English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For a reduced hyperplane arrangement we prove the analytic TwistedLogarithmic Comparison Theorem, subject to mild combinatorial arithmeticconditions on the weights defining the twist. This gives a quasi-isomorphismbetween the twisted logarithmic de Rham complex and the twisted meromorphic deRham complex. The latter computes the cohomology of the arrangement'scomplement with coefficients from the corresponding rank one local system. Wealso prove the algebraic variant (when the arrangement is central), and theanalytic and algebraic (untwisted) Logarithmic Comparison Theorems. The lastitem positively resolves an old conjecture of Terao and Yuzvinsky. We alsoprove that: every nontrivial rank one local system on the complement can becomputed via these Twisted Logarithmic Comparison Theorems; these computationsare explicit finite dimensional linear algebra. Finally, we give some$\mathscr{D}_{X}$-module applications: for example, we give a sharp restrictionon the codimension one components of the multivariate Bernstein--Sato idealattached to an arbitrary factorization of an arrangement. The bound correspondsto (and, in the univariate case, gives an independent proof of) M. Saito'sresult that the roots of the Bernstein--Sato polynomial of a non-smooth,central, reduced arrangement live in $(-2 + 1/d, 0).$

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