Small rational model of subspace complement

Abstract

This paper concerns the rational cohomology ring of the complement M M of a complex subspace arrangement. We start with the De Concini-Procesi differential graded algebra that is a rational model for M M . Inside it we find a much smaller subalgebra D D quasi-isomorphic to the whole algebra. D D is described by defining a natural multiplication on a chain complex whose homology is the local homology of the intersection lattice L L whence connecting the De Concini-Procesi model with the Goresky-MacPherson formula for the additive structure of H ∗ ( M ) H^*(M) . The algebra D D has a natural integral version that is a good candidate for an integral model of M M . If the rational local homology of L L can be computed explicitly we obtain an explicit presentation of the ring H ∗ ( M , Q ) H^*(M,{\mathbf Q}) . For example, this is done for the cases where L L is a geometric lattice and where M M is a k k -equal manifold.