Let $(X,\omega)$ be a compact Kähler manifold and call $\operatorname{HSC}_{\omega}(x,[v])$ the holomorphic sectional curvature of the Chern connection of $\omega$ at the point $x\in X$ in the direction of $v\in T_{X,x}\setminus\{0\}$. Since $\omega$ is Kähler, it coincides with the usual (Riemannian) sectional curvature of the complex $2$-plane spanned by $(v,Jv)$.

Now, $\operatorname{HSC}_{\omega}$ is said to be *quasi-negative* if it is non positive at every point and every direction, and there exists at least one point $x_0\in X$ such that $\operatorname{HSC}_{\omega}(x_0,[v])<0$ for **every** $v\in T_{X,x_0}\setminus\{0\}$.

Since $X$ is compact, if $\operatorname{HSC}_{\omega}$ is strictly negative, then it is bounded away from zero and it is classically known that $X$ is Kobayashi-hyperbolic.

Also, it has been recently shown that quasi-negative holomorphic sectional curvature (and therefore also strict negativity, but this was shown a little bit earlier by Wu and Yau in the projective case and extended to the Kähler setting by Tosatti and Yang) implies the ampleness of the canonical bundle.

**Question.** Does anybody know an example of a compact Kähler manifold which admits a Kähler metric with quasi-negative holomorphic sectional curvature but that is **not** Kobayashi hyperbolic?

Of course, such an example would have dimension at least two and would also give an example of a compact Kähler manifold admitting a Kähler metric with quasi-negative holomorphic sectional curvature but with no (even) Hermitian metrics with negative holomorphic sectional curvature.

Here is where I naturally stared to look at, but I wasn't able (or didn't have enough perseverance...) to push the computations till the end.

**(Possible beginning of an) Example.** Take a three dimensional abelian variety $Y$ containing a (say smooth) elliptic curve $E$. Now, cut $Y$ with a high degree projective hypersurface containing $E$, and call this section $X$. If the hypersurface is general enough and its degree is high enough, some refined version of Bertini's theorem ensures that $X$ is in fact smooth. Since $X$ contains an elliptic curve, it cannot be hyperbolic. Now, endow $X$ with the Kähler metric $\omega$ induced by the flat metric on $Y$. Thanks to the decreasing property of the holomorphic (bi)sectional curvature, we surely have $\operatorname{HSC}_{\omega}\le 0$. Moreover, by adjunction, $K_X$ is ample. Can we find a point of $X$ where $\operatorname{HSC}_{\omega}$ is strictly negative in all directions?

Please observe that there might be some subtleties to overcome: for instance, if one takes the embedding of a compact Riemann surface of higher genus $C$ into its Jacobian, then the induced metric in this case is not necessarily strictly negative! If the Riemann surface is hyperelliptic, then this induced metric will have zero curvature exactly on the Weierstrass points of $C$...

Of course any other (hopefully even more trivial...) example (or counterexample...) is very welcome!