SPECTRUM OF SINGULARITIES, ANALYTIC TORSION AND CALABI-YAU DEGENERATIONS

Abstract

We relate the spectrum of singularities to the geometry of degenerating Calabi-Yau manifolds through holomorphic analytic torsion. We express the leading asymptotic coefficients of the analytic torsion of holomorphic differential forms in terms of Hodge-theoretic invariants of vanishing cycles. For isolated hypersurface singularities, these formulas lead to the formulation of new higher Durfee-Saito conjectures bounding Hodge-filtration dimensions by the Milnor number and Eulerian numbers. We also prove that, for isolated quasi-homogeneous singularities, the spectral measure is dominated in convex order by the corresponding Irwin-Hall distribution. Applying the torsion formulas to the BCOV invariant yields a computable obstruction to birational smooth fillings. For isolated singularities, its local contribution is expressed in terms of Hertling's spectral variance and a Bernoulli sum determined by monodromy. As applications, we rule out birational smooth fillings for projective Calabi-Yau degenerations with smooth total space and singular central fiber having only ADE or terminal Brieskorn-Pham singularities, in relative dimension at least three, extending results of Voisin.

Degenerations of exotic Calabi-Yau metrics through Atiyah's flop
Thibault Langlais, Viktor F. Majewski
arXiv8 Sep 2026PreprintCC BY 4.0Full textAnalysis & PDEs Geometry & topologyarXiv
Calabi-Yau Threefold Singularities and their Universal Quilts
Elizabeth Gasparim, Carlos Simpson, Bruno Suzuki +1
arXiv1 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Mathematical physicsarXiv
Deformations of Compact Calabi--Yau and Fano Varieties with Isolated Singularities
Yohsuke Imagi
arXiv1 Sep 2026PreprintCC BY 4.0Full textAlgebra & number theory Analysis & PDEsarXiv
Chern scalar curvature and finite-time singularities of the conformal Chern--Calabi flow
Hongyu Chen, Li Ma, Li Sheng
arXiv1 Nov 2026PreprintarXiv licenceFull textAnalysis & PDEs Geometry & topologyarXiv
Equivariant analytic torsion forms
Yuxiang Li
arXiv13 Oct 2026PreprintarXiv licenceFull textGeometry & topologyarXiv

Cite

BibTeX RIS

BibTeX

@misc{eriksson2026spectrum,
  title = {{SPECTRUM OF SINGULARITIES, ANALYTIC TORSION AND CALABI-YAU DEGENERATIONS}},
  author = {Dennis Eriksson and Gerard Freixas Montplet},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12849},
  eprint = {2609.12849},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05796659v1},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - SPECTRUM OF SINGULARITIES, ANALYTIC TORSION AND CALABI-YAU DEGENERATIONS
AU  - Dennis Eriksson
AU  - Gerard Freixas Montplet
PY  - 2026
DA  - 2026/09/11
PB  - HAL
DO  - 10.48550/arxiv.2609.12849
UR  - https://hal.science/hal-05796659v1
AB  - We relate the spectrum of singularities to the geometry of degenerating Calabi-Yau manifolds through holomorphic analytic torsion. We express the leading asymptotic coefficients of the analytic torsion of holomorphic differential forms in terms of Hodge-theoretic invariants of vanishing cycles. For isolated hypersurface singularities, these formulas lead to the formulation of new higher Durfee-Saito conjectures bounding Hodge-filtration dimensions by the Milnor number and Eulerian numbers. We also prove that, for isolated quasi-homogeneous singularities, the spectral measure is dominated in convex order by the corresponding Irwin-Hall distribution. Applying the torsion formulas to the BCOV invariant yields a computable obstruction to birational smooth fillings. For isolated singularities, its local contribution is expressed in terms of Hertling's spectral variance and a Bernoulli sum determined by monodromy. As applications, we rule out birational smooth fillings for projective Calabi-Yau degenerations with smooth total space and singular central fiber having only ADE or terminal Brieskorn-Pham singularities, in relative dimension at least three, extending results of Voisin.
ER  -

Abstract

Related

Cite