52,707 research outputs found
Gerbes, simplicial forms and invariants for families of foliated bundles
The notion of a gerbe with connection is conveniently reformulated in terms
of the simplicial deRham complex. In particular the usual Chern-Weil and
Chern-Simons theory is well adapted to this framework and rather easily gives
rise to `characteristic gerbes' associated to families of bundles and
connections. In turn this gives invariants for families of foliated bundles. A
special case is the Quillen line bundle associated to families of flat
SU(2)-bundlesComment: 28 page
Relative cohomology of bi-arrangements
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.Comment: 43 pages; minor correction
The Orlik-Solomon model for hypersurface arrangements
We develop a model for the cohomology of the complement of a hypersurface
arrangement inside a smooth projective complex variety. This generalizes the
case of normal crossing divisors, discovered by P. Deligne in the context of
the mixed Hodge theory of smooth complex varieties. Our model is a global
version of the Orlik-Solomon algebra, which computes the cohomology of the
complement of a union of hyperplanes in an affine space. The main tool is the
complex of logarithmic forms along a hypersurface arrangement, and its weight
filtration. Connections with wonderful compactifications and the configuration
spaces of points on curves are also studied.Comment: 23 pages; presentation simplified, results unchange
Purity, formality, and arrangement complements
We prove a "purity implies formality" statement in the context of the
rational homotopy theory of smooth complex algebraic varieties, and apply it to
complements of hypersurface arrangements. In particular, we prove that the
complement of a toric arrangement is formal. This is analogous to the classical
formality theorem for complements of hyperplane arrangements, due to Brieskorn,
and generalizes a theorem of De Concini and Procesi.Comment: 9 pages; minor changes, references adde
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