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

## TL;DR

This paper proves that the map assigning cohomology classes to nondegenerate holomorphic maps from an open Riemann surface to a special domain is a Serre fibration, unifying several classical and modern results in complex analysis and minimal surface theory.

## Contribution

It establishes that the cohomology class map is a Serre fibration, generalizing classical period and divisor prescription results and parametric h-principles in minimal surface theory.

## Key findings

- The map $ ext{π}$ is a Serre fibration.
- Generalizes Kusunoki and Sainouchi's theorem on holomorphic forms.
- Unifies results in complex analysis and minimal surface theory.

## Abstract

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.