Some Adjunction Properties of Ample Vector Bundles

Let E be an ample vector bundle of rank r on a projective variety X with only log-terminal singularities. We consider the nefness of adjoint divisors KX + (t − r) detE when t ≥ dimX and t > r. As a corollary, we classify pairs (X, E) with cr-sectional genus zero. Introduction. Let X be a smooth projective variety and KX the canonical bundle of X . For the study of X , it is useful to consider adjoint bundles KX + tL, where t is a positive integer and L is an ample line bundle on X . We refer to the books [BS] and [F0] for the properties of KX + tL; it is powerful when t is close to dimX . Recently, as a natural generalization of adjoint bundles, many authors have considered KX + det E , where E is an ample vector bundle on X . (We say that a vector bundle E is ample if OP(E)(1) is ample on P(E).) In particular, Ye and Zhang [YZ] have given a classification for pairs (X, E) when rank E ≥ n−1 and KX +det E is not nef. Many other results on KX + det E are obtained when rank E is close to dimX . It seems to be difficult to study the nefness of KX + det E when rank E is small as compared with dimX . To overcome this difficulty, in the present paper, we consider the nefness of KX + (t−r) det E when t ≥ n = dimX . We mainly use vanishing theorems and an estimate of the length of extremal rays, hence our argument works on projective varieties X with at worst log-terminal singularities. Our main result is Theorem 2.5 in which we show that KX + (n− r) det E is nef unless (X, E) ∼= (P ,O(1)) when 1 < r < n− 1. As a corollary, we see that the cr-sectional genus of the pairs (X, E) is non-negative and we obtain a classification of (X, E) with cr-sectional genus zero in case X is logterminal. We note that cr-sectional genus is introduced in [I] and studied in case X is smooth (see also [FuI]). 1991 Mathematics Subject Classification. Primary 14J60; Secondary 14C20, 14F05, 14J40.