A criterion for a degree-one holomorphic map to be a biholomorphism

TL;DR

The paper proves that any surjective holomorphic map of degree one between certain compact complex manifolds is a biholomorphism, removing previous extra assumptions by showing they are automatically satisfied.

Contribution

It establishes a criterion for degree-one holomorphic maps to be biholomorphisms without additional conditions, generalizing previous results.

Findings

Any surjective degree-one holomorphic map between the manifolds is a biholomorphism.

The condition on the dimensions of the first cohomology groups is automatically satisfied.

The result applies to compact connected complex manifolds with equal second Betti numbers.

Abstract

Let $X$ and $Y$ be compact connected complex manifolds of the same dimension with $b_2(X)= b_2(Y)$. We prove that any surjective holomorphic map of degree one from $X$ to $Y$ is a biholomorphism. A version of this was established by the first two authors, but under an extra assumption that $\dim H^1(X {\mathcal O}_X)\,=\,\dim H^1(Y {\mathcal O}_Y)$. We show that this condition is actually automatically satisfied.