Skip to main content
SpringerLink
Log in

Aspects of the Geometry of Varieties with Canonical Singularities

  • Conference paper
Foliation Theory in Algebraic Geometry

Part of the book series: Simons Symposia ((SISY))

  • 1205 Accesses

Abstract

This survey reports on recent developments regarding the global structure of complex varieties which occur in the minimal model program.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Vanishing of a local intersection cohomology group.

  2. 2.

    Observe that the sheaf \(\mathcal{T}_{X}\) need not be locally free. We refer the reader to [Anc82, Section 2] for the definition of ampleness for arbitrary coherent sheaves.

References

  1. Aprodu M, Kebekus S, Peternell T (2008) Galois coverings and endomorphisms of projective varieties. Math Z 260(2):431–449

    Article  MathSciNet  MATH  Google Scholar 

  2. Aluffi P (2006) Classes de Chern des variétés singulières, revisitées. C R Math Acad Sci Paris 342(6):405–410

    Article  MathSciNet  Google Scholar 

  3. Akizuki Y, Nakano S (1954) Note on Kodaira-Spencer’s proof of Lefschetz theorems. Proc Jpn Acad 30:266–272

    Article  MathSciNet  MATH  Google Scholar 

  4. Ancona V (1982) Faisceaux amples sur les espaces analytiques. Trans Am Math Soc 274(1):89–100

    Article  MathSciNet  MATH  Google Scholar 

  5. Arapura D (1988) Vanishing theorems for V -manifolds. Proc Am Math Soc 102(1):43–48

    MathSciNet  MATH  Google Scholar 

  6. Artin M (1969) Algebraic approximation of structures over complete local rings. Inst Hautes Études Sci Publ Math (36):23–58

    Article  MathSciNet  MATH  Google Scholar 

  7. Birkar C, Cascini P, Hacon CD, McKernan J (2010) Existence of minimal models for varieties of log general type. J AMS 23:405–468. DOI:10.1090/S0894-0347-09-00649-3

    Google Scholar 

  8. Beauville A (1983) Variétés Kähleriennes dont la première classe de Chern est nulle. J Differ Geom 18(4):755–782 (1984)

    Google Scholar 

  9. Bogomolov FA (1974) Kähler manifolds with trivial canonical class. Izv Akad Nauk SSSR Ser Mat 38:11–21

    MathSciNet  Google Scholar 

  10. Bando S, Siu Y-T (1994) Stable sheaves and Einstein-Hermitian metrics. Geometry and analysis on complex manifolds. World Scientific, River Edge, pp 39–50

    MATH  Google Scholar 

  11. Campana F (1995) Fundamental group and positivity of cotangent bundles of compact Kähler manifolds. J Algebraic Geom 4(3):487–502

    MathSciNet  MATH  Google Scholar 

  12. Campana F, Demailly J-P, Peternell T (2014) Rationally connected manifolds and semipositivity of the Ricci curvature. To appear in the proceedings of the conference ‘Recent Advances in Algebraic Geometry’, in honour of Robert Lazarsfeld 60th birthday. London Math Society, Lecture Notes Series, 417:71–91. arXiv:1210.2092., October 2012

  13. Cox DA, Little JB, Schenck HK (2011) Toric varieties, vol 124 of Graduate studies in mathematics. American Mathematical Society, Providence

    MATH  Google Scholar 

  14. Campana F, Peternell T (2011) Geometric stability of the cotangent bundle and the universal cover of a projective manifold. Bull Soc Math France 139(1):41–74

    MathSciNet  MATH  Google Scholar 

  15. Campana F, Paun M (2013) Orbifold generic semi-positivity: an application to families of canonically polarized manifolds. Preprint arXiv:1303.3169, March 2013

  16. Deligne P (1970) Équations différentielles á points singuliers réguliers. Lecture Notes in Mathematics, vol 163. Springer, Berlin

    Book  MATH  Google Scholar 

  17. Druel S (2014) The Zariski-Lipman conjecture for log canonical spaces. Bull London Math Society 46(4):827–835. arXiv:1301.5910, January 2013

    Google Scholar 

  18. Eyssidieux P, Guedj V, Zeriahi A (2009) Singular Kähler-Einstein metrics. J Am Math Soc 22(3):607–639

    Article  MathSciNet  MATH  Google Scholar 

  19. Esnault H, Viehweg E (1986) Logarithmic de Rham complexes and vanishing theorems. Invent Math 86(1):161–194

    Article  MathSciNet  MATH  Google Scholar 

  20. Flenner H (1988) Extendability of differential forms on nonisolated singularities. Invent Math 94(2):317–326

    Article  MathSciNet  MATH  Google Scholar 

  21. Graf P, Kovács SJ (2014) An optimal extension theorem for 1-forms and the Lipman-Zariski conjecture. Doc Math 19:815–830. arXiv:1301.7315, January 2013

  22. Graf P, Kovács SJ (2014) Potentially Du Bois spaces. Journal of Sing 8:117–134. arXiv:1401.4976, January 2014

  23. Greb D, Kebekus S, Kovács SJ (2010) Extension theorems for differential forms, and Bogomolov-Sommese vanishing on log canonical varieties. Compositio Math 146:193–219. DOI:10.1112/S0010437X09004321. A slightly extended version is available as arXiv:0808.3647

    Google Scholar 

  24. Greb D, Kebekus S, Kovács SJ, Peternell T (2011) Differential forms on log canonical spaces. Inst Hautes Études Sci Publ Math 114(1):87–169. DOI:10.1007/s10240-011-0036-0 An extended version with additional graphics is available as arXiv:1003.2913

  25. Greb D, Kebekus S, Peternell T (2011) Singular spaces with trivial canonical class. Preprint arXiv:1110.5250. To appear in Minimal models and extremal rays – proceedings of the conference in honor of Shigefumi Mori’s 60th birthday. Advanced Studies in Pure Mathematics. Kinokuniya Publishing, Tokyo

  26. Greb D, Kebekus S, Peternell T (2014) Reflexive differential forms on singular spaces – Geometry and Cohomology. J Reine Angew Math Published online 09Jan13. To appear in print. DOI:10.1515/crelle-2012-0097. J Reine Angew Math 697:57–89. arXiv:1202.3243

  27. Greb D, Kebekus S, Peternell T (2013) Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties. to appear in Duke Math. Journal. arXiv:1307.5718, July 2013

  28. Greb D, Kebekus S, Peternell T (2014) Movable curves and semistable sheaves. Int Math Research Notes. DOI: 10.1093/imrn/mv126. arXiv:1408.4308, July 2014

  29. Goresky M, MacPherson RD (1988) Stratified Morse theory, vol 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer, Berlin

    Google Scholar 

  30. Graf P (2015) Bogomolov-Sommese vanishing on log canonical pairs. J Reine Angew Math 0:1–34. Published electronically in May 2013. To appear in print. DOI:10.1515/crelle-2013-0031. J Reine Angew Math 702 (2015), 109–142. arXiv:math/1210.0421

  31. Hartshorne R (1977) Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer, New York

    Google Scholar 

  32. Hacon CD, McKernan J (2007) On Shokurov’s rational connectedness conjecture. Duke Math J 138(1):119–136

    Google Scholar 

  33. Hacon CD, McKernan J, Xu C (2012, to appear) ACC for log canonical thresholds. Preprint arXiv:1208.4150. Ann Math

  34. Hochster M (1975) The Zariski-Lipman conjecture for homogeneous complete intersections. Proc Am Math Soc 49:261–262

    MathSciNet  MATH  Google Scholar 

  35. Jörder C, Huber A (2014) Differential forms in the h-topology. Algebr Geom 1(4):449–478. arXiv:1305.7361. Algebraic Geom

  36. Jabbusch K, Kebekus S (2011) Families over special base manifolds and a conjecture of Campana. Math Z 269(3–4):847–878

    Article  MathSciNet  MATH  Google Scholar 

  37. Jörder C (2013) A weak version of the Lipman-Zariski conjecture. Preprint arXiv:1311.5141, November 2013

  38. Jörder C (2014) On the Poincaré Lemma for reflexive differential forms. PhD thesis, Albert-Ludwigs-Universität Freiburg, Math Z 278(3-4):893–899. January 2014. arXiv:1401.7495

  39. Kawamata Y (1985) Minimal models and the Kodaira dimension of algebraic fiber spaces. J Reine Angew Math 363:1–46. DOI:10.1515/crll.1985.363.1

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Kebekus S (2013) Differential forms on singular spaces, the minimal model program, and hyperbolicity of moduli stacks. In: Farkas G, Morrison I (eds) Handbook of Moduli, II, vol 25 of Advanced lectures in mathematics. International Press, Beijing

    Google Scholar 

  42. Kebekus S (2013) Pull-back morphisms for reflexive differential forms. Adv Math 245:78–112. Preprint arXiv:1210.3255

    Google Scholar 

  43. Kebekus S, Kovács SJ (2010) The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties. Duke Math J 155(1):1–33. Preprint arXiv:0707.2054

    Google Scholar 

  44. Kollár J, Mori S (1998) Birational geometry of algebraic varieties, vol 134 of Cambridge tracts in mathematics. Cambridge University Press, Cambridge

    Google Scholar 

  45. Kollár J, Miyaoka Y, Mori S (1992) Rationally connected varieties. J Algebraic Geom 1(3):429–448

    MathSciNet  MATH  Google Scholar 

  46. Kobayashi S (1987) Differential geometry of complex vector bundles, vol 15 of Publications of the mathematical society of Japan. Kanô Memorial Lectures, vol 5. Iwanami Shoten and Princeton University Press, Princeton, NJ

    MATH  Google Scholar 

  47. Kollár J (1995) Shafarevich maps and automorphic forms. M. B. Porter Lectures. Princeton University Press, Princeton, NJ

    Book  MATH  Google Scholar 

  48. Kollár J (1996) Rational curves on algebraic varieties, vol 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer, Berlin

    Book  MATH  Google Scholar 

  49. Kollár J (2007) Lectures on resolution of singularities, vol 166 of Annals of mathematics studies. Princeton University Press, Princeton, NJ

    MATH  Google Scholar 

  50. Kollár J (2008) Is there a topological Bogomolov-Miyaoka-Yau inequality? Pure Appl Math Q 4(2, part 1):203–236

    Google Scholar 

  51. Kollár J (2011) New examples of terminal and log canonical singularities. Preprint arXiv:1107.2864, July 2011

  52. Källström R (2011) The Zariski-Lipman conjecture for complete intersections. J Algebra 337:169–180

    Article  MathSciNet  MATH  Google Scholar 

  53. Lascoux A, Berger M (1970) Variétés Kähleriennes compactes. Lecture Notes in Mathematics, vol 154. Springer, Berlin

    Google Scholar 

  54. Lipman J (1965) Free derivation modules on algebraic varieties. Am J Math 87:874–898

    Article  MathSciNet  MATH  Google Scholar 

  55. MacPherson RD (1974) Chern classes for singular algebraic varieties. Ann Math (2) 100:423–432

    Google Scholar 

  56. Miyaoka Y (1987) The Chern classes and Kodaira dimension of a minimal variety. In: Algebraic geometry, Sendai, 1985, vol 10 of Adv. Stud. Pure Math., pp 449–476. North-Holland, Amsterdam

    Google Scholar 

  57. Miyaoka Y (1987) Deformations of a morphism along a foliation and applications. In: Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), vol 46 of Proc. Sympos. Pure Math., pp 245–268. Amer. Math. Soc., Providence, RI

    Chapter  Google Scholar 

  58. Namikawa Y (2001) Extension of 2-forms and symplectic varieties. J Reine Angew Math 539:123–147

    MathSciNet  MATH  Google Scholar 

  59. Nakayama N, Zhang D-Q (2010) Polarized endomorphisms of complex normal varieties. Math Ann 346(4):991–1018

    Article  MathSciNet  MATH  Google Scholar 

  60. Ou W (2014) Singular rationally connected surfaces with non-zero pluri-forms. Michigan Math Journal 63(4):725–745. arXiv:1301.0915, January 2013

    Google Scholar 

  61. Ou W (2014) Singular rationally connected threefolds with non-zero pluri-forms. Preprint arXiv:1401.2014, January 2014

  62. Patakfalvi Z (2012) Viehweg’s hyperbolicity conjecture is true over compact bases. Adv Math 229(3):1640–1642

    Article  MathSciNet  MATH  Google Scholar 

  63. Peternell T (2006) Kodaira dimension of subvarieties. II. Int J Math 17(5):619–631

    Article  MathSciNet  MATH  Google Scholar 

  64. Ramanujam CP (1972) Remarks on the Kodaira vanishing theorem. J Ind Math Soc (N.S.) 36:41–51

    Google Scholar 

  65. Rossi H (1968) Picard variety of an isolated singular point. Rice Univ Stud 54(4):63–73

    MathSciNet  MATH  Google Scholar 

  66. Shepherd-Barron NI, Wilson PMH (1994) Singular threefolds with numerically trivial first and second Chern classes. J Algebraic Geom 3(2):265–281

    MathSciNet  MATH  Google Scholar 

  67. Simpson CT (1992) Higgs bundles and local systems. Inst Hautes Études Sci Publ Math (75):5–95

    Article  MathSciNet  MATH  Google Scholar 

  68. Scheja G, Storch U (1972) Differentielle Eigenschaften der Lokalisierungen analytischer Algebren. Math Ann 197:137–170

    Article  MathSciNet  MATH  Google Scholar 

  69. Steenbrink JHM, van Straten D (1985) Extendability of holomorphic differential forms near isolated hypersurface singularities. Abh Math Sem Univ Hamburg 55:97–110

    Article  MathSciNet  MATH  Google Scholar 

  70. Takayama S (2003) Local simple connectedness of resolutions of log-terminal singularities. Int J Math 14(8):825–836. DOI:10.1142/S0129167X0300196X

    Article  MathSciNet  MATH  Google Scholar 

  71. Totaro B (2012) Algebraic surfaces and hyperbolic geometry. In: Current developments in algebraic geometry, vol 59 of Math. Sci. Res. Inst. Publ., pp 405–426. Cambridge University Press, Cambridge. Preprint arXiv:1008.3825

  72. Uhlenbeck K, Yau S-T (1986) On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Comm Pure Appl Math 39(S, suppl.):S257–S293. Frontiers of the mathematical sciences: 1985 (New York, 1985)

    Google Scholar 

  73. Verdier J-L (1976) Stratifications de Whitney et théorème de Bertini-Sard. Invent Math 36:295–312

    Article  MathSciNet  MATH  Google Scholar 

  74. Xu C (2014) Finiteness of algebraic fundamental groups. Compos Math 150(3):409–414. arXiv:1210.5564, October 2012

    Google Scholar 

  75. Yau ST (1978) On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm Pure Appl Math 31(3):339–411

    Article  MATH  Google Scholar 

Download references

Acknowledgements

Both authors found the conference on “Foliation theory” exceptionally fruitful. They would like to thank the organisers for the invitation, and the Simons Foundation for financing this event. They also thank Behrouz Taji, who pointed them to a mistake in the first version of this paper.

This overview article summarises the content of the research articles [GKKP11, GKP11, GKP13a, GKP13b] as well as some recent developments by Graf, Jörder and others, and aims to put them into perspective. The results presented here are therefore not new. The exposition frequently follows the original articles. Proper references will be given throughout.

Both authors were supported in part by the DFG-Forschergruppe 790 “Classification of Algebraic Surfaces and Compact Complex Manifolds”. Stefan Kebekus also acknowledges support through a joint fellowship of the Freiburg Institute of Advanced Studies (FRIAS) and the University of Strasbourg Institute for Advanced Study (USIAS). Thomas Peternell thanks FRIAS for providing excellent working conditions.

Author information

Authors and Affiliations

  1. Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstrasse 1, 79104, Freiburg im Breisgau, Germany

    Stefan Kebekus

  2. University of Strasbourg Institute for Advanced Study (USIAS), Strasbourg, France

    Stefan Kebekus

  3. Mathematisches Institut, Universität Bayreuth, 95440, Bayreuth, Germany

    Thomas Peternell

Authors
  1. Stefan Kebekus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Peternell
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan Kebekus .

Editor information

Editors and Affiliations

  1. Mathematics Department, Imperial College of London, London, UK

    Paolo Cascini

  2. Department of Mathematics, UC San Diego, La Jolla, CA, USA

    James McKernan

  3. IMPA, Rio de Janeiro, Brazil

    Jorge Vitório Pereira

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Kebekus, S., Peternell, T. (2016). Aspects of the Geometry of Varieties with Canonical Singularities. In: Cascini, P., McKernan, J., Pereira, J.V. (eds) Foliation Theory in Algebraic Geometry. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-319-24460-0_4

Download citation

Publish with us

Policies and ethics

Search

Navigation