Classification of generalized polarized manifolds by their nef values

Abstract

Abstract Let (M, ℰ) be a generalized polarized manifold, i.e., a pair of an n-dimensional smooth projective variety M and an ample vector bundle ℰ of rank r on M. Let τ be the nef value of a polarized manifold (M, det ℰ), i.e., the minimum of the set of real numbers t such that K M + t det ℰ is nef; we have τr ≤ n + 1 by Mori's theory. In this paper we classify the pairs (M, ℰ) in the following two cases: (1) n − 2 ≤ τr and τ ≥ 1; (2) n + 1 − τr < τ ≤ 1.