On contractions of extremal rays of Fano manifolds.

*(English)*Zbl 0721.14023Let X be a complex projective manifold and \(\phi: X\to Y\) a morphism onto a normal projective variety Y such that all fibers of \(\phi\) are connected. If all curves contracted by \(\phi\) are numerically proportional and have negative intersection with the canonical divisor \(K_ X\) then such a map is called a contraction of an extremal ray.

In the present paper the geometric structure of such contractions is examined. For example, the following inequality is proved \[ \dim(\text{exceptional locus of }\phi)+\dim(\text{non-trivial fiber of }\phi)\geq \dim(X)-1+\ell, \] where the integer \(\ell\) is defined as the minimal intersection number \(-K_ X.C\) among rational curves contracted by \(\phi\).

Most of the results of the paper concern the case when X is a Fano manifold, i.e. \(-K_ X\) is ample. In particular it is examined when a contraction of a Fano manifold leads to a non-Fano manifold. For example, it is shown that a conic bundle over a 3-fold is a Fano manifold only if its base is; for higher dimensions it is not true as an example shows.

In the present paper the geometric structure of such contractions is examined. For example, the following inequality is proved \[ \dim(\text{exceptional locus of }\phi)+\dim(\text{non-trivial fiber of }\phi)\geq \dim(X)-1+\ell, \] where the integer \(\ell\) is defined as the minimal intersection number \(-K_ X.C\) among rational curves contracted by \(\phi\).

Most of the results of the paper concern the case when X is a Fano manifold, i.e. \(-K_ X\) is ample. In particular it is examined when a contraction of a Fano manifold leads to a non-Fano manifold. For example, it is shown that a conic bundle over a 3-fold is a Fano manifold only if its base is; for higher dimensions it is not true as an example shows.

Reviewer: J.A.Wiśniewski (Warszawa)