1 results
On
${\mathcal{D}}$-modules related to the 
$b$-function and Hamiltonian flow
- Part of
-
- Journal:
- Compositio Mathematica / Volume 154 / Issue 11 / November 2018
- Published online by Cambridge University Press:
- 12 October 2018, pp. 2426-2440
- Print publication:
- November 2018
-
- Article
- Export citation
-
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.
