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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)

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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).

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  • $\begingroup$ Thank you so much! $\endgroup$
    – AmorFati
    Oct 5, 2020 at 10:22

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