For projective pairs with nondegenerate log conformal tensors, if the log canonical class is not nef then the pair is a quadric, projective space with hyperplane, or has an isotropic fibration; if numerically zero then under holonomy assumptions it is a toroidal compactification of a semi-abelian.
Math.41(2004), no
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Log-Conformal Projective Manifolds
For projective pairs with nondegenerate log conformal tensors, if the log canonical class is not nef then the pair is a quadric, projective space with hyperplane, or has an isotropic fibration; if numerically zero then under holonomy assumptions it is a toroidal compactification of a semi-abelian.