Contractions on a manifold polarized by an ample vector bundle

Abstract:

A complex manifold $X$ of dimension $n$ together with an ample vector bundle $E$ on it will be called a generalized polarized variety. The adjoint bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. We study the positivity (the nefness or ampleness) of the adjoint bundle in the case $r := rank (E) = (n-2)$. If $r\geq (n-1)$ this was previously done in a series of papers by Ye and Zhang, by Fujita, and by Andreatta, Ballico and Wisniewski.

If $K_X+detE$ is nef then, by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map $\pi :X \longrightarrow W$ from $X$ onto a normal projective variety $W$ with connected fiber and such that $K_X + det(E) = \pi ^*H$, for some ample line bundle $H$ on $W$. We describe those contractions for which $dimF \leq (r-1)$. We extend this result to the case in which $X$ has log terminal singularities. In particular this gives Mukai’s conjecture 1 for singular varieties. We consider also the case in which $dimF = r$ for every fiber and $\pi$ is birational.