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Proof of Miyanishi's conjecture on endomorphisms of varieties

Supravat Sarkar

Abstract

If $X$ is a quasi-projective variety over a field $k$ and $φ$ a birational endomorphism of $X$ that is injective outside a closed subset of codimension $\geq 2$, we prove that $φ$ is an automorphism. This generalizes an old theorem of Ax and proves a conjecture of Miyanishi. A key step in our proof is a finiteness result on class groups, which is of interest in its own right.

Proof of Miyanishi's conjecture on endomorphisms of varieties

Abstract

If is a quasi-projective variety over a field and a birational endomorphism of that is injective outside a closed subset of codimension , we prove that is an automorphism. This generalizes an old theorem of Ax and proves a conjecture of Miyanishi. A key step in our proof is a finiteness result on class groups, which is of interest in its own right.
Paper Structure (3 sections, 3 theorems, 3 equations)

This paper contains 3 sections, 3 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. Acknowledgement

Key Result

Theorem A

Let $X$ be a quasi-projective variety over any field $k$, and $\phi:X\to X$ a birational morphism that is injective outside a closed subset of codimension $\geq 2$ in $X$. Then $\phi$ is an isomorphism.

Theorems & Definitions (7)

  • Conjecture 1.1
  • Theorem A
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3