I am a little confused by the different definitions for toric manifolds/varieties. Depending on the definition of toric manifolds and principal torus bundles that one chooses, when is a toric manifold equivalent to principal torus bundle? If they are not the same by any definition, what is the obstruction preventing one from being the other (i.e. when is a toric manifold also a principal torus bundle)?
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1$\begingroup$ What is your definition of "principal torus bundle"? Are you asking about the description of toric varieties via the moment map, whose generic fiber is diffeomorphic to a product of copies of the circle? $\endgroup$– Jason StarrCommented Nov 4, 2015 at 12:09
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1$\begingroup$ Principal torus bundle $P$ over $X$ in the sense that there is a group action of $T^n$ on $P$ and a projection $P \to X$ ($T^n$ preserves the fiber etc. etc.). Then yes, my point is that via the moment map these two definitions seem to coincide? $\endgroup$– harryCommented Nov 4, 2015 at 12:19
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The only space which is both a toric variety and a principal torus bundle is the complex torus itself. Any other toric variety necessarily contains a non-free torus orbit, and thus is not a principal torus bundle.
answered Nov 4, 2015 at 13:12