  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.
