Let $E\to X$ be a complex vector bundle over a compact Kahler surface $X$. Assume $c\_{i}(E)\in H^{i,i}(X)$ for all i. Does the bundle $E$ admit a holomorphic structure?



On holomorphic vector bundles over compact Kahler surfaces
For line bundles, this is the Lefschetz (1,1)-theorem which says that the map $H^1(X,\mathcal{O}^\times)\to H^2(X,\mathbb{Z})$ from the exponential sequence surjects onto the $H^{1,1}(X)$-part.


For higher rank bundles, the answer is positive whenever $X$ is additionally assumed to be projective. In this case, a theorem of Schwarzenberger states that a complex vector bundle is algebraizable if and only if its determinant is algebraizable, see Theorem 9 in (R.L.E. Schwarzenberger: Vector bundles on algebraic surfaces. Proc. London Math. Soc. (3) 11 (1961), 601--622).


