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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Bernstein–Sato varieties and annihilation of powers


Author: Daniel Bath
Journal: Trans. Amer. Math. Soc. 373 (2020), 8543-8582
MSC (2010): Primary 14F10; Secondary 32S40, 32S05, 32S22, 55N25, 32C38
DOI: https://doi.org/10.1090/tran/8192
Published electronically: September 29, 2020
MathSciNet review: 4177268
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Abstract:

Given a complex germ $f$ near the point $\mathfrak {x}$ of the complex manifold $X$, equipped with a factorization $f = f_{1} \cdots f_{r}$, we consider the $\mathscr {D}_{X,\mathfrak {x}}[s_{1}, \dots , s_{r}]$-module generated by $F^{S} \coloneq f_{1}^{s_{1}} \cdots f_{r}^{s_{r}}$. We show for a large class of germs that the annihilator of $F^{S}$ is generated by derivations and this property does not depend on the chosen factorization of $f$.

We further study the relationship between the Bernstein–Sato variety attached to $F$ and the cohomology support loci of $f$, via the $\mathscr {D}_{X,\mathfrak {x}}$-map $\nabla _{A}$. This is related to multiplication by $f$ on certain quotient modules. We show that for our class of divisors the injectivity of $\nabla _{A}$ implies its surjectivity. Restricting to reduced, free divisors, we also show the reverse, using the theory of Lie–Rinehart algebras. In particular, we analyze the dual of $\nabla _{A}$ using techniques pioneered by Narváez–Macarro.

As an application of our results we establish a conjecture of Budur in the tame case: if $\mathrm {V}(f)$ is a central, essential, indecomposable, and tame hyperplane arrangement, then the Bernstein–Sato variety associated to $F$ contains a certain hyperplane. By the work of Budur, this verifies the Topological Mulivariable Strong Monodromy Conjecture for tame arrangements. Finally, in the reduced and free case, we characterize local systems outside the cohomology support loci of $f$ near $\mathfrak {x}$ in terms of the simplicity of modules derived from $F^{S}.$


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Additional Information

Daniel Bath
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: dbath@purdue.edu

Keywords: Bernstein–Sato, b-function, hyperplane, arrangement, D-module, tame, free divisors, logarithmic, differential, annihilator, Spencer, Lie–Rinehart, Milnor fiber, cohomology support, local system, monodromy, characteristic variety, zeta function, monodromy conjecture
Received by editor(s): July 18, 2019
Received by editor(s) in revised form: March 9, 2020
Published electronically: September 29, 2020
Additional Notes: This work was in part supported by the NSF through grant DMS-1401392 and by the Simons Foundation Collaboration Grant for Mathematicians #580839.
Article copyright: © Copyright 2020 American Mathematical Society