Rigidity of holomorphic maps between fiber spaces

TL;DR

This paper investigates the rigidity of holomorphic maps between complex manifolds, establishing conditions under which such maps are biholomorphisms and characterizing maps between product spaces.

Contribution

It provides new criteria for when degree-one holomorphic maps are biholomorphic and describes the structure of holomorphic maps between product manifolds.

Findings

Degree-one holomorphic maps between diffeomorphic manifolds are biholomorphic under certain conditions.

Rigidity results for holomorphic self-maps of fiber spaces.

Characterization of non-constant holomorphic maps from product spaces involving Riemann surfaces.

Abstract

In the study of holomorphic maps, the term "rigidity" refers to certain types of results that give us very specific information about a general class of holomorphic maps owing to the geometry of their domains or target spaces. Under this theme, we begin by studying when, given two compact connected complex manifolds $X$ and $Y$, a degree-one holomorphic map $f: Y\to X$ is a biholomorphism. Given that the real manifolds underlying $X$ and $Y$ are diffeomorphic, we provide a condition under which $f$ is a biholomorphism. Using this result, we deduce a rigidity result for holomorphic self-maps of the total space of a holomorphic fiber space. Lastly, we consider products $X=X_1\times X_2$ and $Y=Y_1\times Y_2$ of compact connected complex manifolds. When $X_1$ is a Riemann surface of genus $\geq 2$, we show that any non-constant holomorphic map $F:Y\to X$ is of a special form.