Torus Actions, Equivariant Moment-Angle Complexes, and Coordinate Subspace Arrangements

Abstract

We show that the cohomology algebra of the complement of a coordinate subspace arrangement in the m-dimensional complex space is isomorphic to the cohomology algebra of the Stanley―Reisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) After that we calculate the latter cohomology algebra by means of the standard Koszul resolution of a polynomial ring. To prove these facts, we construct a homotopy equivalence (equivariant with respect to the torus action) between the complement of a coordinate subspace arrangement and the moment-angle complex defined by a simplicial complex. The moment-angle complex is a certain subset of the unit polydisk in the m-dimensional complex space invariant with respect to the action of the m-dimensional torus. This complex is a smooth manifold provided that the simplicial complex is a simplicial sphere; otherwise, the complex has a more complicated structure. Then we investigate the equivariant topology of the moment-angle complex and apply the Eilenberg―Moore spectral sequence. We also relate our results with well-known facts in the theory of toric varieties and symplectic geometry. Bibliography: 23 titles.