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Relative cohomology of bi-arrangements


Author: Clément Dupont
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 14C30, 14F05, 14F25, 52C35
DOI: https://doi.org/10.1090/tran/6904
Published electronically: May 30, 2017
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Abstract: A 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 motive. The main reason 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.


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Clément Dupont
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
Address at time of publication: Institut Montpelliérain Alexander Grothendieck, CNRS, Université Montpellier, France
Email: clement.dupont@umontpellier.fr

DOI: https://doi.org/10.1090/tran/6904
Keywords: Arrangements, relative cohomology, mixed Hodge structures
Received by editor(s): March 12, 2015
Received by editor(s) in revised form: January 18, 2016, and January 19, 2016
Published electronically: May 30, 2017
Article copyright: © Copyright 2017 American Mathematical Society