Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms

Let X be a connected open Riemann surface. Let Y be an Oka domain in the smooth locus of an analytic subvariety of C-n, n >= 1, such that the convex hull of Y is all of C-n. Let O*( X, Y) be the space of nondegenerate holomorphic maps X -> Y. Take a holomorphic 1-form theta on X, not identically zero, and let pi : O*( X, Y) -> H-1( X, C-n) send a map g to the cohomology class of g theta. Our main theorem states that pi is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the divisor of a holomorphic form on X can be prescribed arbitrarily. It also subsumes two parametric h-principles in minimal surface theory proved by Forstneric and Larusson in 2016.