480 research outputs found
Matroids, Feynman categories, and Koszul duality
We show that various combinatorial invariants of matroids such as Chow rings
and Orlik--Solomon algebras may be assembled into "operad-like" structures.
Specifically, one obtains several operads over a certain Feynman category which
we introduce and study in detail. In addition, we establish a Koszul-type
duality between Chow rings and Orlik--Solomon algebras, vastly generalizing a
celebrated result of Getzler. This provides a new interpretation of
combinatorial Leray models of Orlik--Solomon algebras.Comment: Should be an almost final versio
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
Algebras related to posets of hyperplanes
We compare two noncommutative algebras which are related to arrangements of hyperplanes. For three special arrangements the induced approximately finite dimensional -algebra and the graded Orlik-Solomon-algebra are investigated
On the homotopy Lie algebra of an arrangement
Let A be a graded-commutative, connected k-algebra generated in degree 1. The
homotopy Lie algebra g_A is defined to be the Lie algebra of primitives of the
Yoneda algebra, Ext_A(k,k). Under certain homological assumptions on A and its
quadratic closure, we express g_A as a semi-direct product of the
well-understood holonomy Lie algebra h_A with a certain h_A-module. This allows
us to compute the homotopy Lie algebra associated to the cohomology ring of the
complement of a complex hyperplane arrangement, provided some combinatorial
assumptions are satisfied. As an application, we give examples of hyperplane
arrangements whose complements have the same Poincar\'e polynomial, the same
fundamental group, and the same holonomy Lie algebra, yet different homotopy
Lie algebras.Comment: 20 pages; accepted for publication by the Michigan Math. Journa
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