Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Log canonical singularities are Du Bois

Authors: János Kollár and Sándor J Kovács
Journal: J. Amer. Math. Soc. 23 (2010), 791-813
MSC (2010): Primary 14J17, 14B07, 14E30, 14D99
Published electronically: February 22, 2010
Previous version: Original version posted February 12, 2010
Corrected version: Current version corrects publisher's introduction of inconsistent rendering of script O.
MathSciNet review: 2629988
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A recurring difficulty in the Minimal Model Program is that while log terminal singularities are quite well behaved (for instance, they are rational), log canonical singularities are much more complicated; they need not even be Cohen-Macaulay. The aim of this paper is to prove that log canonical singularities are Du Bois. The concept of Du Bois singularities, introduced by Steenbrink, is a weakening of rationality. We also prove flatness of the cohomology sheaves of the relative dualizing complex of a projective family with Du Bois fibers. This implies that each connected component of the moduli space of stable log varieties parametrizes either only Cohen-Macaulay or only non-Cohen-Macaulay objects.

References [Enhancements On Off] (What's this?)

  • [AK80] A. B. Altman and S. L. Kleiman, Compactifying the Picard scheme, Adv. in Math. 35 (1980), no. 1, 50-112. MR 555258 (81f:14025a)
  • [Amb03] F. Ambro, Quasi-log varieties, Tr. Mat. Inst. Steklova 240 (2003), no. Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 220-239. MR 1993751 (2004f:14027)
  • [AM69] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802 (39:4129)
  • [BCHM06] C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type, Journal of the AMS 23 (2010), 405-468.
  • [CD89] F. R. Cossec and I. V. Dolgachev, Enriques surfaces. I, Progress in Mathematics, vol. 76, Birkhäuser Boston Inc., Boston, MA, 1989. MR 986969 (90h:14052)
  • [Du81] P. Du Bois, Complexe de de Rham filtré d'une variété singulière, Bull. Soc. Math. France 109 (1981), no. 1, 41-81. MR 613848 (82j:14006)
  • [DJ74] P. Dubois and P. Jarraud, Une propriété de commutation au changement de base des images directes supérieures du faisceau structural, C. R. Acad. Sci. Paris Sér. A 279 (1974), 745-747. MR 0376678 (51:12853)
  • [Fuj00] O. Fujino, Abundance theorem for semi log canonical threefolds, Duke Math. J. 102 (2000), no. 3, 513-532. MR 1756108 (2001c:14032)
  • [Fuj08] O. Fujino, Introduction to the log minimal model program for log canonical pairs, unpublished manuscript, version 4.02, December 19, 2008.
  • [GNPP88] F. Guillén, V. Navarro Aznar, P. Pascual Gainza, and F. Puerta, Hyperrésolutions cubiques et descente cohomologique, Lecture Notes in Mathematics, vol. 1335, Springer-Verlag, Berlin, 1988. Papers from the Seminar on Hodge-Deligne Theory held in Barcelona, 1982. MR 972983 (90a:14024)
  • [Har66] R. Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin, 1966. MR 0222093 (36:5145)
  • [Har77] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
  • [Ish85] S. Ishii, On isolated Gorenstein singularities, Math. Ann. 270 (1985), no. 4, 541-554. MR 776171 (86j:32024)
  • [Ish87a] S. Ishii, Du Bois singularities on a normal surface, Complex analytic singularities, Adv. Stud. Pure Math., vol. 8, North-Holland, Amsterdam, 1987, pp. 153-163. MR 894291 (88f:14033)
  • [Ish87] S. Ishii, Isolated $ Q$-Gorenstein singularities of dimension three, Complex analytic singularities, Adv. Stud. Pure Math., vol. 8, North-Holland, Amsterdam, 1987, pp. 165-198. MR 894292 (89d:32016)
  • [Kaw07] M. Kawakita, Inversion of adjunction on log canonicity, Invent. Math. 167 (2007), no. 1, 129-133. MR 2264806 (2008a:14025)
  • [Kaw97] Y. Kawamata, On Fujita's freeness conjecture for $ 3$-folds and $ 4$-folds, Math. Ann. 308 (1997), no. 3, 491-505. MR 1457742 (99c:14008)
  • [Kol86a] J. Kollár, Higher direct images of dualizing sheaves. I, Ann. of Math. (2) 123 (1986), no. 1, 11-42. MR 825838 (87c:14038)
  • [Kol86b] J. Kollár, Higher direct images of dualizing sheaves. II, Ann. of Math. (2) 124 (1986), no. 1, 171-202. MR 847955 (87k:14014)
  • [Kol95] J. Kollár, Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1995. MR 1341589 (96i:14016)
  • [Kol08] J. Kollár, Exercises in the birational geometry of algebraic varieties, preprint, 2008. arXiv:0809.2579v2 [math.AG]
  • [KM98] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original. MR 1658959 (2000b:14018)
  • [Kol92] J. Kollár et. al, Flips and abundance for algebraic threefolds, Société Mathématique de France, Paris, 1992. Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991. Astérisque No. 211 (1992). MR 1225842 (94f:14013)
  • [Kov99] S. J. Kovács, Rational, log canonical, Du Bois singularities: On the conjectures of Kollár and Steenbrink, Compositio Math. 118 (1999), no. 2, 123-133. MR 1713307 (2001g:14022)
  • [Kov00a] S. J. Kovács, A characterization of rational singularities, Duke Math. J. 102 (2000), no. 2, 187-191. MR 1749436 (2002b:14005)
  • [Kov00b] S. J. Kovács, Rational, log canonical, Du Bois singularities. II. Kodaira vanishing and small deformations, Compositio Math. 121 (2000), no. 3, 297-304. MR 1761628 (2001m:14028)
  • [KS09] S. J. Kovács and K. E. Schwede, Hodge theory meets the minimal model program: A survey of log canonical and Du Bois singularities, 2009. arXiv:0909.0993v1 [math.AG]
  • [KSS08] S. J. Kovács, K. E. Schwede, and K. E. Smith, The canonical sheaf of Du Bois singularities, to appear in Advances in Math., 2008. arXiv:0801.1541v1 [math.AG]
  • [PS08] C. A. M. Peters and J. H. M. Steenbrink, Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 52, Springer-Verlag, Berlin, 2008. MR 2393625
  • [Sai00] M. Saito, Mixed Hodge complexes on algebraic varieties, Math. Ann. 316 (2000), no. 2, 283-331. MR 1741272 (2002h:14012)
  • [Sch06] K. Schwede, On Du Bois and F-injective singularities, Ph.D. thesis, University of Washington, 2006.
  • [Sch07] K. Schwede, A simple characterization of Du Bois singularities, Compos. Math. 143 (2007), no. 4, 813-828. MR 2339829
  • [Sch08] K. Schwede, Centers of F-purity, preprint, 2008. arXiv:0807.1654v3 [math.AC]
  • [Sch09] K. Schwede, $ F$-injective singularities are Du Bois, Amer. J. Math. 131 (2009), no. 2, 445-473. MR 2503989
  • [Ste83] J. H. M. Steenbrink, Mixed Hodge structures associated with isolated singularities, Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 513-536. MR 713277 (85d:32044)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 14J17, 14B07, 14E30, 14D99

Retrieve articles in all journals with MSC (2010): 14J17, 14B07, 14E30, 14D99

Additional Information

János Kollár
Affiliation: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544-1000

Sándor J Kovács
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195-4350

Received by editor(s): April 27, 2009
Received by editor(s) in revised form: November 30, 2009
Published electronically: February 22, 2010
Additional Notes: The first author was supported in part by NSF Grant DMS-0758275.
The second author was supported in part by NSF Grants DMS-0554697 and DMS-0856185 and the Craig McKibben and Sarah Merner Endowed Professorship in Mathematics
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society