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Degree of Ball Maps with Maximum Geometric Rank

Abdullah Al Helal

Abstract

This work focuses on the degree bound of maps between balls with maximum geometric rank and minimum target dimension where this geometric rank occurs. Specifically, we show that rational proper maps between $\mathbb{B}_n$ and $\mathbb{B}_N$ with $n \geq 2$, $N = \frac{n(n+1)}{2}$, and geometric rank $n-1$ cannot have a degree of more than $n+1$.

Degree of Ball Maps with Maximum Geometric Rank

Abstract

This work focuses on the degree bound of maps between balls with maximum geometric rank and minimum target dimension where this geometric rank occurs. Specifically, we show that rational proper maps between and with , , and geometric rank cannot have a degree of more than .

Paper Structure

This paper contains 12 sections, 9 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Normal Forms, Geometric Rank, and Degree Bound
  3. Associated Maps
  4. The Geometric Rank
  5. A Degree Result
  6. Proof of the Main Result
  7. Partial Normalization
  8. A Degree Estimate
  9. A Linear Algebraic Proof of the Claims
  10. Matrix Structures
  11. Determinants and Adjugates
  12. Claims

Key Result

Theorem 1.1

Let $F \colon \mathbb{B}_n \to \mathbb{B}_N$ be a proper holomorphic map that is $C^3$-smooth up to the boundary with geometric rank $\kappa_0$, $n \geq 2$, and $N = n + \frac{\kappa_0 (2n - \kappa_0 - 1)}{2}$. Then $F$ is rational with $\deg F \leq \kappa_0 + 2$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 2.1: Geometric Rank
  • Definition 2.2: Degree of a Rational Map
  • Proposition 2.3
  • Corollary 2.4
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 5 more