Abstract
Let \(X\) be a projective variety which is algebraic Lang hyperbolic. We show that Lang’s conjecture holds (one direction only): \(X\) and all its subvarieties are of general type and the canonical divisor \(K_X\) is ample at smooth points and Kawamata log terminal points of \(X\), provided that \(K_X\) is \(\mathbb {Q}\)-Cartier, no Calabi–Yau variety is algebraic Lang hyperbolic and a weak abundance conjecture holds.
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Acknowledgments
We would like to thank the referee for very careful reading, many suggestions to improve the paper and the insistence on clarity of exposition, and Kenji Matsuki for bringing [15] to our attention where Theorem 1.4 was proved in dimension 3. The last named-author is partially supported by an ARF of NUS.
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Hu, F., Meng, S. & Zhang, DQ. Ampleness of canonical divisors of hyperbolic normal projective varieties. Math. Z. 278, 1179–1193 (2014). https://doi.org/10.1007/s00209-014-1351-1
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DOI: https://doi.org/10.1007/s00209-014-1351-1