Ngaiming Mok, HKU, Hong Kong

On the projective connection on Kähler manifolds of constant holomorphic sectional curvature

Abstract:

The complex projective space Pⁿ, the Euclidean space Cⁿ and the complex hyperbolic space form, equivalently the complex unit ball Bⁿ, share the common feature of possessing a complete Kähler metric of constant holomorphic sectional curvature.  The Levi-Civita connections of these canonical metrics agree with the natural holomorphic projective connections.  In particular, the second fundamental form on a complex submanifold is holomorphic.  We examine applications of this elementary but basic fact which underlies the geometries of Pⁿ, Cⁿ and Bⁿ and their quotient manifolds.  An example is the proof, by means of harmonic forms, that the tangent bundle of the ambient space form splits holomorphically on a compact complex submanifold S if and only if S is totally geodesic, a well-known result in the case of Pⁿ due to Van de Ven. (The Euclidean case was a recent result established by Jahnke.)  Another application is to the study of holomorphic mappings between compact complex hyperbolic space forms.