Tjurina spectrum and graded symmetry of missing spectral numbers

Abstract

For a hypersurface isolated singularity defined by $f$, the Steenbrink spectrum can be obtained as the Poincaré polynomial of the graded quotients of the $V$-filtration on the Jacobian ring of $f$. The Tjurina subspectrum is defined by replacing the Jacobian ring with its quotient by the image of the multiplication by $f$. We prove that their difference (consisting of missing spectral numbers) has a canonical graded symmetry. This follows from the self-duality of the Jacobian ring, which is compatible with the action of $f$ as well as the $V$-filtration. It implies for instance that the number of missing spectral numbers which are smaller than $(n+1)/2$ (with $n$ the ambient dimension) is bounded by $[(μ-τ)/2]$. We can moreover improve the estimate of Briançon-Skoda exponent in the semisimple monodromy case.