The moment map and line bundles over presymplectic toric manifolds (Q1312135)

Let \(M\) be a presymplectic manifold with (possibly degenerate) closed 2- form \(\omega\), let a compact torus \(T\) act on \(M\) effectively preserving \(\omega\) and let \((M,T,\omega)\) admit a moment map \(\Phi: M \to t^*\). Consider the push-forward of Liouville measure, \(\Phi_ *\omega^ n\). The authors give an explicit description of \(\Phi_ *\omega^ n\). It is a ``twisted polytope'' in \(t^*\) which is determined by the winding of numbers of a map \(S^{n-1} \to t^*\) around points in \(t^*\). The index of an equivariant, holomorphic line bundle with curvature \(\omega\) is a virtual \(T\)-representation easily described by means of this ``twisted polytope''.