Ample vector bundles on singular varieties II

Abstract

Theorem 0 Let X be a n-dimensional smooth projective variety over complex number C and let E be an ample vector bundle on X o f rank n + 1 such that c l (X) = cl(E). Then (X, E) ~ (~n, Qn+16~,.(1))" This also had been proved independently by Peternell [P1]. The theorem gives us a characterization of the projective spaces via ample vector bundles. The methods we used in our proof depend heavily on Mori ' s theory of the extremal rays. In particular, a theorem of Lazarsfeld [L] played an important role: I f X is smooth and f : ~ ~ X is surjective, then X ~ 1~. However, this is no longer true if X has singularities as showed by the following example: Example. Let Y = ~2 and let ~: [x, y, z ] --, [ x , y , z] be an involution on Y. Then X = Y / has a rational double point. The quotient map Y --, X is of degree two which ramified along I? 1. In this paper, our aim is to generalize Mukai ' s conjecture to projective varieties having at most log-terminal singularities. The precise statements of our results are as follows: