Algebraizability of Vector Bundles over Real Algebraic Varieties

Let $X$ be an affine smooth real algebraic variety (in the sense of Bochnak, Coste, and Roy) and let $V$ be a topological vector bundle over $X(\mathbb{R})$. We investigate the problem of deciding whether $V$ is topologically isomorphic to an algebraic vector bundle using motivic homotopy theory. We prove that if $\dim X\leq 3$, then the algebraicity of Stiefel-Whitney classes is a necessary and sufficient condition for $V$ to be algebraizable. Next, we show that when $\dim X=4$ and $X(\mathbb{R})$ is compact, even if the characteristic classes of $V$ are algebraic, there is still an obstruction to algebraizing $V$ related to the Pontryagin class $p_1$ and the Stiefel-Whitney class $w_4$. Then we give some applications of this result. Namely, we give an example where this obstruction is nontrivial, and we investigate the group $\mathrm{K}_0(X)$.