On uniqueness of the equivariant smooth structure on a real moment-angle manifold
Nikolai Erokhovets, Elena Erokhovets
TL;DR
The paper addresses the uniqueness of the equivariant smooth structure on real moment-angle manifolds $\mathbb{R}\mathcal{Z}_P$ for simple polytopes, extending the complex-case results of Bosio and Meersseman to the real setting. By leveraging a given $\mathbb{T}^m$-equivariant diffeomorphism between complex moment-angle manifolds and translating it into a $\mathbb{Z}_2^m$-equivariant diffeomorphism between the real loci, the authors also construct a diffeomorphism between the underlying polytopes viewed as manifolds with corners, showing that linear and quadratic smooth structures on the polytope are diffeomorphic. They prove the induced maps are smooth and deduce the uniqueness of the smooth structure on $\mathbb{R}\mathcal{Z}_P$ making the $\mathbb{Z}_2^m$-action smooth, connecting with recent results in Kuroki–Masuda–Yu. This work strengthens the real-case analogue of equivariant smooth rigidity in toric topology and ties together the geometry of moment-angle manifolds with polytope smooth structures and small covers.
Abstract
The paper is devoted to the well-known problem of smooth structures on moment-angle manifolds. Each real or complex moment-angle manifold has an equivariant smooth structure given by an intersection of quadrics corresponding to a geometric realisation of a polytope. In 2006 F.Bosio and L.Meersseman proved that complex moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic. Using arguments from calculus we derive from this result that real moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic and the polytopes are diffeomorphic as manifolds with corners.