We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
${\mathcal{D}}$-modules related to the 
$b$-function and Hamiltonian flow
Let 
$f$ be a quasi-homogeneous polynomial with an isolated singularity in 
$\mathbf{C}^{n}$. We compute the length of the 
${\mathcal{D}}$-modules 
${\mathcal{D}}f^{\unicode[STIX]{x1D706}}/{\mathcal{D}}f^{\unicode[STIX]{x1D706}+1}$ generated by complex powers of 
$f$ in terms of the Hodge filtration on the top cohomology of the Milnor fiber. When 
$\unicode[STIX]{x1D706}=-1$ we obtain one more than the reduced genus of the singularity (
$\dim H^{n-2}(Z,{\mathcal{O}}_{Z})$ for 
$Z$ the exceptional fiber of a resolution of singularities). We conjecture that this holds without the quasi-homogeneous assumption. We also deduce that the quotient 
${\mathcal{D}}f^{\unicode[STIX]{x1D706}}/{\mathcal{D}}f^{\unicode[STIX]{x1D706}+1}$ is nonzero when 
$\unicode[STIX]{x1D706}$ 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 
${\mathcal{D}}$-modules to those defined by Etingof and the second author which represent invariants under Hamiltonian flow.
