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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bauer, T., Campana, F., Eckl, T., Kebekus, S., Peternell, T., Rams, S., Szemberg, T., Wotzlaw, L.: A Reduction Map for Nef Line Bundles, Complex Geometry (Göttingen, 2000). Springer, Berlin, pp. 27–36 (2002)
Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23, 405–468 (2010)
Boucksom, S., Demailly, J.-P., Paun, M., Peternell, T.: The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. J. Algebr. Geom. 22(2), 201–248 (2013)
Fujino, O.: Fundamental theorems for the log minimal model program. Publ. Res. Inst. Math. Sci. 47(3), 727–789 (2011)
Fujino, O.: On maximal Albanese dimensional varieties. Proc. Jpn Acad. Ser. A Math. Sci. 89(8), 92–95 (2013)
Hacon, C.D., McKernan, J.: On Shokurov’s rational connectedness conjecture. Duke Math. J. 138(1), 119–136 (2007)
Heier, G., Steven, SYLu: On the canonical line bundle and negative holomorphic sectional curvature. Math. Res. Lett. 17(6), 1101–1110 (2010)
Kawamata, Y.: Characterization of abelian varieties. Compos. Math. 43(2), 253–276 (1981)
Kawamata, Y.: Minimal models and the Kodaira dimension of algebraic fiber spaces. J. Reine Angew. Math. 363, 1–46 (1985)
Kawamata, Y.: Pluricanonical systems on minimal algebraic varieties. Invent. Math. 79(3), 567–588 (1985)
Kawamata, Y.: Crepant blowing-ups of \(3\)-dimensional canonical singularities and its application to degeneration of surfaces. Ann. Math. 127, 93–163 (1988)
Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math., vol. 134. Cambridge University Press, Cambridge (1998)
Lang, S.: Hyperbolic and Diophantine analysis. Bull. Am. Math. Soc. (N.S.) 14, 159–205 (1986)
Lu, S.S.Y., Zhang, D.-Q.: Positivity criteria for log canonical divisors and hyperbolicity. arXiv:1207.7346
Matsuki, K.: An approach to the abundance conjecture for 3-folds. Duke Math. J. 61(1), 207–220 (1990)
Mori, S., Mukai, S.: The uniruledness of the moduli space of curves of genus 11, Algebraic Geometry (Tokyo/Kyoto, 1982). Lecture Notes in Math., vol. 1016, Springer, Berlin, pp. 334–353 (1983)
Nakayama, N.: Zariski-decomposition and abundance. MSJ Memoirs, vol. 14. Mathematical Society of Japan, Tokyo (2004)
Peternell, T.: Calabi–Yau manifolds and a conjecture of Kobayashi. Math. Z. 207(2), 305–318 (1991)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-014-1351-1