Ampleness conjectures for compact complex manifolds with hyperbolicity or negative holomorphic sectional curvature

Let $M^n$ be a compact complex manifold. A Hermitian metric has quasi-negative holomorphic sectional curvature when its holomorphic sectional curvature is nonpositive everywhere and negative somewhere.

Ampleness conjectures.

• If $M$ is Kobayashi hyperbolic, then its canonical line bundle $K_M$ is ample.
• If $M$ admits a Hermitian metric with quasi-negative holomorphic sectional curvature, then $K_M$ is ample.
• If $M$ admits a Hermitian metric with negative holomorphic sectional curvature, then $K_M$ is ample.

These conjectures extend results on the positivity of the canonical bundle from projective Kähler manifolds to compact complex and Hermitian manifolds. The source explicitly describes them as still open.