A Factorization Theorem for Curves with Vanishing Self-Intersection

Let C be a connected divisor in a compact Kähler manifold such that the self-intersection of C, computed with respect to a Kähler metric, vanishes. Assume that the normal closure of the image of \(\pi _{1}(C)\) in \(\pi _{1}(Y)\) has infinite index. Then there exists a holomorphic map f from Y to a curve B such that C is a fiber. The conclusion holds if one assumes that the image of \(\pi _{1}(C)\) is amenable but \(\pi _{1}(Y)\) is non-amenable.