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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 $\mathbb{Z} \hookrightarrow \mathbb{C}$?

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    $\begingroup$ On a smooth variety every Weil divisor defines an integral cohomology class. $\endgroup$
    – byu
    Commented Aug 18, 2016 at 18:25

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Yes, and in fact the classes $\mathfrak{z}_F$ generate the integral cohomology ring. See, for instance, Section 5.7 of these notes by Nick Proudfoot. He proves a stronger result: these classes generate the $T$-equivariant integral cohomology ring.

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  • $\begingroup$ Thanks Max! So apparently, the assumption of nonsingularity in my question is not necessary. $\endgroup$
    – user94803
    Commented Aug 20, 2016 at 3:15
  • $\begingroup$ The classes $\mathfrak{z}_F$ should always be integral. As far as generating the cohomology ring goes, you do need nonsingularity (all of Proudfoot's toric varieties are nonsingular). This can be relaxed a little, but you need to be careful, as $H^*(X; \mathbb{Z})$ can certainly have torsion. In the case where $X$ is complete (projective) and simplicial, the classes $\mathfrak{z}_F$ will generate the rational cohomology $H^*(X; \mathbb{Q})$. I am not sure what you can say when $X$ is not simplicial or not complete. $\endgroup$
    – Max Kutler
    Commented Aug 25, 2016 at 22:47
  • $\begingroup$ Actually, you can relax the simplicial requirement if you work with Chow cohomology $A^*(X)$ instead of ordinary (singular) cohomology. (This also allows you to consider toric varieties defined over fields other than $\mathbb{C}$.) If $X$ is complete, then $A^*(X)$ has a system of generators indexed by codimension $1$ torus-invariant subvarieties. These subvarieties correspond faces of the polytope $P$ (or, more generally, to rays of the fan of $X$). See, e.g., Fulton-Sturmfels "Intersection theory on toric varieites" arxiv.org/abs/alg-geom/9403002. $\endgroup$
    – Max Kutler
    Commented Aug 25, 2016 at 22:58

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