Journal of Mathematics of Kyoto University
L e t g be a vector bundle of rank r on a com pact complex manifold M of dim ension n. L et P = P (‘) be the associated Pr 1 -bundle and le t H = H (g) be the tautological line bundle (9(1) on P . g is said to be ample if so is H on P . In this case A = det (6) is also am ple on M . The c r sectional genus g of g is defined to b e g(M, A ), w hich is determ ined by the formula 2g(M, A) — 2 = (K + (n — 1)A)AH1 , where K is the canonical bundle o f M . Then g(M, A ) is a non-negative in teg e r b y [F 5 ]. In this paper we establish a classification theory of the case g(M, A) 2. T h e c a s e r = 1 was treated in [F 6 ] a n d w e study here the case r > 1. In §1, we study the case g = 0 or 1 . The case g = 2 is studied in §2. The main theorem is in (2.25). I n § 3 , we give a classification according to the sectional genus of (P, H). T h e author thanks Professor S . M u k a i fo r helpful comments during the preparation of this paper. W e employ similar notation to th a t in our previous papers o n polarized manifolds.