Initial monomial invariants of holomorphic maps

TL;DR

This paper introduces a new biholomorphic invariant for holomorphic maps based on generic initial ideals, enabling classification and distinction of maps between complex domains, especially spheres and hyperquadrics.

Contribution

It develops a novel invariant using generic initial ideals for holomorphic maps, providing a computable tool for classifying maps between complex domains.

Findings

Generic initial monomials distinguish all four inequivalent rational proper maps from the 2-ball to the 3-ball.

The invariant is computable for rational maps of spheres and hyperquadrics.

The generic initial monomial subspace remains invariant under biholomorphic transformations and certain automorphisms.

Abstract

We study a new biholomorphic invariant of holomorphic maps between domains in different dimensions based on generic initial ideals. We start with the standard generic monomial ideals to find invariants for rational maps of spheres and hyperquadrics, giving a readily computable invariant in this important case. For example, the generic initial monomials distinguish all four inequivalent rational proper maps from the two to the three dimensional ball. Next, we associate to each subspace $X \subset {\mathcal O}(U)$ a generic initial monomial subspace, which is invariant under biholomorphic transformations and multiplication by nonzero functions. The generic initial monomial subspace is a biholomorphic invariant for holomorphic maps if the target automorphism is linear fractional as in the case of automorphisms of spheres or hyperquadrics.