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On Fujita’s log spectrum conjecture

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Abstract

We prove Fujita’s log spectrum conjecture. It follows from the ACC of a suitable set of pseudo-effective thresholds.

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References

  1. Alexeev, V., Hacon, C., Kawamata, Y.: Termination of (many) 4-dimensional log flips. Invent. Math. 168(2), 433–448 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Araujo, C.: The cone of pseudo-effective divisors on log varieties after Batyrev. Math. Zeit. 264(1), 179–193 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Batyrev, V., Tschinkel, Y.: Tamagawa numbers of polarized algebraic varieties. Astérisque 251, 299–340 (1998)

    MathSciNet  MATH  Google Scholar 

  4. Beltrametti, M.C., Sommese, A.J.: The adjunction theory of complex projective varieties. In: de Gruyter Expositions in Mathematics, vol. 16. Walter de Gruyter & Co., Berlin (1995)

  5. Birkar, C., Cascini, P., Hacon, C., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23, 405–468 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Demailly, J.-P., Hacon, C., Păun, M.: Extension theorems, non-vanishing and the existence of good minimal models (2010). arXiv:1012.0493v2

  7. Di Cerbo, G.: Uniform bounds for the Iitaka fibration. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13(4), 1133–1143 (2014)

  8. Fujino, O.: Effective base point free theorem for log canonical pairs, II. Angehrn-Siu type theorems. Michigan Math. J. 59(2), 303–312 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fujino, O., Mori, S.: A canonical bundle formula. J. Differ. Geom. 56(1), 167–188 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Fujita, T.: On Kodaira energy and adjoint reduction of polarized manifolds. Manuscripta Math. 76(1), 59–84 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fujita, T.: On Kodaira energy and classification of polarized varieties. Sugaku Expositions 8(2), 183–196 (1995)

    MathSciNet  Google Scholar 

  12. Fujita, T.: On Kodaira energy of polarized log varieties. J. Math. Soc. Japan 48(1), 1–12 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fujita, T.: On Kodaira energy and adjoint reduction of polarized threefolds. Manuscripta Math. 94(2), 211–229 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gongyo, Y.: Remarks on the non-vanishing conjecture (2012). arXiv:1201.1128v2

  15. Hacon, C., McKernan, J., Xu, C.: ACC for log canonical thresholds (2012). arXiv:1208.4150v1

  16. Jiang, X.: On the pluricanonical maps of varieties of intermediate Kodaira dimension (2012). arXiv:1012.3817v2

  17. Kollár, J.: Effective base point freeness. Math. Ann. 296, 595–605 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kollár, J.: Singularities of pairs. In: Algebraic Geometry, Santa Cruz 1995, Proc. Symp. Pure Math, vol. 62, pp. 221–287. Amer. Math. Soc., Providence (1997)

  19. Kollár, J.: Singularities of the minimal model program, with a collaboration of Sándor Kovács. In: Cambridge Tracts in Mathematics, vol. 200. Cambridge University Press, Cambridge (2013)

  20. Kollár, J., Mori, S.: Birational geometry of algebraic varieties. In: Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998)

  21. Lohmann, D.: Families of canonically polarized manifolds over log Fano varieties (2011). arXiv:1107.4545v1

  22. Tschinkel, Y.: Fujita’s program and rational points. In: Higher Dimensional Varieties and Rational Points (Budapest, 2001). Bolyai Soc. Math. Stud., vol. 12, pp. 283–310. Springer, Berlin (2003)

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Acknowledgments

I would like to express my gratitude to Professor János Kollár for his constant support and many constructive comments. I also would like to thank Professor Christopher Hacon for generously sharing his insight with me. In particular, his ideas helped me in the proof of Theorem 1.2 when M is not globally generated. Part of this work was written while the author visited the University of Utah.

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Correspondence to Gabriele Di Cerbo.

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Di Cerbo, G. On Fujita’s log spectrum conjecture. Math. Ann. 366, 447–457 (2016). https://doi.org/10.1007/s00208-015-1333-6

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  • DOI: https://doi.org/10.1007/s00208-015-1333-6

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