I am interested in universal covers of complex manifolds. Are the following true?
- The universal cover of a complex manifold is a complex manifold?
- The deck transformations of the universal cover are biholomorphic?
Coming from non-complex differential geometry/topology I only know so far that the universal cover of a manifold is a manifold and that the Deck transformations are smooth.
I am quite sure that the complex structure should lift to the universal cover. But what about the Deck transformations. In my book they are defined to be biholomorphic maps $f: X\to X$ with $p=p\circ f$, where $X$ is the universal cover and $p:X\to Y$ is the projection.
But is this definition a restriction? Are there maps $g$ satisfying $p=p\circ g$ which are not biholomorphic. In particular, is the group of Deck transformations still the fundamental group when using the definition as above (i.e. requesting biholomorphicy)?
Depending how hard this is to prove, a (good) reference would be enough.