Let $X$ be a nonsingular complex toric variety with moment map $\mu : X \to P$ over a convex polytope $P$. Given a facet $F$ of $P$, its preimage $\mu^{-1}(F)$ is a complex codimension 1 subvariety of $X$, so by Poincare duality, induces a cohomological class $\mathfrak{z}\_F \in H^2(X; \mathbb{C})$.


Is $\mathfrak{z}\_F$ integral, i.e. does it lie in the image of the map $H^2(X; \mathbb{Z}) \to H^2(X; \mathbb{C})$ induced by the inclusion of coefficient rings $\mathbb{Z} \hookrightarrow \mathbb{C}$?



In a nonsingular complex toric variety, is an algebraic cycle over a facet of the quotient polytope integral?
Yes, and in fact the classes $\mathfrak{z}\_F$ generate the integral cohomology ring. See, for instance, Section 5.7 of [these notes](http://pages.uoregon.edu/njp/sg.pdf) by Nick Proudfoot. He proves a stronger result: these classes generate the $T$-equivariant integral cohomology ring.


