Relative cohomology of bi-arrangements

TL;DR

This paper introduces a new combinatorial and cohomological framework for studying the motives of bi-arrangements of hyperplanes, generalizing classical arrangements, and applies it to compute motives related to multiple zeta values and mixed Tate motives.

Contribution

It develops the Orlik-Solomon bi-complex for bi-arrangements, providing a systematic method to compute their motives, extending classical arrangement theory.

Findings

The motive of an exact bi-arrangement is computed by its Orlik-Solomon bi-complex.

The formalism enables explicit calculations of motives related to multiple zeta values.

Application to periods of mixed Tate motives demonstrates the framework's versatility.

Abstract

A bi-arrangement of hyperplanes in a complex affine space is the data of two sets of hyperplanes along with a coloring information on the strata. To such a bi-arrangement, one naturally associates a relative cohomology group, that we call its motive. The motivation for studying such relative cohomology groups comes from the notion of motivic period. More generally, we suggest the systematic study of the motive of a bi-arrangement of hypersurfaces in a complex manifold. We provide combinatorial and cohomological tools to compute the structure of these motives. Our main object is the Orlik-Solomon bi-complex of a bi-arrangement, which generalizes the Orlik-Solomon algebra of an arrangement. Loosely speaking, our main result states that "the motive of an exact bi-arrangement is computed by its Orlik-Solomon bi-complex", which generalizes classical facts involving the Orlik-Solomon algebra of an arrangement. We show how this formalism allows us to explicitly compute motives arising from the study of multiple zeta values and sketch a more general application to periods of mixed Tate motives.