# Curvature of varieties of log general type

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)

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.