Generalized polarized manifolds with low second class

Abstract

On a smooth complex projective variety X of dimension n, consider an ample vector bundle E of rank r a n À 2 and an ample line bundle H. A numerical character m 2 ¼ m 2 ðX ; E; HÞ of the triplet ðX ; E; HÞ is defined, extending the well-known second class of a polarized manifold ðX ; HÞ, when either n ¼ 2 or H is very ample. Under some additional assumptions on F :¼ E l H lðnÀrÀ2Þ , triplets ðX ; E; HÞ as above whose m 2 is small with respect to the invariants d :¼ c nÀ2 ðFÞH 2 and g :Á H are studied and classified.