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

  • Dennis Eriksson

    Chalmers University of Technology and University of Gothenburg, Sweden
  • Gerard Freixas i Montplet

    CNRS – École Polytechnique – IP Paris, Palaiseau, France
The spectral genus of an isolated hypersurface singularity and its relation to the Milnor number and analytic torsion cover
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Abstract

In this paper, we introduce the notion of spectral genus of a germ of an isolated hypersurface singularity , defined as a sum of small exponents of monodromy eigenvalues. The number of these is equal to the geometric genus , and hence can be considered as a secondary invariant to it. We then explore a secondary version of the Durfee conjecture on , and we predict an inequality between and the Milnor number , to the effect that

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.

Cite this article

Dennis Eriksson, Gerard Freixas i Montplet, The spectral genus of an isolated hypersurface singularity and its relation to the Milnor number and analytic torsion. Doc. Math. (2025), published online first

DOI 10.4171/DM/1013