Families of proper holomorphic maps

TL;DR

This paper constructs a continuous family of proper holomorphic maps from a surface with varying complex structures into 2, demonstrating how such maps can depend continuously on parameters.

Contribution

It introduces a method to produce continuous families of proper holomorphic maps from surfaces with varying complex structures into 2.

Findings

Established the existence of continuous families of proper holomorphic maps

Demonstrated dependence of maps on parameter space

Extended previous results to more general surface families

Abstract

Given a smooth, open, oriented surface $X$ endowed with a family of complex structures $\{J_b\}_{b\in B}$ depending continuously on the parameter $b$ in a metrisable space $B$, we construct a continuous family of proper holomorphic maps $F_{b}:(X,J_b)\to\mathbb C^{2}$, $b\in B$.