Hyperplane Arrangements Satisfy (Un)Twisted Logarithmic Comparison Theorems, Applications to $\mathscr {D}_{X}$ -modules

• Published in: Forum of mathematics. Pi Vol. 12
• Main Author: Bath, Daniel
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.01.2024
• Subjects: 13D45; 14B15; 14F10; 14F40; 32S20; 32S22; 32S40; Algebraic and Complex Geometry; 32S22; 32S20; 14F40; 32S40; 14B15; 14F10; 13D45
• ISSN: 2050-5086; 2050-5086
• DOI: 10.1017/fmp.2024.17

For a reduced hyperplane arrangement, we prove the analytic Twisted Logarithmic Comparison Theorem, subject to mild combinatorial arithmetic conditions on the weights defining the twist. This gives a quasi-isomorphism between the twisted logarithmic de Rham complex and the twisted meromorphic de Rham complex. The latter computes the cohomology of the arrangement’s complement with coefficients from the corresponding rank one local system. We also prove the algebraic variant (when the arrangement is central), and the analytic and algebraic (untwisted) Logarithmic Comparison Theorems. The last item positively resolves an old conjecture of Terao. We also prove that: Every nontrivial rank one local system on the complement can be computed via these Twisted Logarithmic Comparison Theorems; these computations are explicit finite-dimensional linear algebra. Finally, we give some $\mathscr {D}_{X}$ -module applications: For example, we give a sharp restriction on the codimension one components of the multivariate Bernstein–Sato ideal attached to an arbitrary factorization of an arrangement. The bound corresponds to (and, in the univariate case, gives an independent proof of) M. Saito’s result that the roots of the Bernstein–Sato polynomial of a non-smooth, central, reduced arrangement live in $(-2 + 1/d, 0).$