Let $X$ be a smooth projective toric variety. The Chern character gives an isomorphism of rings:

$$operatorname{Ch}:K_{0}(X)otimesmathbb{Q} to A(X)otimes mathbb{Q} $$

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.