If $L$ is a complex line bundle on a topological space $X$, let $c_1(L)$ denote the image of its Chern class in $H^2(X;C)$. A complex manifold structure on $X$ splits $H^2(X,C)$ into $H^{2,0}$, $H^{1,1}$, and $H^{0,2}$. If $L$ can be given a compatible complex structure, then $c_1(L)$ belongs to $H^{1,1}$. It's rewarding to study the moduli space of isomorphism classes of compatible complex structures on a given $L$--for instance, this is another finite-dimensional complex manifold in a natural way.

One not-very-specific way to explain what a "compatible complex structure" is, is to say that it's a tensor on the total space of L, satisfying a differential equation. Is there a natural kind of differential-geometric structure like this that one can put on a line bundle that guarantees that c_1(L) belongs to $H^{2,0}$ or $H^{0,2}$? Is there an interesting moduli space of such structures?

You could ask a similar question about Chern classes of higher-dimensional vector bundles on complex manifolds.