On a conjecture of Bitoun and Schedler

TL;DR

This paper investigates a conjecture relating the length of a specific D-module to the reduced genus of a hypersurface singularity, proving it in certain cases and providing counterexamples.

Contribution

It proves a lower bound for the D-module length conjecture, characterizes when equality holds, and identifies cases where the conjecture is valid or fails.

Findings

Proved the length is always ≥ g_P(Z)+2.

Equality holds iff 1/f is in the D-module generated by I_0(f)·1/f.

Counterexample shows the conjecture does not always hold.

Abstract

Suppose that $X$ is a smooth complex algebraic variety of dimension $\geq 3$ and $f$ defines a hypersurface $Z$ in $X$, with a unique singular point $P$. Bitoun and Schedler conjectured that the ${\mathcal D}$-module generated by $\tfrac{1}{f}$ has length equal to $g_P(Z)+2$, where $g_{P}(Z)$ is the reduced genus of $Z$ at $P$. We prove that this length is always $\geq g_P(Z)+2$ and equality holds if and only if $\tfrac{1}{f}$ lies in the ${\mathcal D}$-module generated by $I_0(f)\tfrac{1}{f}$, where $I_0(f)$ is the multiplier ideal ${\mathcal J}(f^{1-\epsilon})$, with $0<\epsilon\ll 1$. In particular, we see that the conjecture holds if the pair $(X,Z)$ is log canonical. We can also recover, with an easy proof, the result of Bitoun and Schedler saying that the conjecture holds for weighted homogeneous isolated singularities. On the other hand, we give an example (a polynomial in $3$ variables with an ordinary singular point of multiplicity $4$) for which the conjecture does not hold.