TITLE: Classification of holomorphic disc bundles
QUESTION [10 upvotes]: I've had difficulty finding sources which treat the classification of holomorphic disc bundles over (compact and noncompact) Riemann surfaces.  Note that by "bundle", I mean a holomorphic fiber bundle, which means it is locally holomorphically trivial.     
I'm really just looking for information about holomorphic disc bundles, but if you want a specific question: 
What is the classification of holomorphic disc bundles over a Riemann surface?

REPLY [3 votes]: If I don't misunderstand the question, I don't  believe that any kind of such classification can exists in the case when the normal bundle of the surface has positive degree. This moduli is infinite-dimensional. 
Even in the case of negative degree there are problems. If we take a disk bundle over $\mathbb CP^1$ of degree -1, and contract $\mathbb CP^1$ we will get in particular plenty of complex balls in $\mathbb C^2$. Can one classify them up to biholomorphism?