Bernstein–Sato varieties and annihilation of powers
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- by Daniel Bath PDF
- Trans. Amer. Math. Soc. 373 (2020), 8543-8582 Request permission
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
- 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.
- © Copyright 2020 American Mathematical Society
- 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
- MathSciNet review: 4177268