Skip to main content
SpringerLink
Log in

On the singularity of Quillen metrics

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

Let π: X → S be a holomorphic map from a compact Kähler manifold (X,g X ) to a compact Riemann surface S. Let Σπ be the critical locus of π and let Δ  =  π(Σπ) be the discriminant locus. Let (ξ, h ξ) be a holomorphic Hermitian vector bundle on X. We determine the singularity of the Quillen metric on det Rπ*ξ near Δ with respect to g X | TX/S and h ξ.

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. Barlet, D.: Développement asymptotique des fonctions obtenues par intégration sur les fibres. Invent. Math. 68, 129–174 (1982)

    Google Scholar 

  2. Bismut J.-M. (1990) Superconnection currents and complex immersions. Invent. Math. 99, 59–113

    Article  MATH  MathSciNet  Google Scholar 

  3. Bismut J.-M. (1997) Quillen metrics and singular fibers in arbitrary relative dimension. J. Algebr. Geom. 6, 19–149

    MATH  MathSciNet  Google Scholar 

  4. Bismut J.-M., Bost J.-B. (1990) Fibrés déterminants, métriques de Quillen et dégénérescence des courbes. Acta Math. 165, 1–103

    Article  MATH  MathSciNet  Google Scholar 

  5. 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 

  6. Bismut J.-M., Gillet H., Soulé C. (1990) Complex immersions and Arakelov geometry. In: Cartier P., etal. (eds) The Grothendieck Festschrift. Birkhäuser, Boston, pp. 249–331

    Google Scholar 

  7. Bismut J.-M., Lebeau G. (1991) Complex immersions and Quillen metrics. Publ. Math. IHES 74, 1–297

    MATH  Google Scholar 

  8. Bost J.-B., Gillet H., Soulé C. (1994) Hights of projective varieties and positive Green forms. J. Am. Math. Soc. 7, 903–1027

    Article  MATH  Google Scholar 

  9. Fang, H., Lu, Z., Yoshikawa, K.-I.: Analytic torsion for Calabi–Yau threefolds. E-print  math.DG/0601411 (2006)

  10. Gillet H., Soulé C. (1990) Characteristic classes for algebraic vector bundles with hermitian metric, I,II. Ann. Math. 131, 163–238

    Article  MATH  Google Scholar 

  11. Gillet H., Soulé C. (1990) Arithmetic intersection theory. Publ. Math. IHES 72, 93–174

    MATH  Google Scholar 

  12. Knudsen F.F., Mumford D. (1976) The projectivity of the moduli space of stable curves, I. Math. Scand. 39, 19–55

    MATH  MathSciNet  Google Scholar 

  13. Noguchi, J., Ochiai, T.: Geometric function theory in several complex variables. Am. Math. Soc. (1990)

  14. Quillen D. (1985) Determinants of Cauchy–Riemann operators over a Riemann surface. Funct. Anal. Appl. 14, 31–34

    Article  Google Scholar 

  15. Soulé C., et al. (1992) Lectures on Arakelov Geometry. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  16. Yoshikawa K.-I. (1998) Smoothing of isolated hypersurface singularities and Quillen metrics. Asian J. Math. 2, 325–344

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo, 153-8914, Japan

    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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yoshikawa, KI. On the singularity of Quillen metrics. Math. Ann. 337, 61–89 (2007). https://doi.org/10.1007/s00208-006-0027-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-006-0027-5

Keywords

Search

Navigation