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 ${\mathbb{C}}^n$, $n\ge 1$, such that the convex hull of $Y$ is all of ${\mathbb{C}}^n$. Let ${\mathcal{O}}_{\ast }(X,Y)$ be the space of nondegenerate holomorphic maps $X\to Y$. Take a holomorphic $1$-form $\theta$ on $X$, not identically zero, and let $\pi:\mathscr O_*(X,Y) \to H^1(X,\mathbb C^n)$sendamap$g$tothecohomologyclassof$g\theta$. 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.