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.