=0,1,...,2n.\)
The author proves Theorem 1. (a) \({\mathcal E}_{n,2n}={\mathcal E}_{n,2n,\infty}\cup {\mathcal E}_{n,2n,0}\cup {\mathcal E}_{n,2n,\bar k}\) holds for every \(k=1,...,n\). In other words, each hypersurface D of degree 2n in \({\mathbb{P}}^ n\), not containing lines, intersects some projective line \(l_ D\) at most in two points in type \(\bar k.\) (b) There exist a subset \({\mathcal H}_ k\subset {\mathcal E}_{n,2n,\bar k}\) for every \(k=1,...,n\) open in \({\mathcal E}_{n,2n}\) by Zarisky topology, and a natural number \(r=r(n,k)\) such that the set \(\{l_ D\}\) of lines, intersecting a given hypersurface \(D\in {\mathcal H}_ k\) in type \(\bar k,\) consists of r lines. Moreover, this \(\{l_ D\}\) is an r-valued holomorphic function on \(H_ k\), i.e. any of coordinates of this r lines is a holomorphic function of the coordinates of D (locally on \({\mathcal H}_ k\) in some coordinate representation).
A consequence of this theorem contains the author’s main result in the title. We recall that hyperbolicity of \(U={\mathbb{P}}^ n\setminus D\) means that the pseudo-metric \(k_ U\) of Kobayashi is not degenerate and \(k_ U\) majorizes \(c\cdot h\), where \(c>0\) and h is the Fubini-Study metric on \({\mathbb{P}}^ n\).
Reviewer: E.Molnár

MSC:

32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32H25 Picard-type theorems and generalizations for several complex variables
14C20 Divisors, linear systems, invertible sheaves
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