A compact Kähler manifold carrying a degree-2 hyperbolic cohomology class with nonzero top self-intersection must have nonzero holomorphic sections of adjoint bundles for sufficiently positive bundles, cannot be uniruled, and in dimension two must be of general type.
The pseudo-effective cone of a compact K \"ahler manifold and varieties of negative K odaira dimension
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Geometric effects of hyperbolic cohomology classes on K\"ahler manifolds (with an appendix by Beno\^it Claudon)
A compact Kähler manifold carrying a degree-2 hyperbolic cohomology class with nonzero top self-intersection must have nonzero holomorphic sections of adjoint bundles for sufficiently positive bundles, cannot be uniruled, and in dimension two must be of general type.