Skip to main content
SpringerLink
Log in

K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space III: the case r(M) ≥ 18

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We prove the automorphic property of the invariant of K3 surfaces with involution, which we obtained using equivariant analytic torsion, in the case where the dimension of the moduli space is less than or equal to 2.

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

Access this article

Log in via an institution

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. Bierstone E., Milman P.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128, 207–302 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bismut J.-M.: Equivariant immersions and Quillen metrics. J. Differ. Geom. 41, 53–157 (1995)

    MathSciNet  MATH  Google Scholar 

  3. Bismut, J.-M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles I,II,III. Commun. Math. Phys. 115, 49–78, 79–126, 301–351 (1988)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Demailly J.-P., Lempert L., Shiffman B.: Algebraic approximations of holomorphic maps from Stein domains to projective manifolds. Duke Math. J. 76, 333–363 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dolgachev I.: Mirror symmetry for lattice polarized K3 surfaces. J. Math. Sci. 81, 2599–2630 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Köhler K., Roessler D.: A fixed point formula of Lefschetz type in Arakelov geometry I. Invent. Math. 145, 333–396 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ma X.: Submersions and equivariant Quillen metrics. Ann. Inst. Fourier 50, 1539–1588 (2000)

    Article  MATH  Google Scholar 

  9. Mumford D.: Hirzebruch’s proportionality theorem in the non-compact case. Invent. Math. 42, 239–272 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  10. Nikulin V.V.: Integral Symmetric bilinear forms and some of their applications. Math. USSR Izv. 14, 103–167 (1980)

    Article  MATH  Google Scholar 

  11. Nikulin V.V.: Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections. J. Soviet Math. 22, 1401–1476 (1983)

    Article  MATH  Google Scholar 

  12. Ray D.B., Singer I.M.: Analytic torsion for complex manifolds. Ann. Math. 98, 154–177 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  13. Serre J.-P.: Cours d’arithmétique. Presses Universitaires de France, Paris (1970)

    MATH  Google Scholar 

  14. Siu Y.-T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53–156 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  15. Takegoshi K.: Higher direct images of canonical sheaves tensorized with semi-positive vector bundles by proper Kähler morphisms. Math. Ann. 303, 389–416 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yoshikawa K.-I.: K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space. Invent. Math. 156, 53–117 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yoshikawa K.-I.: On the singularity of Quillen metrics. Math. Ann. 337, 61–89 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yoshikawa, K.-I.: K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space II: a structure theorem for r(M) > 10. J. Reine Angew. Math. (2011, to appear)

  19. Yoshikawa, K.-I.: Singularities and analytic torsion. arXiv:1007.2835v1 (2010, preprint)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-8502, Japan

    Ken-Ichi Yoshikawa

  2. Korea Institute for Advanced Study, Hoegiro 87, Dongdaemun-gu, Seoul, 130-722, Korea

    Ken-Ichi Yoshikawa

Authors
  1. Ken-Ichi Yoshikawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ken-Ichi Yoshikawa.

Additional information

The author is partially supported by the Grants-in-Aid for Scientific Research (B) 19340016 and (S) 22224001, JSPS.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yoshikawa, KI. K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space III: the case r(M) ≥ 18. Math. Z. 272, 175–190 (2012). https://doi.org/10.1007/s00209-011-0928-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-011-0928-1

Keywords

Search

Navigation