On D-modules related to the b-function and Hamiltonian flow

TL;DR

This paper investigates D-modules associated with quasi-homogeneous polynomials with isolated singularities, linking their structure to Hodge theory and Hamiltonian invariants, and explores properties related to the b-function.

Contribution

It computes the length of specific D-modules generated by complex powers of f in terms of Hodge filtration, and relates these to Hamiltonian flow invariants, extending known results.

Findings

Length of D-modules expressed via Hodge filtration

Non-vanishing of quotient when c is a root of the b-function

Comparison with Hamiltonian flow invariants

Abstract

Let f be a quasi-homogeneous polynomial with an isolated singularity. We compute the length of the D-modules $Df^c/Df^{c+1}$ generated by complex powers of f in terms of the Hodge filtration on the top cohomology of the Milnor fiber. For 1/f we obtain one more than the reduced genus of the singularity. We conjecture that this holds without the quasi-homogeneous assumption. We also deduce that the aforementioned quotient is nonzero when c is a root of the b-function of f (which Saito recently showed fails to hold in the inhomogeneous case). We obtain these results by comparing these D-modules to those defined by Etingof and the second author which represent invariants under Hamiltonian flow.