Holomorphic Gromov's partial order

TL;DR

This paper investigates holomorphic maps between compact complex manifolds, proving that degree-one maps are biholomorphic and establishing the holomorphic Gromov relation as a partial order under certain conditions.

Contribution

It demonstrates that holomorphic maps of degree one are biholomorphic and confirms the holomorphic Gromov relation as a partial order under mild restrictions.

Findings

Holomorphic maps of degree one are biholomorphic.

The holomorphic Gromov relation '>_' is a partial order under certain conditions.

Provides a criterion for when the relation is a partial order.

Abstract

As in [5], we study holomorphic maps of positive degree between compact complex manifolds, and prove that any holomorphic map of degree one from a compact complex manifold to itself is biholomorphic. This conclusion confirms that under a mild restriction the holomorphic Gromov relation ">_" is indeed a partial order.