Relative cohomology of bi-arrangements

bi-arrangement of hyperplanes in a complex affine space is the data of two sets of hyperplanes along with coloring information on the strata. To such a bi-arrangement one naturally associates a relative cohomology group that we call its . The main reason for studying such relative cohomology groups comes from the notion of . More generally, we suggest the systematic study of the motive of a in a complex manifold. We provide combinatorial and cohomological tools to compute the structure of these motives. Our main object is the 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.]]>