Embedding theorems for quasi-toric manifolds given by combinatorial data (Q2832317)

The well-known Mostow-Palais theorem states that for every compact smooth manifold \(M\) equipped with a smooth action of a compact Lie group \(G\) there is a smooth embedding of \(M\) into \(\mathbb R^N\), which is equivariant with respect to some representation \(G\rightarrow\mathrm{GL}_N(\mathbb R)\).NEWLINENEWLINEThe paper under review considers and explicitly constructs equivariant embeddings of quasi-toric manifolds. A quasi-toric manifold \(M\) is equipped with an action of the (\(n\)-dimensional) torus \(T^n\). Such a manifold can be described by its orbit polytope \(P\subset\mathbb R^n\) and its characteristic matrix -- an integer matrix which satisfies a certain combinatorial condition. The embeddings constructed in the paper are determined by these combinatorial data. They are of the form \(M\hookrightarrow\mathbb R^n\times\mathbb C^q\) and \(M\hookrightarrow\mathbb R^n\times\mathbb C\mathrm P^{q-1}\) and they are equivariant with respect to a representation \(T^n\rightarrow T^q\) (the torus \(T^q\) acts trivially on \(\mathbb R^n\) and naturally on \(\mathbb C^q\) and \(\mathbb C\mathrm P^{q-1}\)). The first coordinate map of these embeddings is the projection (the moment map) \(M\rightarrow P\subset\mathbb R^n\), and the number of edges of the orbit polytope \(P\) is shown to be an upper bound for the integer \(q\) in the first embedding.