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Let $X$ be a $n$-dimensional toric variety associate to a fan $\Sigma$. It is already known that for any curve (i.e. $1$-cycle) $C$ on $X$, $C$ is linearly equivalent to $$\sum_{ w \in \Sigma(n-1)} a_w V(w)$$ where $\Sigma(n-1)$ is the set of $n-1$ dimensional cone in the fan $\Sigma$, and $V(w)$ is the corresponding 1 dimensional toric invariant subvariety. The same thing also holds for a $(n-1)$-cycle (i.e. a Weil divisor).

Question: is it true that for any $k$-cycle $D$ on $X$, $D$ is linearly equivalent to a summation of toric invariant subvarieties of dimensional $k$? That is $$D \sim \sum_{w\in \Sigma(n-k)} a_w V(w)?$$

Besides, if linearly equivalent is too strong, I am fine with a weaker condition like numerically equivalent.

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"Linearly equivalent" is an equivalence relation on divisors only. The right notion for arbitrary cycles is "rationally equivalent".

With this adjustment the answer to your question is yes. A reference is Proposition on p.96 of Fulton, Toric Varieties (where you should already have looked!)

The idea of the proof is very simple: when you delete all the torus-invariant $k$-cycles from $X$, what is left over is a disjoint union of orbits, i.e. open subsets of affine space. Then the usual exact sequence for the inclusion of a closed subscheme gives the result you want.

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  • $\begingroup$ I see, thank you!! Since the curve case is proved using very complicated method in Reid's paper of MMP of toric variety, I thought it might be even harder to prove in the general case... $\endgroup$
    – Li Yutong
    Commented Feb 6, 2015 at 14:03

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