Bernstein--Sato polynomials of locally quasi-homogeneous divisors in ${\mathbb{C}}^3$

Abstract

We consider the Bernstein--Sato polynomial of a locally quasi-homogeneous polynomial $f\in R=\mathbb{C}[x_1,x_2,x_3]$. We construct, in the analytic category, a complex of ${\mathcal{D}}_X[s]$-modules that can be used to compute the ${\mathcal{D}}_X[s]$-dual of ${\mathcal{D}}_X[s]f^{s-1}$ as the middle term of a short exact sequence where the outer terms are well understood. This extends a result by Narváez Macarro where a freeness assumption was required. We derive many results about the zeroes of the Bernstein--Sato polynomial. First, we prove each nonvanishing degree of the zeroeth local cohomology of the Milnor algebra $H_{\mathfrak{m}}^0(R/(\partial f))$ contributes a root to the Bernstein--Sato polynomial, generalizing a result of M. Saito's (where the argument cannot weaken homogeneity to quasi-homogeneity). Second, we prove the zeroes of the Bernstein--Sato polynomial admit a partial symmetry about $-1$, extending a result of Narváez Macarro that again required freeness. We give applications to very small roots, the twisted Logarithmic Comparison Theorem, and more precise statements when $f$ is additionally assumed to be homogeneous. Finally, when $f$ defines a hyperplane arrangement in ${\mathbb{C}}^3$ we give a complete formula for the zeroes of the Bernstein--Sato polynomial of $f$. We show all zeroes except the candidate root $-2+(2/\text{deg}(f))$ are (easily) combinatorially given; we give many equivalent characterizations of when the only non-combinatorial candidate root $-2+(2/\text{deg}(f))$ is in fact a zero of the Bernstein--Sato polynomial. One equivalent condition is the nonvanishing of $H_{\mathfrak{m}}^0(R/(\partial f){)}_{\text{deg}(f)-1}$.

Title change due to terminology change. No results changed, though some quality of life improvements occurred (e.g. some arguments in Section 2 greatly simplified). Final version to appear in Compositio Mathematica