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Rational curves on fibered Calabi-Yau manifolds. (English) Zbl 1445.14060

Let \(X\) be a normal projective variety of dimension \(n\) which admits a fibration \(f\) onto a base \(B\) such that the canonical bundle of the general fiber is numerically trivial. It is natural to ask under which conditions the variety \(X\) does contain rational curves. Before this paper, only very particular cases for low dimensions were already studied. The most manageable cases for arbitrary dimension are when \(B\) is a curve and when \(\operatorname{dim}(B)=n-1\), i.e. \(f\) is a genus-one fibration. Also to face off the case of a genus-one fibration, it is difficult to use just the condition on the triviality of the canonical bundle of the general fiber. A more manageable condition, used by the authors, is that the canonical bundle of \(X\) is linearly equivalent to the pullback of a line bundle on \(B\).
In this paper, the authors proved that, if \(X\) is a smooth projective variety with finite fundamental group that admits a genus-one fibration \(f\) such that \(K_X\sim f^*L\) for some Cartier divisor on the base, then \(X\) does contain a rational curve. More precisely this curve is a vertical curve that maps onto the singular values of \(f\).
This result is a first step towards the understanding of the rational curves on genus-one fibration, and it is particularly important in the case the canonical bundle of \(X\) is globally numerically trivial. For our convenience, we say that a Calabi-Yau manifold is a smooth projective variety with finite fundamental group and numerically trivial canonical bundle. For a Calabi-Yau manifold \(X\), as a consequence of the first result of the authors, they show, passing through the global index-one cover of \(X\), that it contains a rational curve.
In the second part of the paper, the authors prove that a Calabi-Yau manifold \(X\) that admits an abelian fibration onto a smooth curve \(C\) does contain a rational curve. In this case, it is not clear where rational curves should be. The idea of the authors is to assume by contradiction that \(X\) does not contain rational curves, and then construct from the divisor given by a generic fiber, another \(\mathbb{Q}\)-Cartier \(\mathbb{Q}\)-divisor that must be semiample of Iitaka dimension \(n-1\). As a consequence they apply the first theorem of the paper, reaching a contradiction.

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32Q55 Topological aspects of complex manifolds
14E30 Minimal model program (Mori theory, extremal rays)
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