Curvature characterization of hyperquadrics

Abstract

After proving the dimension two case jointly with Andreotti, Frankel [3] conjectured that a compact K/ihler manifold of positive sectional curvature is biholomorphic to the complex projective space. Mabuchi [8] verified the case of dimension three by using the result of Kobayashi-Ochiai [6]. Very recently by using the methods of algebraic geometry of positive characteristic Mori [10] proved that a compact Kihler manifold with ample tangent bundle must be biholomorphic to the complex projective space. By methods of Kihler geometry Siu-Yau [12] proved that a compact K/ihler manifold of positive holomorphic bisectional curvature must be biholomorphic to the complex projective space. Frankel’s conjecture is a special case of these more general results. It is reasonable to conjecture that there are similar curvature characterizations for other irreducible compact symmetric K/ihler manifolds. In this paper we obtain such a curvature characterization for the complex hyperquadric. Definition. Let M be a K/ihler manifold and P M. If the holomorphic bisectional curvature of M is nonnegative at P, then for a nonzero element (a) of the holomorphic tangent space T,M of M at P, the curvature null space at P in the direction of , denoted by N,(O, is defined as the set of all