TITLE: Integral isomorphism between $K_0(X)$ and $A(X)$ for toric varieties
QUESTION [13 upvotes]: Let $X$ be a smooth projective toric variety. The Chern character gives an isomorphism of rings:
$$\operatorname{Ch}:K_{0}(X)\otimes\mathbb{Q} \to A(X)\otimes \mathbb{Q} $$
where $K_{0}(X)$ is the Grothendieck group of vector bundles on $X$ and $A(X)$ is the Chow ring of $X$. This map seems only well-defined over $\mathbb{Q}$, but I was wondering (likely naively) if there is possibly an integral isomorphism (i.e. without tensor with $\mathbb{Q}$)?
Why might we hope for such a map? Fulton and Strumfels showed that there exists an isomorphism $\mathcal{D}_{X}:A^{k}(X)\to \operatorname{Hom}(A_{K}(X),\mathbb{Z})$ where $A^{k}(X)$ and $A_{k}(X)$ are the Chow cohomology and homology groups respectively. In particular, this means that the Chow ring $A(X)$ of a smooth toric variety is torsion free. In the couple very (very) simple examples I've done $K_{0}(X)$ also seems torsion free, although I am unsure whether this is true generally.
Of course, even if both $K_{0}(X)$ and $A(X)$ are torsion free there need not be an isomorphism between them, but one can hope.

REPLY [5 votes]: For the permutahedral toric variety associated to the type A hyperplane arrangement, there is such an isomorphism -- see Section 10 of Berget-Eur-Spink-Tseng https://arxiv.org/abs/2103.08021 . I believe this isomorphism also descends to toric varieties with coarser fans, and many small examples (like e.g. projective space) are special cases of this.