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

Abstract

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 $${\tilde{p}}_g\le \frac{\mu -1}{(n+2)!}.$$ 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.

29 pages. Added a relation with the conjecture of Durfee, and a more detailed discussion about determinants of Laplacians of curves. Also minor improvements of language and presentation