Polarized orbifolds associated to quantized Hamiltonian torus actions

Abstract

Suppose given a holomorphic and Hamiltonian action of a compact torus T on a polarized Hodge manifold M. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation of T on the associated Hardy space. If in addition the moment map is nowhere zero, for each weight ν the ν-th isotypical component in the Hardy space of the polarization is finite-dimensional. Assuming that the moment map is transverse to the ray through ν, we give a geometric interpretation of the isotypical components associated to the weights kν, k→+∞, in terms of certain polarized orbifolds associated to the Hamiltonian action and the weight. These orbifolds are generally not reductions of M in the usual sense, but arise rather as quotients of certain loci in the unit circle bundle of the polarization; this construction generalizes the one of weighted projective spaces as quotients of the unit sphere, viewed as the domain of the Hopf map.