Timeline for In a nonsingular complex toric variety, is an algebraic cycle over a facet of the quotient polytope integral?
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| Aug 25, 2016 at 22:58 | comment | added | Max Kutler | 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. | |
| Aug 25, 2016 at 22:47 | comment | added | Max Kutler | 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. | |
| Aug 20, 2016 at 3:15 | comment | added | user94803 | Thanks Max! So apparently, the assumption of nonsingularity in my question is not necessary. | |
| Aug 19, 2016 at 1:19 | vote | accept | CommunityBot | ||
| Aug 18, 2016 at 18:35 | history | answered | Max Kutler | CC BY-SA 3.0 |