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On the behavior of analytic torsion for twisted canonical bundles under degenerations

Ken-Ichi Yoshikawa

Abstract

Consider a degeneration of projective algebraic manifolds equipped with a compact group action over a curve. Suppose that the total space carries a Nakano semi-positive vector bundle, which is equivariant with respect to this action. We consider the relative canonical bundle twisted by this bundle. Under this setting, we prove that the logarithm of the equivariant analytic torsion of the regular fibers for this coefficient admits an asymptotic expansion near the discriminant locus. The leading term is given by a logarithmic singularity, while the subdominant term is given by a loglog-type singularity. In the non-equivariant case, we provide a formula for the coefficient of the leading term in terms of an integral of characteristic classes associated with the semi-stable reduction of the family. To establish these results, we prove the existence of an asymptotic expansion for both the equivariant Quillen metrics and the $L^{2}$-metrics in the above setting. We also calculate the leading term of the fiber integral of the Bott-Chern classes associated with the degeneration.

On the behavior of analytic torsion for twisted canonical bundles under degenerations

Abstract

Consider a degeneration of projective algebraic manifolds equipped with a compact group action over a curve. Suppose that the total space carries a Nakano semi-positive vector bundle, which is equivariant with respect to this action. We consider the relative canonical bundle twisted by this bundle. Under this setting, we prove that the logarithm of the equivariant analytic torsion of the regular fibers for this coefficient admits an asymptotic expansion near the discriminant locus. The leading term is given by a logarithmic singularity, while the subdominant term is given by a loglog-type singularity. In the non-equivariant case, we provide a formula for the coefficient of the leading term in terms of an integral of characteristic classes associated with the semi-stable reduction of the family. To establish these results, we prove the existence of an asymptotic expansion for both the equivariant Quillen metrics and the -metrics in the above setting. We also calculate the leading term of the fiber integral of the Bott-Chern classes associated with the degeneration.
Paper Structure (45 sections, 59 theorems, 284 equations)

This paper contains 45 sections, 59 theorems, 284 equations.

Table of Contents

  1. Equivariant analytic torsion and equivariant Quillen metrics
  2. Equivariant analytic torsion and equivariant Quillen metrics
  3. Characteristic forms
  4. Smoothness of equivariant Quillen metrics
  5. The isotypic decomposition of the direct image sheaves
  6. Equivariant determinant of the cohomology
  7. Smoothness of equivariant Quillen metrics
  8. Nakano semi-positive vector bundles
  9. Equivariant Quillen metrics under degenerations
  10. Set up
  11. Gauss map and its equivariant resolution
  12. The singularity of equivariant Quillen metrics
  13. The case of twisted relative canonical bundle
  14. Asymptotic behavior of $L^{2}$-metrics under degenerations
  15. Semi-stable reduction
  16. ...and 30 more sections

Key Result

Theorem 1

Let $g \in G$. Near $s=0$, $\log \tau_{G}(X_{s},K_{X_{s}}(\xi_{s}))(g)$ is expressed as where $c_{g} \in {\mathbf C}$, $\{ r_{i} \}_{i \in I} \subset {\mathbf Q}\cap (0,1]$ is a finite set of positive rational numbers, $(a_{0,W}^{q}, \ldots, a_{nh^{q}_{W},W}^{q}) \not= (0,\ldots,0)$ is a non-zero real vector, the $\phi_{i,m,g}(s)$ are smooth ${\mathbf C}$-valued functions on $S$, and t Then there

Theorems & Definitions (132)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 1.1
  • proof
  • Theorem 1.2
  • proof
  • Theorem 2.1: Takegoshi95MourouganeTakayama08
  • proof
  • ...and 122 more