A factorisation theorem for curve with vanishing self-intersection
  Let C be a curve in a compact Kahler surface Y. Assume that the self-intersection of C is vanishing, and the image of the fundamental group of C in Y is of infinite index in the fundamental group of Y. Does it follow that C is a fiber of a holomorphic map from Y to a curve ?   We give, in some cases, a positive answer to the question. Our strategy is to solve a Poincare-Lelong equation on some infinite covering of Y. 
Pascal Dingoyan
math.CV