On the curvature of the blowup of a point

Abstract

We consider the blowup of a point of a compact K\"ahler manifold and a metric of the form $\mu^*h + t b$ on it, where $h$ is a K\"ahler metric on the original manifold and $b$ is Hermitian form that looks like the Fubini--Study metric near the exceptional divisor. We calculate the curvature tensor of this metric on the exceptional divisor and show that its holomorphic sectional curvature is negative in some directions for all small enough $t$, which torpedos a natural approach to showing that blowups of manifolds of positive holomorphic sectional curvature have positive curvature.