Hyperbolicity for log canonical pairs and the cone theorem

Abstract

Given a log canonical pair $(X,\Delta )$, we show that $K_X+\Delta$ is nef assuming there is no non-constant map from the affine line with values in the open strata of the stratification induced by the non-klt locus of $(X,\Delta )$. This implies a generalization of the Cone Theorem where each $K_X+\Delta$-negative extremal ray is spanned by a rational curve that is the closure of a copy of the affine line contained in one of the open strata of $\mathrm{Nklt}(X,\Delta )$. Moreover, we give a criterion of Nakai type to determine when under the above condition $K_X+\Delta$ is ample and we prove some partial results in the case of arbitrary singularities.

v3: final accepted version. v2: title was changed; typos fixed; improved presentation. 21 pages. comments are welcome!

References in corpus (2)

• Introduction to the log minimal model program for log canonical pairs
• On a generalized canonical bundle formula and generalized adjunction

Cited by in corpus (5)

• On the connectedness principle and dual complexes for generalized pairs
• Birational boundedness of rationally connected Calabi-Yau 3-folds
• Local and global applications of the Minimal Model Program for co-rank one foliations on threefolds
• Cone theorem and Mori hyperbolicity
• Subadjunction for quasi-log canonical pairs and its applications