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

Journal article, 2025

In this paper, we introduce the notion of spectral genus (p) over tilde (g) 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 p(g), and hence (p) over tilde (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 (p) over tilde (g) and the Milnor number mu, to the effect that (p) over tilde (g) <= 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.