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 question:

(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