Skip to main content
Log in

The Kodaira dimension of the moduli of K3 surfaces

  • Published:
Inventiones mathematicae Aims and scope

Abstract

The global Torelli theorem for projective K3 surfaces was first proved by Piatetskii-Shapiro and Shafarevich 35 years ago, opening the way to treating moduli problems for K3 surfaces. The moduli space of polarised K3 surfaces of degree 2d is a quasi-projective variety of dimension 19. For general d very little has been known hitherto about the Kodaira dimension of these varieties. In this paper we present an almost complete solution to this problem. Our main result says that this moduli space is of general type for d>61 and for d=46, 50, 54, 57, 58, 60.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ash, A., Mumford, D., Rapoport, M., Tai, Y.: Smooth compactification of locally symmetric varieties. In: Lie Groups: History, Frontiers and Applications, vol. IV. Math. Sci. Press, Brookline, Mass. (1975)

  2. Baily, W.L., Jr., Borel, A.: Compactification of arithmetic quotients of bounded symmetric domains. Ann. Math. (2) 84, 442–528 (1966)

    Google Scholar 

  3. Borcherds, R.E.: Automorphic forms on O s+2,2(ℝ) and infinite products. Invent. Math. 120, 161–213 (1995)

    Article  MATH  Google Scholar 

  4. Borcherds, R.E.: Automorphic forms with singularities on Grassmannians. Invent. Math. 132, 491–562 (1998)

    Article  MATH  Google Scholar 

  5. Borcherds, R.E., Katzarkov, L., Pantev, T., Shepherd-Barron, N.I.: Families of K3 surfaces. J. Algebr. Geom. 7, 183–193 (1998)

    MATH  Google Scholar 

  6. Bourbaki, N.: Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines. Éléments de mathématique. Fasc. XXXIV. Actualités Scientifiques et Industrielles, No. 1337 Hermann, Paris (1968)

  7. Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups. Grundlehren Math. Wiss., vol. 290. Springer, New York (1988)

    MATH  Google Scholar 

  8. Eichler, M.: Quadratische Formen und orthogonale Gruppen. Grundlehren Math. Wiss., vol. 63. Springer, Berlin Göttingen Heidelberg (1952)

    MATH  Google Scholar 

  9. Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Prog. Math., vol. 55. Birkhäuser, Boston, Mass. (1985)

    MATH  Google Scholar 

  10. Faltings, G., Chai, C.-L.: Degeneration of Abelian Varieties. Ergeb. Math. Grenzgeb. (3), vol. 22. Springer, Berlin (1990)

    MATH  Google Scholar 

  11. Freitag, E.: Siegelsche Modulfunktionen. Grundlehren Math. Wiss., vol. 254. Springer, Berlin Göttingen Heidelberg (1983)

    MATH  Google Scholar 

  12. Gritsenko, V.: Modular forms and moduli spaces of abelian and K3 surfaces. Algebra Anal. 6, 65–102 (1994) (English translation in St. Petersbg. Math. J. 6, 1179–1208 (1995))

    MATH  Google Scholar 

  13. Gritsenko, V., Hulek, K.: Appendix to the paper: Irrationality of the moduli spaces of polarized abelian surfaces. In: Abelian Varieties. Proceedings of the International Conference Held in Egloffstein, pp. 83–84. Walter de Gruyter, Berlin, (1995)

  14. Gritsenko, V., Hulek, K.: Minimal Siegel modular threefolds. Math. Proc. Camb. Philos. Soc. 123, 461–485 (1998)

    Article  MATH  Google Scholar 

  15. Gritsenko, V., Hulek, K., Sankaran, G.K.: The Hirzebruch–Mumford volume for the orthogonal group and applications. Preprint (2005), arXiv:math.NT/0512595

  16. Gritsenko, V., Hulek, K., Sankaran, G.K.: Hirzebruch–Mumford proportionality and locally symmetric domains of orthogonal type. Preprint (2006), arXiv:math.AG/0609744

  17. Gritsenko, V., Nikulin, V.V.: Automorphic forms and Lorentzian Kac–Moody algebras. II. Int. J. Math. 9, 201–275 (1998)

    Article  MATH  Google Scholar 

  18. Gritsenko, V., Sankaran, G.K.: Moduli of abelian surfaces with a (1,p 2) polarisation. Izv. Ross. Akad. Nauk, Ser. Mat. 60, 19–26 (1996); reprinted in Izv. Math. 60, 893–900 (1996)

  19. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1979)

    MATH  Google Scholar 

  20. Iwaniec, H.: Topics in Classical Automorphic Forms. Grad. Stud. Math., vol. 17. Am. Math. Soc., Providence, RI (1997)

  21. Kazhdan, D.: On the connection of the dual space of a group with the structure of its closed subgroups. Funkts. Anal. Prilozh. 1, 71–74 (1967)

    Google Scholar 

  22. Kneser, M.: Quadratische Formen. Springer, Berlin (2002)

    MATH  Google Scholar 

  23. Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Grad. Texts Math., vol. 97. Springer, New York (1984)

    MATH  Google Scholar 

  24. Kondo, S.: On the Kodaira dimension of the moduli space of K3 surfaces. Compos. Math. 89, 251–299 (1993)

    MATH  Google Scholar 

  25. Kondo, S.: On the Kodaira dimension of the moduli space of K3 surfaces. II. Compos. Math. 116, 111–117 (1999)

    Article  MATH  Google Scholar 

  26. Mukai, S.: Curves, K3 surfaces and Fano 3-folds of genus ≤10. In: Algebraic Geometry and Commutative Algebra, vol. I, pp. 357–377. Kinokuniya, Tokyo (1988)

  27. Mukai, S.: Polarized K3 surfaces of genus 18 and 20. Complex Projective Geometry (Trieste, 1989/Bergen, 1989), pp. 264–276. Lond. Math. Soc. Lect. Note Ser., vol. 179. Cambridge Univ. Press, Cambridge (1992)

  28. Mukai, S.: Curves and K3 surfaces of genus eleven. In: Moduli of Vector Bundles (Sanda, 1994; Kyoto, 1994). Lect. Notes Pure Appl. Math., vol. 179, pp. 189–197. Dekker, New York (1996)

  29. Mukai, S.: Polarized K3 surfaces of genus thirteen. In: Moduli Spaces and Arithmetic Geometry (Kyoto 2004). Adv. Stud. Pure Math., vol. 45, pp. 315–326. Math. Soc. Japan, Tokyo (2007)

    Google Scholar 

  30. Mumford, D.: Hirzebruch’s proportionality theorem in the noncompact case. Invent. Math. 42, 239–272 (1977)

    Article  MATH  Google Scholar 

  31. Nikulin, V.V.: Finite automorphism groups of Kähler K3 surfaces. Tr. Mosk. Mat. O.-va. 38, 75–137 (1979) (English translation in Trans. Mosc. Math. Soc. 2, 71–135 (1980))

    MATH  Google Scholar 

  32. Nikulin, V.V.: Integral symmetric bilinear forms and some of their applications. Izv. Akad. Nauk SSSR, Ser. Mat. 43, 111–177 (1979) (English translation in Math. USSR, Izv. 14, 103–167 (1980))

    MATH  Google Scholar 

  33. Nikulin, V.: Factor groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections. Algebro-geometric applications. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 18, 3–114 (1981) (English translation in J. Sov. Math. 22, 1401–1475 (1983))

    Google Scholar 

  34. Oda, T.: Convex Bodies and Algebraic Geometry. Ergeb. Math. Grenzgeb. (3), vol. 15. Springer, Berlin (1988)

    MATH  Google Scholar 

  35. Piatetskii-Shapiro, I., Shafarevich, I.: A Torelli theorem for algebraic surfaces of type K3. Izv. Akad. Nauk SSSR, Ser. Mat. 35, 530–572 (1971) (English translation in Math. USSR, Izv. 5, 547–588 (1971/1972))

    Google Scholar 

  36. Reid, M.: Canonical 3-folds. In: Journées de Géometrie Algébrique d’Angers 1979, pp. 273–310. Sijthoff & Noordhoff, Alphen aan den Rijn (1980)

  37. Scattone, F.: On the Compactification of Moduli Spaces of Algebraic K3 Surfaces. Mem. Am. Math. Soc., vol. 70, no. 374 (1987)

  38. Shepherd-Barron, N.I.: Perfect forms and the moduli space of abelian varieties. Invent. Math. 163, 25–45 (2006)

    Article  MATH  Google Scholar 

  39. Snurnikov, V.: Quotients of canonical toric singularities. Ph.D. thesis, Cambridge (2002)

  40. Tai, Y.: On the Kodaira dimension of the moduli space of abelian varieties. Invent. Math. 68, 425–439 (1982)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to K. Hulek.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gritsenko, V., Hulek, K. & Sankaran, G. The Kodaira dimension of the moduli of K3 surfaces. Invent. math. 169, 519–567 (2007). https://doi.org/10.1007/s00222-007-0054-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-007-0054-1

Keywords

Navigation