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?