The spectral genus of an isolated hypersurface singularity and a conjecture relating to the Milnor number

In this paper, we introduce the notion of spectral genus pg 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 pg 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 pg and the Milnor number μ, to the effect that pg≤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.