Characterizations of complex projective spaces and hyperquadrics

Abstract

O. Introduction Hirzebruch and K o daira [4] have given characterizations of the complex projective spaces. A similar characterization for the complex hyperquadrics has been given by B rie s k o rn [1 ]. (See also a recent paper o f Morrow [8 ] on these topics.) The purpose of the present paper is to give slightly different characterizations o f these spaces. Our motive is to give characterizations which will be useful in differential geometry of compact }Mier manifolds of positive curvature. Our results are expressed in terms of the first Chern class of a mani-fo ld. The first Chern class is closely related to the Ricci curvature of a m anifold. We refer the reader to the paper [6 ] fo r a n application o f results o f this paper to 3-dimensional compact Kdhler mani-folds of positive curvature. A similar characterization has been used recently by Howard [ 5 ] in his work on positively pinched Kdhler manifolds. Results which can be found in Hirzebruch's book [3 ] are used freely often without explicit references. The cohomology o f M with coefficients in the sheaf a (F) of germs of holomorphic sections of a line bundle (or a vector bundle)