Classification of locally standard torus actions

Abstract

An action of a torus T on a manifold M is locally standard if, at each point, the stabilizer is a sub-torus and the non-zero isotropy weights are a basis to its weight lattice. The quotient M/T is then a manifold-with-corners, decorated by a so-called unimodular labelling, which keeps track of the isotropy representations in M, and by a degree two cohomology class with coefficients in the integral lattice of the Lie algebra of T, which encodes the "twistedness" of M over M/T. We classify locally standard smooth actions of T, up to equivariant diffeomorphisms, in terms of triples (Q,lambda,c), where Q is a manifold-with-corners, lambda is a unimodular labelling, and c is a degree two cohomology class with coefficients in the integral lattice.