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 \tilde{p}_{g} of a germ of an isolated hypersurface singularity (\mathbb{C}^{n+1},0)\to (\mathbb{C},0) , defined as a sum of small exponents of monodromy eigenvalues. The number of these is equal to the geometric genus p_{g} , and hence \tilde{p}_{g} can be considered as a secondary invariant to it. We then explore a secondary version of the Durfee conjecture on p_{g} , and we predict an inequality between \tilde{p}_{g} and the Milnor number \mu , to the effect that \tilde{p}_{g}\leq\frac{\mu-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.