发表机构
Chalmers University of Technology and University of Gothenburg; CNRS – Centre de Mathématiques Laurent Schwartz - École Polytechnique - Institut Polytechnique de Paris（查尔姆斯理工大学和哥德堡大学; 法国国家科学研究中心-洛朗·施瓦茨数学中心-巴黎综合理工学院-巴黎理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过解析挠率将奇点谱与Calabi--Yau退化几何相联系，提出高阶Durfee--Saito猜想，并利用挠率公式给出双有理光滑填充的障碍，推广了Voisin的结果。
AI 中文摘要
我们通过全纯解析挠率将奇点谱与退化Calabi--Yau流形的几何联系起来。我们用消失环的Hodge理论不变量表达了全纯微分形式的解析挠率的前导渐近系数。对于孤立超曲面奇点，这些公式导致了新的更高阶Durfee--Saito猜想的提出，该猜想用Milnor数和Eulerian数来界定Hodge滤过维数。我们还证明了，对于孤立拟齐次奇点，谱测度在凸序意义下被相应的Irwin--Hall分布所控制。将挠率公式应用于BCOV不变量，产生了一个可计算的双有理光滑填充的障碍。对于孤立奇点，其局部贡献用Hertling谱方差和由单值化决定的Bernoulli和来表示。作为应用，我们排除了射影Calabi--Yau退化（其全空间光滑且中心纤维奇异，仅具有ADE或末端Brieskorn--Pham奇点，相对维数至少为三）的双有理光滑填充，推广了Voisin的结果。
英文摘要
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.
