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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?

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  • $\begingroup$ The answer to your first question is positive, and well-known. You can just take for example the proof in Thm 2.7 in arxiv.org/pdf/2011.11379.pdf and see that it applies also under your hypotheses. Or you can refer to Kobayashi's book "Hyperbolic complex spaces" Thm 3.7.8 which gives even stronger information $\endgroup$
    – YangMills
    Apr 8, 2021 at 14:30
  • $\begingroup$ Namely, if you just assume that HSC$\leq 0$ and $M$ is not hyperbolic, then you can find a nonconstant entire curve $f:\mathbb{C}\to M$ such that HSC vanishes along $f(\mathbb{C})$ (in the direction of the entire curve). $\endgroup$
    – YangMills
    Apr 8, 2021 at 18:17
  • $\begingroup$ About your second question, not all metrics have negative scalar curvature, but you can often say something about the total scalar curvature (=integral of the Chern scalar curvature): a compact hyperbolic manifold is not uniruled; assume it is also projective, then by BDPP its canonical bundle is pseudoeffective, then use Thm 1.1 arxiv.org/pdf/1705.02672.pdf to conclude that either $M$ is Calabi-Yau (in the sense that $c_1^{\mathbb{R}}(K_X)=0$) in which case all Hermitian metrics have vanishing total scalar curvature, or else all Hermitian metrics have negative total scalar curvature $\endgroup$
    – YangMills
    Apr 8, 2021 at 18:44
  • $\begingroup$ @YangMills You sir, are incredible! You are a treasure to this community! Thank you very much :) $\endgroup$ Apr 8, 2021 at 21:33

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