Let f : X → Y be a
C1-submersion from a complex manifold X to a real C1-manifold Y . One main
object of this paper is to prove two sets of necessary and sufficient conditions which
will guarantee that Y can be equipped with a complex structure making f
holomorphic. We provide a generic counterexample to show the essential nature of
the conditions we establish.
The second set of conditions (Theorem 3) apply even when X is a complex
Banach manifold. This theorem is then used to prove the existence of the natural
complex structure on the Teichmuller spaces of Riemann surfaces of finite
type.
