Let $X$ be a projective manifold and $\Delta$ a divisor with simple normal crossings. Consider $X$ as the compactification of a quasi-projective variety $X\_0$ with boundary $\Delta$, i.e. $X\_0 = X \backslash \Delta$. Suppose that $(X,\Delta)$ is of log general type, i.e. $K\_X+D$ is big.


A theorem of Cadorel [Cad16] tells us that a projective log smooth pair $(X,\Delta)$ with $X\_0$ admitting a Kähler metric of non-positive bisectional curvature and negative holomorphic sectional curvature, then the sheaf of logarithmic differentials $\Omega\_X(\log(\Delta))$ is big. If one also assumes that $\omega$ is bounded near $D$, this can be strengthened to $\Omega\_X$ being big.


Guenancia [Gue18], in a similar spirit, has shown that if $(X, \Delta)$ is log smooth with the holomorphic sectional curvature of $\omega$ on $X\_0$ bounded above by a negative constant, then $(X,\Delta)$ is of log general type.


Is the converse true? That is, given a log smooth pair $(X, \Delta)$ of log general type, is the sectional curvature bounded above by a negative (nonpositive?) constant on $X\_0$; is the bisectional curvature nonpositive on $X\_0$?


[Cad16] -- Cadoral, B., *Symmetric differentials on complex hyperbolic manifolds with cusps*, arXiv:1606.05470, (2016)


[Gue18] -- Guenancia, H., *Quasi-projective manifolds with negative holomorphic sectional curvature*, arXiv:1808.01854v3, (2018)



Curvature of varieties of log general type
No, this is not true, even for $\Delta=\emptyset$. If $X$ admits a Kähler metric with negative holomorphic bisectional curvature, then so do all its subvarieties; in particular, all its subvarieties are of general type.


However, many varieties of general type admit subvarieties with non-maximal Kodaira dimension (blow up any general type variety along a smooth center for instance).


Even in the minimal case, there are hypersurfaces of the projective space with ample canonical bundle that admit rational curves (e.g. some Fermat hypersurfaces).


