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Hyperbolicity for log canonical pairs and the cone theorem

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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.

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Acknowledgements

The author wants to thank his Ph.D. advisor, Professor James McKernan, for offering constant support and encouragement during the development of this project. He thanks John Lesieutre and Tiankai Liu for useful discussions and exchange of ideas, as well as Professors Gabriele di Cerbo, Antonio Lerario, and Jacopo Stoppa for reading preliminary versions of this work and for their encouragement and Prof. Steven Lu for his interest in this work. He also wishes to thank the anonymous referees for the beneficial suggestions related to the structure of the paper. Part of this work was done while the author was supported by NSF DMS #0701101 and #1200656. The author would also like to thank MIT where he was a graduate student when most of this work was completed. During the final revision of this work the author was visiting Professor Jacopo Stoppa at SISSA, Trieste. He would to thank Jacopo Stoppa and SISSA for providing such a congenial place to work. The visit was supported by funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 307119. This work is part of the author’s Ph.D. thesis which received the 2016 Federigo Enriques Prize, awarded by the Unione Matematica Italiana and Centro Studi Enriques. The author wishes to thank these institutions for such honor.

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  1. EPFL SB MATH MATH-GE, MA B1 497 (Bâtiment MA), Station 8, 1015, Lausanne, Switzerland

    Roberto Svaldi

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  1. Roberto Svaldi
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Svaldi, R. Hyperbolicity for log canonical pairs and the cone theorem. Sel. Math. New Ser. 25, 67 (2019). https://doi.org/10.1007/s00029-019-0512-9

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