I am currently learning intersection theory of smooth algebraic varieties and I have the following question.
Let $X$ be a smooth projective variety and $\mathcal{F}$ a vector bundle on $X$. Then the $i$th Chern class of $\mathcal{F}$ is an element of the $i$th Chow group $A^i(X)$ of $X$. What about the converse? Can every class in $A^i(X)$ be realized as the $i$th Chern class if a vector bundle? Clearly, this is true for $A^1(X)$ and it is true for all $i$ if $X$ is the projective space. Is it true in general? If not, is it true for some nice varieties, for example for Grassmannians?
It is not true in general. The counterexample that I know is $X$ a general hyperplane section of $LGr(3,6)$ and the class of a line on $X$.