The weak and strong spectral genus conjectures for isolated hypersurface singularities

Let $f\colon (\mathbb C^{n+1},0)\to(\mathbb C,0)$ define an $n$-dimensional isolated hypersurface singularity, with $n\geq 1$. Let $\mu$ be its Milnor number, and let $\widetilde{p}_g$ be its spectral genus, defined as the sum of the spectral contributions in $[0,1)$. Spectral genus conjecture. The following weak and strong inequalities are conjectured:

• Weak form:

$$\widetilde{p}_g<\frac{\mu}{(n+2)!}.$$

• Strong form:

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

This is a secondary analogue of the Durfee-type inequality for the geometric genus. The proposed bounds are motivated by the distribution of spectral numbers and are expected to be sharp; the source gives no resolution of either form.