Skip to main content

Around and Beyond the Canonical Class

  • Chapter
  • First Online:

Abstract

This survey is an invitation to recent developments in higher dimensional birational geometry.

Mathematics Subject Classification codes (2000): 14E30

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   159.99
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

Learn about institutional subscriptions

References

  1. C. Birkar, P. Cascini, C. D. Hacon, and J. M c Kernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23, no. 2, 405–468, (2010).

    Google Scholar 

  2. P. Cascini and V. Lazić, The Minimal Model Program revisited, Contributions to Algebraic Geometry (Piotr Pragacz, ed.), EMS Series of Congress Reports, EMS Publishing House, Zürich, 2012, pp. 169–187.

    Google Scholar 

  3. P. Cascini and V. Lazić, New outlook on the Minimal Model Program, I, Duke Math. J. 161, no. 12, 2415–2467, (2012).

    Google Scholar 

  4. A. Corti and V. Lazić, New outlook on the Minimal Model Program, II, Math. Ann. (2012), DOI:10.1007/s00208-012-0858-1.

    Google Scholar 

  5. A. Corti, Finite generation of adjoint rings after Lazić: an introduction, Classification of Algebraic Varieties, EMS Series of Congress Reports, EMS Publishing House, 2011, 197–220.

    Google Scholar 

  6. L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye, and M. Popa, Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble) 56, no. 6, 1701–1734, (2006).

    Google Scholar 

  7. Y. Hu and S. Keel, Mori dream spaces and GIT, Michigan Math. J. 48, 331–348, (2000).

    Google Scholar 

  8. C. D. Hacon and S. J. Kovács, Classification of higher dimensional algebraic varieties, Oberwolfach Seminars, vol. 41, Birkhäuser Verlag, Basel, 2010.

    Google Scholar 

  9. B. Hassett and Y. Tschinkel, Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces, J. Inst. Math. Jussieu 9, 125–153, (2010).

    Google Scholar 

  10. J. Kollár et al., Flips and abundance for algebraic threefolds, Astérisque 211, Soc. Math. France, Paris, 1992.

    Google Scholar 

  11. Y. Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2) 127 (1988), no. 1, 93–163.

    Google Scholar 

  12. Y. Kawamata, On the cone of divisors of Calabi-Yau fiber spaces, Internat. J. Math. 8, 665–687, (1997).

    Google Scholar 

  13. A.-S. Kaloghiros, A. Küronya, and V. Lazić, Finite generation and geography of models, to appear in “Minimal models and extremal rays”, Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo, arXiv:1202.1164, (2012).

    Google Scholar 

  14. J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998.

    Google Scholar 

  15. R. Lazarsfeld, Positivity in algebraic geometry. I, II, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 48, 49, Springer-Verlag, Berlin, 2004.

    Google Scholar 

  16. V. Lazić, Adjoint rings are finitely generated, arXiv:0905.2707, (2009).

    Google Scholar 

  17. V. Lazić and Th. Peternell, On the Cone conjecture for Calabi-Yau manifolds with Picard number two, arXiv:1207.3653, (2012).

    Google Scholar 

  18. E. Markman, A survey of Torelli and monodromy results for holomorphic-symplectic varieties, Complex and Differential Geometry (W. Ebeling, K. Hulek, and K. Smoczyk, eds.), Springer Proceedings in Mathematics, vol. 8, Springer Berlin Heidelberg, 2011, 257–322.

    Google Scholar 

  19. J. M c Kernan, Mori dream spaces, Japan. J. Math. 5, no. 1, 127–151, (2010).

    Google Scholar 

  20. S. Mori, Classification of higher-dimensional varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, 269–331.

    Google Scholar 

  21. S. Mori, Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1, no. 1, 117–253, (1988).

    Google Scholar 

  22. D. Morrison, Compactifications of moduli spaces inspired by mirror symmetry, Astérisque 218, 243–271, (1993).

    Google Scholar 

  23. N. Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004.

    Google Scholar 

  24. Y. Namikawa, Periods of Enriques surfaces, Math. Ann. 270 (1985), 201–222.

    Google Scholar 

  25. K. Oguiso, Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups, arXiv:1107.5862, (2011).

    Google Scholar 

  26. K. Oguiso, Automorphism groups of Calabi-Yau manifolds of Picard number two, arXiv:1206.1649, (2012).

    Google Scholar 

  27. A. Prendergast-Smith, The Cone Conjecture for Abelian Varieties, J. Math. Sci. Univ. Tokyo 19, no. 2, 243–261, (2012).

    Google Scholar 

  28. M. Reid, Canonical 3-folds, Journées de Géométrie Algébrique d’Angers (A. Beauville, ed.), Sijthoof and Nordhoof, Alphen aan den Rijn, 1980, 273–310.

    Google Scholar 

  29. M. Reid, Twenty-five years of 3-folds–an old person’s view, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 313–343.

    Google Scholar 

  30. V. V. Shokurov and S. R. Choi, Geography of log models: theory and applications, Cent. Eur. J. Math. 9 (2011), no. 3, 489–534.

    Google Scholar 

  31. A. J. Sommese, On the adjunction theoretic structure of projective varieties, Complex analysis and algebraic geometry (Göttingen, 1985), Lecture Notes in Math., vol. 1194, Springer, Berlin, 1986, 175–213.

    Google Scholar 

  32. H. Sterk, Finiteness results for algebraic K3 surfaces, Math. Z. 189, 507–513, (1985).

    Google Scholar 

  33. B. Totaro, The cone conjecture for Calabi-Yau pairs in dimension two, Duke Math. J. 154, 241–263, (2010).

    Google Scholar 

  34. B. Totaro, Algebraic surfaces and hyperbolic geometry, Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ., vol. 59, Cambridge Univ. Press, Cambridge, 2012, 405–426.

    Google Scholar 

  35. O. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. 76, no. 3, 560–615, (1962), with an appendix by David Mumford.

    Google Scholar 

Download references

Acknowledgements

Many thanks to P. Cascini, A. Corti, K. Frantzen, D. Greb, A.-S. Kaloghiros, J. Kollár, A. Küronya, Th. Peternell, and S. Weigl for many useful comments and discussions, and to the referee for many useful remarks. I was supported by the DFG-Forschergruppe 790 “Classification of Algebraic Surfaces and Compact Complex Manifolds”.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Vladimir Lazić .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Lazić, V. (2013). Around and Beyond the Canonical Class. In: Bogomolov, F., Hassett, B., Tschinkel, Y. (eds) Birational Geometry, Rational Curves, and Arithmetic. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6482-2_9

Download citation

Publish with us

Policies and ethics