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)