Math. Ann. 189, 1--4 (1970) © by Springer-Verlag 1970 On Holomorphic Sections of Certain Hermitian Vector Bundles SHOSHICHI KOBAYASHI* and HUNG-HSI Wu** I. Introduction Let E be a holomorphic vector bundle over a compact complex manifold M. Let h be a hermitian fibre metric in E. We shall show that if h has negative curvature (in the sense to be made precise below), then E admits no nonzero holomorphic cross sections. We use notations in [4; Chapter IX, Section 10]. Let z 1..... z" b~ a local coordinate system in M and sl ..... sr holomorphic local cross sections of E which form a basis of the fibre at each point. Let h,~, (a, fl = 1 ..... r), be the components ofh with respect to s 1..... s,. Let K~G, (a, fl = 1 ..... r and i,j = t .... ..., n), be the components of the curvature of the hermitian connection with respect to z 1, ..., z" and sl, ..., sr. We set K~l~i] = ~h~TK~iy. The theorem to be proven reads as follows: Theorem. Let E be a holomorphic vector bundle over a compact complex manifold M with a hermitian fibre metric h such that (~tK~ai~) is a negative definite hermitian matrix at each point of M. Then E admits no nonzero holomor- phic sections. We state its immediate corollaries. Some are new but some are known (see the remarks below). Corollary 1. Let E be a complex line bundle over a compact complex manifold M such that its Chern class Cl(E) can be represented by a (1, 1)-form whose trace is negative. Then E admits no nonzero holomorphic cross sections. Corollary 2. Let M be a compact hermitian manifold whose scalar curvature is positive. Then M admits no nonzero holomorphic n-forms, (n = dimM). Corollary 3. Let M be a compact hermitian manifold. Let r 1.... , r. be the eigen-values of the Ricci tensor. If ri~ + " " + rip > 0 for all i 1 <.,. < ip, then M admits no nonzero holomorphic p-forms. Corollary 4. A compact complex manifold M with positive first Chern class admits no nonzero holomorphic p-forms for p > 1. * Partially supported by NSF Grant GP 8008. ** Partially supported by NSF Grant GP 13348. 1 Math. Ann. 189