The spectral genus of an isolated hypersurface singularity and its relation to the Milnor number and analytic torsion

In this paper, we introduce the notion of spectral genus p̃g of a germ of an isolated hypersurface singularity (Cⁿ⁺¹,0)→ (C,0) , defined as a sum of small exponents of monodromy eigenvalues. The number of these is equal to the geometric genus pg , and hence p̃g can be considered as a secondary invariant to it. We then explore a secondary version of the Durfee conjecture on pg , and we predict an inequality between p̃g and the Milnor number μ , to the effect that p̃g≤μ-1/(n+2)!. We provide evidence by confirming our conjecture in several cases, including homogeneous singularities and singularities with large Newton polyhedra, and quasi-homogeneous or irreducible curve singularities. We also show that a weaker inequality follows from Durfee’s conjecture, and hence holds for quasi-homogeneous singularities and curve singularities.Our conjecture is shown to relate closely to the asymptotic behavior of the holomorphic analytic torsion of the sheaf of holomorphic functions on a degeneration of projective varieties, potentially indicating deeper geometric and analytic connections.