Logarithmic Comparison Theorems for Hyperplane Arrangements, Twisted or Otherwise
In the 1990s, Terao and Yuzvinsky conjectured that reduced hyperplane arrangements satisfy the Logarithmic Comparison Theorem, asserting that the logarithmic de Rham complex computes the cohomology of the arrangement's complement. Essentially, this replaces the Brieskorn algebra in Brieskorn's Theorem with the logarithmic de Rham complex. We prove this conjecture by, among other things, sharply bounding the Castelnuovo--Mumford regularity of logarithmic j-forms of a central, essential, reduced arrangement. Time permitting we will discuss how to extend this untwisted Logarithmic Comparison Theorem to a twisted version. Here the twisted logarithmic de Rham complex computes the cohomology of the arrangement's complement with coefficients the rank one local system corresponding to the twist. Unlike the twisted Orlik--Solomon algebra, which can only compute a subset of the rank one local systems on the complement, this generalization computes all such rank one local systems.