Bernstein-Sato ideals and local systems

Abstract

The topology of smooth quasi-projective complex varieties is very restrictive. One aspect of this statement is the fact that natural strata of local systems, called cohomology support loci, have a rigid structure: they consist of torsion-translated subtori in a complex torus. We propose and partially confirm a relation between Bernstein-Sato ideals and local systems. This relation gives yet a different point of view on the nature of the structure of cohomology support loci of local systems. The main result is a partial generalization to the case of a collection of polynomials of the theorem of Malgrange and Kashiwara which states that the Bernstein-Sato polynomial of a hypersurface recovers the monodromy eigenvalues of the Milnor fibers of the hypersurface. We also address a multi-variable version of the Monodromy Conjecture, prove that it follows from the usual single-variable Monodromy Conjecture, and prove it in the case of hyperplane arrangements.

Local systems on analytic germ complements

“…Remark 5.3. Note that, as pointed out by Liu-Maxim [20], in all the statements in [4] where the uniform support Supp unif x (ψ F C) of the Sabbah specialization complex appears, the unif should be dropped to conform to what is proven in [4]. Indeed, the support Supp x (ψ F C) needs no uniformization.…”

“…Proof of Theorem 1.4. Again, [4,Lemma 3.34] and [4,Proposition 3.31] reduce the statement to the case when f = j f i defines a reduced germ and the f i define the mutually distinct analytic branches. In this case, as above, [4,Theorem 4] gives that Supp 0 (ψ F C) = V(U), and claim then follows from Theorem 5.1.…”

Cohomology support loci of local systems

“…In this section we recall a few facts about Sabbah's specialization complex from [Sab90,Bud15,LiMa14].…”

“…Here, res −1 y (V(U y )) is what was called in [Bud15,LiMa14] the uniformization V(U y ) unif of V(U y ) with respect to M B (U x ), for y ∈ f −1 (0) close to x. As pointed out in [LiMa14], in all the statements in [Bud15] where the uniform support Supp unif x (ψ F (C X )) appears, the unif should be dropped to conform to what is proven in [Bud15]. Indeed, the support Supp x (ψ F (C X )) needs no uniformization.…”

“…be the map which pulls back local systems from (C * ) r to U via F , where U = X \f −1 (0). Then, by [Bud15,LiMa14],…”