  Let $X$ be a connected open Riemann surface. Let $Y$ be an Oka domain in the
smooth locus of an analytic subvariety of $\mathbb C^n$, $n\geq 1$, such that
the convex hull of $Y$ is all of $\mathbb C^n$. Let $\mathscr O_*(X, Y)$ be the
space of nondegenerate holomorphic maps $X\to Y$. Take a holomorphic $1$-form
$\theta$ on $X$, not identically zero, and let $\pi:\mathscr O_*(X,Y) \to
H^1(X,\mathbb C^n)$ 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.
