On the behavior of analytic torsion for twisted canonical bundles under degenerations
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912907660361728 |
|---|---|
| author | Yoshikawa, Ken-Ichi |
| author_facet | Yoshikawa, Ken-Ichi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14797 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the behavior of analytic torsion for twisted canonical bundles under degenerations Yoshikawa, Ken-Ichi Algebraic Geometry Differential Geometry 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. |
| title | On the behavior of analytic torsion for twisted canonical bundles under degenerations |
| topic | Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2602.14797 |