Publications mathématiques de l'IHÉS

, Volume 114, Issue 1, pp 87–169 | Cite as

Differential forms on log canonical spaces

  • Daniel Greb
  • Stefan Kebekus
  • Sándor J Kovács
  • Thomas Peternell


The present paper is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities. In fact, a much more general theorem for log canonical pairs is established. The proof relies on vanishing theorems for log canonical varieties and on methods of the minimal model program. In addition, a theory of differential forms on dlt pairs is developed. It is shown that many of the fundamental theorems and techniques known for sheaves of logarithmic differentials on smooth varieties also hold in the dlt setting.

Immediate applications include the existence of a pull-back map for reflexive differentials, generalisations of Bogomolov-Sommese type vanishing results, and a positive answer to the Lipman-Zariski conjecture for klt spaces.


Differential Form Extension Theorem Exceptional Divisor Residue Sequence Canonical Singularity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Bar78]
    D. Barlet, Le faisceau \(\omega ^{\cdot }_{X}\) sur un espace analytique X de dimension pure, in Fonctions de plusieurs variables complexes III (Sém. François Norguet, 1975–1977), Lecture Notes in Math., vol. 670, pp. 187—204, Springer, Berlin, 1978. MR0521919 (80i:32037). CrossRefGoogle Scholar
  2. [BS95]
    M. C. Beltrametti and A. J. Sommese, The Adjunction Theory of Complex Projective Varieties, de Gruyter Expositions in Mathematics, vol. 16, de Gruyter, Berlin, 1995. 96f:14004. MATHGoogle Scholar
  3. [BCHM10]
    C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type, J. Am. Math. Soc., 23 (2010), 405–468. doi: 10.1090/S0894-0347-09-00649-3. MathSciNetMATHCrossRefGoogle Scholar
  4. [Cam04]
    F. Campana, Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble), 54 (2004), 499–630. MR2097416 (2006c:14013). MathSciNetMATHCrossRefGoogle Scholar
  5. [Car85]
    J. A. Carlson, Polyhedral resolutions of algebraic varieties, Trans. Am. Math. Soc., 292 (1985), 595–612. MR808740 (87i:14008). MATHCrossRefGoogle Scholar
  6. [{Cor}07]
    A. Corti et al., Flips for 3-Folds and 4-Folds, Oxford Lecture Series in Mathematics and Its Applications, vol. 35, Oxford University Press, Oxford, 2007. MR2352762 (2008j:14031). MATHCrossRefGoogle Scholar
  7. [Del70]
    P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, vol. 163, Springer, Berlin, 1970. 54 #5232. MATHGoogle Scholar
  8. [DB81]
    P. Du Bois, Complexe de de Rham filtré d’une variété singulière, Bull. Soc. Math. Fr., 109 (1981), 41–81. MR613848 (82j:14006). MATHGoogle Scholar
  9. [DBJ74]
    P. Du Bois 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. MR0376678 (51 #12853). MathSciNetMATHGoogle Scholar
  10. [dJS04]
    A. J. de Jong and J. Starr, Cubic fourfolds and spaces of rational curves, Ill. J. Math., 48 (2004), 415–450. MR2085418 (2006e:14007). MATHGoogle Scholar
  11. [EV82]
    H. Esnault and E. Viehweg, Revêtements cycliques, in Algebraic threefolds (Varenna, 1981), Lecture Notes in Mathematics, vol. 947, pp. 241–250. Springer, Berlin, 1982. MR0672621 (84m:14015). CrossRefGoogle Scholar
  12. [EV90]
    H. Esnault and E. Viehweg, Effective bounds for semipositive sheaves and for the height of points on curves over complex function fields, Compos. Math., 76 (1990), 69–85. Algebraic geometry (Berlin, 1988). MR1078858 (91m:14038). MathSciNetMATHGoogle Scholar
  13. [EV92]
    H. Esnault and E. Viehweg, Lectures on Vanishing Theorems, DMV Seminar, vol. 20, Birkhäuser, Basel, 1992. MR1193913 (94a:14017). MATHCrossRefGoogle Scholar
  14. [FGI+05]
    B. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure, and A. Vistoli, Fundamental Algebraic Geometry, Mathematical Surveys and Monographs, vol. 123, American Mathematical Society, Providence, 2005. Grothendieck’s FGA explained. MR2222646 (2007f:14001). MATHGoogle Scholar
  15. [Fle88]
    H. Flenner, Extendability of differential forms on nonisolated singularities, Invent. Math., 94 (1988), 317–326. MR958835 (89j:14001). MathSciNetMATHCrossRefGoogle Scholar
  16. [Fog69]
    J. Fogarty, Invariant Theory, W. A. Benjamin, Inc., New York, 1969. MR0240104 (39 #1458). MATHGoogle Scholar
  17. [God73]
    R. Godement, Topologie algébrique et théorie des faisceaux, Hermann, Paris, 1973. Troisième édition revue et corrigée, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XIII, Actualités Scientifiques et Industrielles, No. 1252. MR0345092 (49 #9831). MATHGoogle Scholar
  18. [Gra72]
    H. Grauert, Über die Deformation isolierter Singularitäten analytischer Mengen, Invent. Math., 15 (1972), 171–198. MR0293127 (45 #2206). MathSciNetMATHCrossRefGoogle Scholar
  19. [GR70]
    H. Grauert and O. Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math., 11 (1970), 263–292. MR0302938 (46 #2081). MathSciNetMATHCrossRefGoogle Scholar
  20. [GKK10]
    D. Greb, S. Kebekus, and S. J. Kovács, Extension theorems for differential forms, and Bogomolov-Sommese vanishing on log canonical varieties, Compos. Math., 146 (2010), 193–219. doi: 10.1112/S0010437X09004321. MathSciNetMATHCrossRefGoogle Scholar
  21. [GKKP10]
    D. Greb, S. Kebekus, S. J. Kovács, and T. Peternell, Differential forms on log canonical varieties, Extended version of the present paper, including more detailed proofs and color figures. arXiv:1003.2913, March 2010.
  22. [GLS07]
    G.-M. Greuel, C. Lossen, and E. Shustin, Introduction to Singularities and Deformations, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR2290112 (2008b:32013). MATHGoogle Scholar
  23. [Gre80]
    G.-M. Greuel, Dualität in der lokalen Kohomologie isolierter Singularitäten, Math. Ann., 250 (1980), 157–173. MR582515 (82e:32009) MathSciNetMATHGoogle Scholar
  24. [Gro60]
    A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math., 4 (1960), 228. MR0217083 (36 #177a). CrossRefGoogle Scholar
  25. [Gro71]
    A. Grothendieck, Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, vol. 224. Springer, Berlin, 1971, Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1), Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud, MR0354651 (50 #7129). MATHGoogle Scholar
  26. [GNPP88]
    F. Guillén, V. Navarro Aznar, P. Pascual Gainza, and F. Puerta, Hyperrésolutions cubiques et descente cohomologique, Lecture Notes in Mathematics, vol. 1335, Springer, Berlin, 1988, Papers from the Seminar on Hodge-Deligne Theory held in Barcelona, 1982. MR972983 (90a:14024). MATHGoogle Scholar
  27. [HK10]
    C. D. Hacon and S. J. Kovács, Classification of Higher Dimensional Algebraic Varieties, Oberwolfach Seminars, Birkhäuser, Boston, 2010. MATHCrossRefGoogle Scholar
  28. [HM07]
    C. D. Hacon and J. McKernan, On Shokurov’s rational connectedness conjecture, Duke Math. J., 138 (2007), 119–136. MR2309156 (2008f:14030). MathSciNetMATHCrossRefGoogle Scholar
  29. [Har77]
    R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer, New York, 1977. MR0463157 (57 #3116). MATHGoogle Scholar
  30. [HM89]
    H. Hauser and G. Müller, The trivial locus of an analytic map germ, Ann. Inst. Fourier (Grenoble), 39 (1989), 831–844. MR1036334 (91m:32035). MathSciNetMATHCrossRefGoogle Scholar
  31. [Hei91]
    P. Heinzner, Geometric invariant theory on Stein spaces, Math. Ann., 289 (1991), 631–662. MR1103041 (92j:32116). MathSciNetMATHCrossRefGoogle Scholar
  32. [Hir62]
    H. Hironaka, On resolution of singularities (characteristic zero), in Proc. Int. Cong. Math., 1962, pp. 507–521. Google Scholar
  33. [Hoc75]
    M. Hochster, The Zariski-Lipman conjecture for homogeneous complete intersections, Proc. Am. Math. Soc., 49 (1975), 261–262. MR0360585 (50 #13033). MathSciNetMATHGoogle Scholar
  34. [Hol61]
    H. Holmann, Quotienten komplexer Räume, Math. Ann., 142 (1960/1961), 407–440. MR0120665 (22 #11414). MathSciNetCrossRefGoogle Scholar
  35. [HL97]
    D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, Aspects of Mathematics, vol. E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR1450870 (98g:14012). MATHGoogle Scholar
  36. [JK09a]
    K. Jabbusch and S. Kebekus, Families over special base manifolds and a conjecture of Campana, Math. Z., to appear. doi: 10.1007/s00209-010-0758-6, arXiv:0905.1746, May 2009.
  37. [JK09b]
    K. Jabbusch and S. Kebekus, Positive sheaves of differentials on coarse moduli spaces, Ann. Inst. Fourier (Grenoble), to appear. arXiv:0904.2445, April 2009.
  38. [Kaw88]
    Y. Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. Math., 127 (1988), 93–163. MR924674 (89d:14023). MathSciNetMATHCrossRefGoogle Scholar
  39. [KK07]
    S. Kebekus and S. J. Kovács, The structure of surfaces mapping to the moduli stack of canonically polarized varieties, preprint (July 2007). arXiv:0707.2054.
  40. [KK08]
    S. Kebekus and S. J. Kovács, Families of canonically polarized varieties over surfaces, Invent. Math., 172 (2008), 657–682. doi: 10.1007/s00222-008-0128-8. MR2393082 MathSciNetMATHCrossRefGoogle Scholar
  41. [KK10a]
    S. Kebekus and S. J. Kovács, The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties, Duke Math. J., 155 (2010), 1–33. MR2730371. arXiv:0812.2305 MathSciNetMATHCrossRefGoogle Scholar
  42. [Kol]
    J. Kollár, Algebraic groups acting on schemes, Undated, unfinished manuscript. Available on the author’s website at
  43. [Kol96]
    J. Kollár, Rational Curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, A Series of Modern Surveys in Mathematics, vol. 32, Springer, Berlin, 1996. MR1440180 (98c:14001). Google Scholar
  44. [Kol07]
    J. Kollár, Lectures on Resolution of Singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, 2007. MR2289519. MATHGoogle Scholar
  45. [KK10b]
    J. Kollár and S. J. Kovács, Log canonical singularities are Du Bois, J. Am. Math. Soc., 23 (2010), 791–813. doi: 10.1090/S0894-0347-10-00663-6. MATHCrossRefGoogle Scholar
  46. [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. 2000b:14018. MATHCrossRefGoogle Scholar
  47. [{Kol}92]
    J. Kollár et al., Flips and Abundance for Algebraic Threefolds, Astérisque, No. 211, 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 (1992). MR1225842 (94f:14013). MATHGoogle Scholar
  48. [KS09]
    S. J. Kovács and K. Schwede, Hodge theory meets the minimal model program: a survey of log canonical and Du Bois singularities, in Topology of Stratified, pp. 51–94, 2001. Google Scholar
  49. [Lau73]
    H. B. Laufer, Taut two-dimensional singularities, Math. Ann., 205 (1973), 131–164. MR0333238 (48 #11563). MathSciNetMATHCrossRefGoogle Scholar
  50. [Lip65]
    J. Lipman, Free derivation modules on algebraic varieties, Am. J. Math., 87 (1965), 874–898. MR0186672 (32 #4130). MathSciNetMATHCrossRefGoogle Scholar
  51. [Loj64]
    S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 18 (1964), 449–474. MR0173265 (30 #3478). MathSciNetMATHGoogle Scholar
  52. [ML95]
    S. Mac Lane, Homology, Classics in Mathematics, Springer, Berlin, 1995, Reprint of the 1975 edition. MR1344215 (96d:18001). MATHGoogle Scholar
  53. [{Mas}1899]
    H. Maschke, Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind, Math. Ann., 52 (1899), 363–368. MR1511061. MathSciNetMATHCrossRefGoogle Scholar
  54. [Nam01]
    Y. Namikawa, Extension of 2-forms and symplectic varieties, J. Reine Angew. Math., 539 (2001), 123–147. MR1863856 (2002i:32011). MathSciNetMATHCrossRefGoogle Scholar
  55. [OSS80]
    C. Okonek, M. Schneider, and H. Spindler, Vector Bundles on Complex Projective Spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, 1980. MR561910 (81b:14001). MATHCrossRefGoogle Scholar
  56. [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 [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 52, Springer, Berlin, 2008. MR2393625. MATHGoogle Scholar
  57. [Pri67]
    D. Prill, Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J., 34 (1967), 375–386. MR0210944 (35 #1829). MathSciNetMATHCrossRefGoogle Scholar
  58. [Rei80]
    M. Reid, Canonical 3-folds, in A. Beauville (ed.) Algebraic Geometry (Angers, 1979), Sijthoff & Noordhoff, Alphen aan den Rijn, 1980. Google Scholar
  59. [Rei87]
    M. Reid, Young person’s guide to canonical singularities, in Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, pp. 345–414, Amer. Math. Soc, Providence, 1987. MR927963 (89b:14016). Google Scholar
  60. [SS72]
    G. Scheja and U. Storch, Differentielle Eigenschaften der Lokalisierungen analytischer Algebren, Math. Ann., 197 (1972), 137–170. MR0306172 (46 #5299). MathSciNetMATHCrossRefGoogle Scholar
  61. [Sei50]
    A. Seidenberg, The hyperplane sections of normal varieties, Trans. Am. Math. Soc., 69 (1950), 357–386. MR0037548 (12,279a). MathSciNetMATHGoogle Scholar
  62. [Sha94]
    I. R. Shafarevich, Basic Algebraic Geometry. 1, 2nd edn., Springer, Berlin, 1994. Varieties in projective space, Translated from the 1988 Russian edition and with notes by Miles Reid. MR1328833 (95m:14001). MATHCrossRefGoogle Scholar
  63. [SvS85]
    J. Steenbrink and D. van Straten, Extendability of holomorphic differential forms near isolated hypersurface singularities, Abh. Math. Semin. Univ. Hamb., 55 (1985), 97–110. MR831521 (87j:32025). MATHCrossRefGoogle Scholar
  64. [Ste85]
    J. H. M. Steenbrink, Vanishing theorems on singular spaces, Astérisque, 130 (1985), 330–341, Differential systems and singularities (Luminy, 1983). MR804061 (87j:14026). MathSciNetGoogle Scholar
  65. [Sza94]
    E. Szabó, Divisorial log terminal singularities, J. Math. Sci. Univ. Tokyo, 1 (1994), 631–639. MR1322695 (96f:14019). MathSciNetMATHGoogle Scholar
  66. [Tei77]
    B. Teissier, The hunting of invariants in the geometry of discriminants, in Real and Complex Singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), pp. 565–678, Sijthoff and Noordhoff, Alphen aan den Rijn, 1977. MR0568901 (58 #27964). Google Scholar
  67. [Ver76]
    J.-L. Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math., 36 (1976), 295–312. MR0481096 (58 #1242) MathSciNetMATHCrossRefGoogle Scholar
  68. [Vie10]
    E. Viehweg, Compactifications of smooth families and of moduli spaces of polarized manifolds, Ann. Math., 172 (2010), 809–910. arXiv:math/0605093. MR2680483. MathSciNetMATHCrossRefGoogle Scholar
  69. [VZ02]
    E. Viehweg and K. Zuo, Base spaces of non-isotrivial families of smooth minimal models, in Complex Geometry (Göttingen, 2000), pp. 279–328, Springer, Berlin, 2002. MR1922109 (2003h:14019). CrossRefGoogle Scholar
  70. [Wah85]
    J. M. Wahl, A characterization of quasihomogeneous Gorenstein surface singularities, Compos. Math., 55 (1985), 269–288. MR799816 (87e:32013) MathSciNetMATHGoogle Scholar

Copyright information

© IHES and Springer-Verlag 2011

Authors and Affiliations

  • Daniel Greb
    • 1
  • Stefan Kebekus
    • 1
  • Sándor J Kovács
    • 2
  • Thomas Peternell
    • 3
  1. 1.Mathematisches InstitutAlbert-Ludwigs-Universität FreiburgFreiburg im BreisgauGermany
  2. 2.Department of MathematicsUniversity of WashingtonSeattleUSA
  3. 3.Matematisches InstitutUniversität BayreuthBayreuthGermany

Personalised recommendations