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

Abstract

Let [Formula: see text] be a connected open Riemann surface. Let [Formula: see text] be an Oka domain in the smooth locus of an analytic subvariety of [Formula: see text], [Formula: see text], such that the convex hull of [Formula: see text] is all of [Formula: see text]. Let [Formula: see text] be the space of nondegenerate holomorphic maps [Formula: see text]. Take a holomorphic 1-form [Formula: see text] on [Formula: see text], not identically zero, and let [Formula: see text] send a map [Formula: see text] to the cohomology class of [Formula: see text]. Our main theorem states that [Formula: see text] 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 [Formula: see text] can be prescribed arbitrarily. It also subsumes two parametric h-principles in minimal surface theory proved by Forstnerič and Lárusson in 2016.