Definition of quasi toric manifolds :

The action of $(S^1)^n$ on $\Bbb C^n$ by pointwise multiplication is called the standard representation. Given a manifold $M^{2n}$ with an $(S^1)^n$-action, a local isomorphism of $M^{2n}$ with the standard representation consists of

  1. an automorphism $\theta:(S^1)^n\to (S^1)^n$
  2. $(S^1)^n$-stable open sets $V$ in $M^{2n}$ and $W$ in $\Bbb C^n$ and
  3. a $\theta$-equivariant homeomorphism $f:V\to W$

One says that $M^{2n}$ is locally isomorphic to standard representation if each point of $M^{2n}$ is in the domain of some local isomorphism.

Let $P^n$ be a simple convex polytope. A quasitoric manifold over $P^n$ is a manifold $M^{2n}$ with an $(S^1)^n$ action that is locally isomorphic to the standard representation with a projection map $\pi:M^{2n}\to P^n$ such that the fibres are the $(S^1)^n$ orbits.

Definition of small covers :

By replacing $\Bbb C^n$ by $\Bbb R^n$, $(S^1)^n$ by $\Bbb Z_2^n$ (where $\Bbb Z_2=\{1,-1\}$) and $M^{2n}$ by $M^n$ in the above definition we get a small cover over $P^n$.

In the Wikipedia article on Quasitoric manifolds in the section "comparison with toric manifolds" they state the following :

  1. Any projective toric manifold is a quasitoric manifold, and in some cases non-projective toric manifolds are also quasitoric manifolds.

  2. Not all quasitoric manifolds are toric manifolds. For example, the connected sum $\mathbb {C} P^{2}\sharp \mathbb {C} P^{2}$ can be constructed as a quasitoric manifold, but it is not a toric manifold.

Question : I was wondering if the same statements can be made for small covers (real analogues of quasitoric manifolds) and the real valued points of a toric manifold?

For example where we have $\mathbb {C} P^{2}\sharp \mathbb {C} P^{2}$ is a quasitoric manifold but not a toric manifold, we have $\mathbb {C} P^{2}\sharp \mathbb {C} P^{2}$ which is the Klein bottle, is a small cover as well as a "real" toric manifold.



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