A Characterization of Seven Compact Kaehler Submanifolds by Holomorphic Pinching

Abstract

From the results of Simons [9] and Chern, Do Carmo and Kobayashi [2], we know that, in the class of compact minimal submanifolds of a sphere, the totally geodesic submanifolds are isolated and that some simple minimal submanifolds can be characterized by suitable pinching on their curvatures. These ideas are extended naturally to compact Kaehler submanifolds of a complex projective space. The problem of the best pinching for the above submanifolds was studied later by several authors, e.g. Yau [11] and Ogiue [4]. For surfaces and hypersurfaces, the problem is completely resolved. However in the general case we have only partial results. Let M' be a compact Kaehler submanifold, of complex dimension n, immersed in the complex projective space CPtm(1) endowed with the Fubini-Study metric of constant holomorphic sectional curvature 1. Let H and K be the holomorphic sectional curvature and the sectional curvature of Mn respectively. Ogiue conjectured the following: (1) If H > , or (2) If n ? 2 and K> , or (3) If m-n n(n + ) and K > 0, 2 then Mn is a linear subvariety of CPm(1). Recently, using natural arguments at the minimum of the function H defined on the unit tangent bundle of MW, the author [7] and Verstraelen and the author [8] resolved the conjectures (1) and (2) respectively. In this paper we obtain the following complete solution of the pinching problem in the Kaehlerian case.