The Orlik-Solomon model for hypersurface arrangements

TL;DR

This paper generalizes the Orlik-Solomon algebra to compute the cohomology of hypersurface arrangement complements in smooth projective varieties, extending known results from hyperplane arrangements and exploring related geometric structures.

Contribution

It introduces a global Orlik-Solomon model for hypersurface arrangements, utilizing logarithmic forms and weight filtrations, expanding the algebraic tools for complex geometric analysis.

Findings

Developed a cohomology model for hypersurface arrangements

Connected the model to wonderful compactifications and configuration spaces

Extended the Orlik-Solomon algebra beyond hyperplanes

Abstract

We develop a model for the cohomology of the complement of a hypersurface arrangement inside a smooth projective complex variety. This generalizes the case of normal crossing divisors, discovered by P. Deligne in the context of the mixed Hodge theory of smooth complex varieties. Our model is a global version of the Orlik-Solomon algebra, which computes the cohomology of the complement of a union of hyperplanes in an affine space. The main tool is the complex of logarithmic forms along a hypersurface arrangement, and its weight filtration. Connections with wonderful compactifications and the configuration spaces of points on curves are also studied.