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

TL;DR

This paper introduces the spectral genus as a new invariant for isolated hypersurface singularities, proposes a conjecture relating it to the Milnor number, and provides evidence supporting this conjecture in various cases.

Contribution

It defines the spectral genus of hypersurface singularities, formulates a new inequality conjecture involving it and the Milnor number, and verifies this in multiple classes of singularities.

Findings

Confirmed the conjecture for homogeneous singularities.

Validated the inequality for singularities with large Newton polyhedra.

Established the relation for quasi-homogeneous and irreducible curve singularities.

Abstract

In this paper, we introduce the notion of spectral genus $\widetilde{p}_{g}$ 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 $\widetilde{p}_g$ 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 $\widetilde{p}_{g}$ and the Milnor number $\mu$, to the effect that $$\widetilde{p}_g\leq\frac{\mu-1}{(n+2)!}.$$ 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.