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Existence of log canonical closures

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Let f:X→U be a projective morphism of normal varieties and (X,Δ) a dlt pair. We prove that if there is an open set U 0⊂U, such that (X,Δ)× U U 0 has a good minimal model over U 0 and the images of all the non-klt centers intersect U 0, then (X,Δ) has a good minimal model over U. As consequences we show the existence of log canonical compactifications for open log canonical pairs, and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness.

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References

  1. Abramovich, D., Karu, K.: Weak semistable reduction in characteristic 0. Invent. Math. 139(2), 241–273 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambro, F.: Quasi-log varieties. Tr. Mat. Inst. Steklova 240, 220–239 (2003). Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry; translation in Proc. Steklov Inst. Math. 240(1), 214–233 (2003)

    MathSciNet  Google Scholar 

  3. Ambro, F.: Shokurov’s boundary property. J. Differ. Geom. 67(2), 229–255 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Ambro, F.: A semiampleness criterion. J. Math. Sci. Univ. Tokyo 12(3), 445–466 (2005)

    MathSciNet  MATH  Google Scholar 

  5. Birkar, C.: Existence of log canonical flips and a special LMMP. arXiv:1104.4981

  6. Birkar, C., Cascini, P., Hacon, C., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Corti, A.: Flips for 3-Folds and 4-Folds. Oxford Lecture Ser. Math. Appl., vol. 35. Oxford University Press, Oxford (2007)

    Book  MATH  Google Scholar 

  8. Corti, A., Lazic, V.: New outlook on Mori Theory II. arXiv:1005.0614v2

  9. Fujino, O.: Abundance theorem for semi log canonical threefolds. Duke Math. J. 102(3), 513–532 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fujino, O.: Higher direct images of log canonical divisors. J. Differ. Geom. 66(3), 453–479 (2004)

    MathSciNet  MATH  Google Scholar 

  11. Fujino, O.: Introduction to the log minimal model program for log canonical pairs. Preprint (2008). http://www.math.kyoto-u.ac.jp/~fujino/MMP21-s.pdf

  12. Fujino, O.: On Kawamata’s theorem. In: Classification of Algebraic Varieties. EMS Ser. of Congr. Rep., pp. 305–315. Eur. Math. Soc., Zürich (2011). Available on arXiv:0910.1156

    Chapter  Google Scholar 

  13. Fujino, O.: Base point free theorems—saturation, B-divisors and canonical bundle formula. Algebra Number Theory, to appear. arXiv:math/0508554v3

  14. Fujino, O., Gongyo, Y.: Log pluricanonical representations and abundance conjecture. arXiv:1104.0361

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

    MathSciNet  MATH  Google Scholar 

  16. Gongyo, Y.: On the minimal model theory for dlt pairs of numerical log Kodaira dimension zero. Math. Res. Lett. 18(5), 991–1000 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Gongyo, Y.: Abundance theorem for numerical trivial log canonical divisors of semi-log canonical pairs. arXiv:1005.2796

  18. Kawamata, Y.: Pluricanonical systems on minimal algebraic varieties. Invent. Math. 79(3), 567–588 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kawamata, Y.: On the length of an extremal rational curve. Invent. Math. 105(3), 609–611 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kawamata, Y.: Subadjunction of log canonical divisors. II. Am. J. Math. 120(5), 893–899 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kawamata, Y.: Remarks on the cone of divisors. In: Classification of Algebraic Varieties. EMS Ser. Congr. Rep., pp. 317–325. Eur. Math. Soc., Zürich (2011)

    Chapter  Google Scholar 

  22. Kawamata, Y.: Variation of mixed Hodge structures and the positivity for algebraic fiber spaces. arXiv:1008.1489

  23. Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings. I. Lecture Notes in Mathematics, vol. 339. Springer, Berlin (1973)

    MATH  Google Scholar 

  24. Kollár, J.: Kodaira’s canonical bundle formula and adjunction. In: Flips for 3-Folds and 4-Folds. Oxford Lecture Ser. Math. Appl., vol. 35, pp. 134–162. Oxford University Press, Oxford (2007)

    Chapter  Google Scholar 

  25. Kollár, J.: Two examples of surfaces with normal crossing singularities. Sci. China Ser. A 54(8), 1707–1712 (2011)

    Article  MATH  Google Scholar 

  26. Kollár, J.: Quotients by finite equivalence relations. In: Current Developments in Algebraic Geometry. Math. Sci. Res. Inst. Publ., vol. 59, pp. 227–256. Cambridge University Press, Cambridge (2011). arXiv:0812.3608

    Google Scholar 

  27. Kollár, J.: Moduli of varieties of general type. arXiv:1008.0621

  28. Kollár, J.: Seminormal log centers and deformations of pairs. arXiv:1103.0528v1

  29. Kollár, J.: Moduli of Higher Dimensional Varieties. Book to appear. Available in March 2010 at http://www.math.princeton.edu/~kollar/

  30. Kollár, J., Kovács, S.: Log canonical singularities are Du Bois. J. Am. Math. Soc. 23(3), 791–813 (2010)

    Article  MATH  Google Scholar 

  31. Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998). With the collaboration of Clemens, C.H. and Corti, A. Translated from the 1998 Japanese original

    Book  MATH  Google Scholar 

  32. Lai, C.: Varieties fibered by good minimal models. Math. Ann. 350(3), 533–547 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Nakayama, N.: Zariski-Decomposition and Abundance. MSJ Memoirs, vol. 14. Mathematical Society of Japan, Tokyo (2004)

    MATH  Google Scholar 

  34. Shokurov, V.: Three-dimensional log perestroikas. Izv. Akad. Nauk SSSR, Ser. Mat. 56(1), 105–203 (1992)

    Google Scholar 

  35. Shokurov, V.: 3-fold log models. Algebraic geometry, 4. J. Math. Sci. 81(3), 2667–2699 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, UT, 84112, USA

    Christopher D. Hacon & Chenyang Xu

  2. Beijing International Center of Mathematics Research, 5 Yiheyuan Road, Haidian District, Beijing, 100871, China

    Chenyang Xu

Authors
  1. Christopher D. Hacon
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  2. Chenyang Xu
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Corresponding author

Correspondence to Christopher D. Hacon.

Additional information

The first author was partially supported by NSF research grant no. 0757897, the second author was partially supported by NSF research grant no. 0969495. We are grateful to O. Fujino, J. Kollár and J. McKernan for many useful comments and suggestions. We are also in debt to J. Kollár for allowing us to use the materials of [29] in Sect. 3.

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Hacon, C.D., Xu, C. Existence of log canonical closures. Invent. math. 192, 161–195 (2013). https://doi.org/10.1007/s00222-012-0409-0

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  • DOI: https://doi.org/10.1007/s00222-012-0409-0

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