# dg.differential geometry – Holomorphic sectional curvature and Kobayashi hyperbolicity

Let $$(M,g)$$ be a compact Hermitian manifold. Let $$text{HSC}(g)$$ denote the holomorphic sectional curvature of $$g$$. The implication $$text{HSC}(g) < 0 implies M text{is Kobayashi hyperbolic}$$ is well-known, as is the failure of the converse to hold by an old example of Demailly (produce a projective surface which is fibered by hyperbolic curves over a hyperbolic curve such that one fiber is sufficiently singular to violate Demailly’s Riemann–Hurwitz-type criterion for the existence of a metric with holomorphic sectional curvature).

It is natural to ask the following:

Suppose that $$text{HSC}(g) <0$$ everywhere except at one, or a finite number of points. Is $$M$$ Kobayashi hyperbolic?

Of course, the more natural question to ask is whether $$text{HSC}_g leq 0$$ implies Kobayashi hyperbolicity, but this is clearly false: Take $$mathbb{C}^n$$ with the flat metric.

Since the converse to the implication mentioned in the beginning is false,

are there known curvature constraints on the Hermitian metrics which live on a compact Kobayashi hyperbolic manifold? Is the scalar curvature necessarily negative, for example?