In Voinsin's book [1], Theorem 11.32 (page 280) says:

"If X is an algebraic variety, these subgroups of $Hdg^{2k}(X) coincide."

However, the proof did not show that the subgroup generated by cycle classes (denoted by $A$) is containded in the subgroup generated by Chern classes of vector bundles (denoted by $B$) when $k>1$. In fact it just claims that $B\subseteq A$.

There are two questions:

(1) How to show $A\subseteq B$?

(2) Why is the condition that $X$ is an algebraic variety necessary?

[1] C. Voisin, Hodge theory and complex algebraic geometry, Vol. I, Cambrige Univ Press, 2002