Seminar of Algebra

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Hyperplane arrangements satisfy (un)twisted logarithmic comparison theorems

Dan Bath (KU Leuven)
Departamento de Álgebra
Thu, 13 apr 2023 12:30
Currently an FWO junior postdoctoral fellow. Interested in all things Bernstein-Sato, rank one local systems, and (non-)combinatorial properties of hyperplane arrangements

In the 1970s Terao conjectured that, for hyperplane arrangements, the logarithmic de Rham complex computes the cohomology of the arrangement's complement. By Grothendieck's and Deligne's comparison theorems, this amounts to the logarithmic de Rham complex being quasi-isomorphic to the meromorphic de Rham complex. We prove this conjecture. We also prove a twisted version: by augmenting the differential with a weighted one-form, the twisted complex computes the cohomology of the corresponding rank one local system on the complement. Unlike the Brieskorn algebra, our twisted logarithmic comparison theorem computes all rank one local systems on the complement (and not just non-torsion translated ones).

We will discuss two proofs as well as various applications. The first proof is by the speaker alone and develops the philosophy of Castro-Jiménez, Mond and Narváez-Macarro used in the free case; the second proof is D-module theoretic and is joint with Morihiko Saito.