Suppose we have a smooth complex algebraic variety $X$. Then in general, $K^a(X)\to K(X^{an})$ is not surjective. Could someone give an example of a topological vector bundle class which contains no algebraic vector bundles?
In a similar spirit, what is an example of a homology class in $H_{2*}(X^{an})$ representable by a combination of submanifolds, but not in the image of $A^*(X)$ (i.e. not representable by a combination of submanifolds which are the analytifications of subvarieties)?