Microlocal Bernstein--Sato polynomials on singular ambient varieties

Abstract

We introduce the microlocal Bernstein--Sato polynomial of a function on a possibly singular ambient variety, extending the theory of Saito. We show that, contrary to the smooth ambient setting, these polynomials are not generally equal to the reduced $b$-functions obtained by removing the trivial root. We define the minimal exponent and use it to study the singularities of the divisor and the Hodge filtration on local cohomology. Our main results include a generalization of Saito's theorem relating the minimal exponent to rational singularities, a characterization of purity of local cohomology, a Thom--Sebastiani formula for the minimal exponent, and a linear combination formula for Bernstein--Sato polynomials of ideals. When the ambient variety is a complete intersection with rational singularities, we provide effective algorithms for these Bernstein--Sato polynomials and implement them in Macaulay2.