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?
Best Answer
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 $X$.