On the sectional curvature of holomorphic curvature operators

Abstract:

The object of this paper is to study the pointwise behaviour of the holomorphic sectional curvature function of a Kähler manifold and its relationship with the Riemannian curvature tensor. We begin by defining an orthogonal projection from the space of positive semidefinite Kähler curvature operators into the space of Kähler curvature operators which satisfy the Bianchi identity and have nonnegative holomorphic sectional curvature. In dimension four we show that this map is onto. As a corollary we obtain a description of the minimum and maximum sets of the holomorphic sectional curvature function. For each higher dimension, we exhibit an example of a Kähler curvature operator which is not the projection of a positive semidefinite one. Specifically, we construct a Kähler curvature operator with nonnegative holomorphic sectional curvature, whose zero set does not admit a description as in dimension four. These examples contradict results in the literature which claim to describe the extremal sets for the holomorphic sectional curvature function for all dimensions.