Torus actions and combinatorics of polytopes (Q2736069)

The authors study smooth manifolds defined by simple convex polytopes. With every \(n\)-dimensional simple polytope \(P\) with \(m\) facets is associated a smooth \((m+n)\)-dimensional manifold \({\mathcal Z}_P\) that admits a canonical action of the compact \(m\)-torus \(T^m\). Many manifolds that are important in topology, algebra and symplectic geometry arise in this context as quotients of manifolds \({\mathcal Z}_P\) by toric subgroups \(T^k\) of \(T^m\) which act freely on \({\mathcal Z}_P\). The class of quasitoric manifolds is obtained when \(k=m-n\), which is the largest possible value for \(k\); this class includes all smooth projective toric varieties (toric manifolds).NEWLINENEWLINENEWLINEThe authors take the following approach to constructing these manifolds. The combinatorial structure of \(P\) determines a certain collection of affine planes in \(\mathbb{C}^m\), whose complement \(U(P)\) in \(\mathbb{C}^m\) admits an action by \((\mathbb{C}^*)^m\). It is always possible to find certain subgroups \(R\) of \((\mathbb{C}^*)^m\) which are isomorphic to \((\mathbb{R}^*_+)^{m-n}\) and act freely on \(U(P)\). The manifolds then are the quotients of \(U(P)\) by such subgroups.NEWLINENEWLINENEWLINEIn particular, the authors investigate the relationships between the combinatorics of simple polytopes and the topology of the manifolds associated with them. It is proved that the cohomology of \({\mathcal Z}_P\) possesses a natural structure as a bigraded algebra. This bigraded cohomology algebra carries all the information about the combinatorics of \(P\). For example, there are appropriate interpretations of the Dehn-Sommerville equations and the upper-bound-theorem for simple convex polytopes.NEWLINENEWLINEFor the entire collection see [Zbl 0967.00102].