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Exhaustion functions and normal forms for proper maps of balls

Jiri Lebl

TL;DR

The paper presents a computable normal form for rational proper maps $f:\mathbb{B}_n\to\mathbb{B}_N$ up to spherical equivalence modulo a finite unitary subgroup, derived from a strongly plurisubharmonic exhaustion function $\Lambda_f$. By locating the unique critical point of $\Lambda_f$ and transporting it to the origin, the denominator can be normalized so its linear terms vanish and its quadratic part diagonalizes as $G_2(z)=\sum_{k=1}^n\sigma_k z_k^2$, with the $\sigma_k$ serving as spherical invariants; when the $\sigma_k$ are distinct and positive, the normal form is unique up to a finite subgroup of $U(n)$. The work also proves the existence of degree-3 maps for small $\sigma_k$ and analyzes when a map is target-equivalent to a polynomial, including a discussion of embedding-dimension and model targets such as $\mathbb{B}_{1,N-1}$ and $\mathbb{H}_N$. Collectively, these results provide a practical framework for classifying and understanding rational proper maps of balls and their polynomial counterparts, with implications for mappings between balls, generalized balls, and Heisenberg realizations.

Abstract

We study a relationship between rational proper maps of balls in different dimensions and strongly plurisubharmonic exhaustion functions of the unit ball induced by such maps. Putting the unique critical point of this exhaustion function at the origin leads to a normal form for rational proper maps of balls. The normal form of the map, which is up to composition with unitaries, takes the origin to the origin, and it normalizes the denominator by eliminating the linear terms and diagonalizing the quadratic part. The singular values of the quadratic part of the denominator are spherical invariants of the map. When these singular values are positive and distinct, the normal form is determined up to a finite subgroup of the unitary group. We also study which denominators arise for cubic maps, and when we do not require taking the origin to the origin, which maps are equivalent to polynomials.

Exhaustion functions and normal forms for proper maps of balls

TL;DR

The paper presents a computable normal form for rational proper maps up to spherical equivalence modulo a finite unitary subgroup, derived from a strongly plurisubharmonic exhaustion function . By locating the unique critical point of and transporting it to the origin, the denominator can be normalized so its linear terms vanish and its quadratic part diagonalizes as , with the serving as spherical invariants; when the are distinct and positive, the normal form is unique up to a finite subgroup of . The work also proves the existence of degree-3 maps for small and analyzes when a map is target-equivalent to a polynomial, including a discussion of embedding-dimension and model targets such as and . Collectively, these results provide a practical framework for classifying and understanding rational proper maps of balls and their polynomial counterparts, with implications for mappings between balls, generalized balls, and Heisenberg realizations.

Abstract

We study a relationship between rational proper maps of balls in different dimensions and strongly plurisubharmonic exhaustion functions of the unit ball induced by such maps. Putting the unique critical point of this exhaustion function at the origin leads to a normal form for rational proper maps of balls. The normal form of the map, which is up to composition with unitaries, takes the origin to the origin, and it normalizes the denominator by eliminating the linear terms and diagonalizing the quadratic part. The singular values of the quadratic part of the denominator are spherical invariants of the map. When these singular values are positive and distinct, the normal form is determined up to a finite subgroup of the unitary group. We also study which denominators arise for cubic maps, and when we do not require taking the origin to the origin, which maps are equivalent to polynomials.
Paper Structure (6 sections, 16 theorems, 60 equations)

This paper contains 6 sections, 16 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on proper maps of balls
  3. The $\Lambda$-function
  4. Normalizing the denominator
  5. Existence of third degree maps
  6. Rational and polynomial maps of maximal embedding dimension

Key Result

Theorem 1.1

Suppose $f \colon {\mathbb{B}}_n \to {\mathbb{B}}_N$ is a rational proper map of degree $d$. Then there exist numbers $0 \leq \sigma_1 \leq \sigma_2 \leq \cdots \leq \sigma_n \leq \frac{d-1}{2}$, and automorphisms $\psi \in \operatorname{Aut}({\mathbb{B}}_n)$ and $\tau \in \operatorname{Aut}({\mathb That is, $G$ has no linear terms, and the quadratic part is diagonalized. The $\sigma_1,\ldots,\sig

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • ...and 20 more