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Calabi-Yau manifolds and a conjecture of Kobayashi. (English) Zbl 0735.14028
Let \(X\) be a Calabi-Yau manifold, i.e. a projective manifold with a trivial canonical bundle and with no holomorphic 1-forms. This paper deals with the basic problem of finding rational curves on \(X\). In dimension 2, \(X\) is a \(K3\) surface and there is nothing more to say about this problem. — In this paper the author proves that, in dimension 3, \(X\) contains a rational curve in each one of the following cases:
(a) there exists a nef divisor \(D\) on \(X\), \(D^ 3=0\), \(Dc_ 2(X)\neq0\), \(D^ 2H>0\), for some ample \(H\);
(b) \(X\) contains an irreducible surface \(S\) as a non ample divisor.
In case (a) a result by P. M. A. Wilson [Invent. Math. 98, No. 1, 139-155 (1989; Zbl 0688.14032){]} says that a multiple of \(D\) gives rise to an elliptic fibre space \(f\): \(X\to S\) onto a normal surface \(S\). \(S\) turns out to be rational and the inverse image via \(f\) of a rational curve on \(S\) is a surface on \(X\), which is shown to contain a rational curve.
As to (b) in most cases the result is got by applying either (a) or similar criteria by Wilson, which show the existence of rational curves if there are certain special divisors \(D\) on \(X\). The proof consists in building those special \(D\)’s.
Next a hyperbolic projective threefold \(Y\) is considered, this meaning that any holomorphic map \(\mathbb{C}\to Y\) is constant. Following a conjecture of Kobayashi, the canonical bundle of \(Y\) is proved to be ample possibly unless \(Y\) is a Calabi-Yau manifold with any effective divisor being ample and \(\rho(X)\leq19\).

MSC:
14J30 \(3\)-folds
14C20 Divisors, linear systems, invertible sheaves
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