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3

Warning: I'm not an expert on logarithmic geometry. According to Proposition 4.1.2. of Ogus' Lectures on Logarithmic Algebraic Geometry, a log smooth morphism (of fine log schemes) is log flat. Log flatness is defined in Definition 4.1.1. Evidently, log flatness does not imply classical flatness. However, the same proposition says that log flatness does ...

2

I think not, my argument is slightly messy but it works I guess. Take the Hirzebruch surface $S=\mathbb{F}_2$ with its standard toric action, it is diffeomorphic to $S'=\mathbb{CP}^1 \times \mathbb{CP}^1$, but I claim they are not equivariantly diffeomorphic (with respect to the standard toric action on $\mathbb{CP}^1 \times \mathbb{CP}^1$). Suppose that ...

2

In the definition of moment-angle manifold in Davis-Januszkiewicz's 1991 paper, the orbit space is a simple convex polytope $P$ which is contractible. We can replace $P$ by an arbitrary smooth nice manifold with corners $Q$ and define the generalized moment-angle manifold $\mathcal{Z}_Q$ in the similar way as usual moment-angle manifold (see https://arxiv....

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