Relative cohomology of bi-arrangements

Abstract

A bi-arrangement of hyperplanes in a complex affine space is data of two sets of hyperplanes along with a coloring information on 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 notion of motivic period. More generally, we suggest systematic study of motive of a bi-arrangement of hypersurfaces in a complex manifold. We provide combinatorial and cohomological tools to compute structure of these motives. Our main object is Orlik-Solomon bi-complex of a bi-arrangement, which generalizes 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 Orlik-Solomon algebra of an arrangement. We show how this formalism allows us to explicitly compute motives arising from study of multiple zeta values and sketch a more general application to periods of mixed Tate motives.