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On Miyanishi conjecture for quasi-projective varieties

Takumi Asano

TL;DR

This work advances the Miyanishi conjecture for quasi-projective varieties in characteristic zero by establishing automorphism results under two broad settings. First, if $X$ sits densely inside a $\\mathbb{Q}$-factorial normal projective $\\overline{X}$ with $\mathrm{codim}(\\overline{X}\\setminus X)\\ge 2$ and either $K_{\\overline{X}}$ or $-K_{\\overline{X}}$ is ample, then every endomorphism injective outside a codimension-2 subset is an automorphism. Second, when $\\overline{X}$ has canonical singularities and its canonical model arises from the MMP with only divisorial contractions, the conjecture also holds; the paper further analyzes threefold flips and shows that, under additional conditions on flips, the conjecture remains valid. The approach hinges on birational geometry techniques, the minimal model program, and the behavior of canonical divisors under resolutions, enabling a reduction to a contradiction via the ampleness of the canonical divisor on the canonical model. The results have implications for the rigidity of endomorphisms on quasi-projective varieties and connect Miyanishi-type questions with modern MMP methods.

Abstract

Miyanishi conjecture claims that for any variety over an algebraically closed field of characteristic zero, any endomorphism of such a variety which is injective outside a closed subset of codimension at least $2$ is bijective. We prove Miyanishi conjecture for any quasi-projective variety $X$ which is a dense open subset of a $\mathbb{Q}$-factorial normal projective variety $\overline{X}$ such that codim $(\overline{X} \setminus X) \ge 2$ with the ample canonical divisor or the ample anti-canonical divisor. Also, we observe Miyanishi conjecture without the conditions of its canonical divisor by using minimal model program. In particular, we prove Miyanishi conjecture in the case that $\overline{X}$ has canonical singularities and $\overline{X}$ has the canonical model which is obtained by divisorial contractions.

On Miyanishi conjecture for quasi-projective varieties

TL;DR

This work advances the Miyanishi conjecture for quasi-projective varieties in characteristic zero by establishing automorphism results under two broad settings. First, if sits densely inside a -factorial normal projective with and either or is ample, then every endomorphism injective outside a codimension-2 subset is an automorphism. Second, when has canonical singularities and its canonical model arises from the MMP with only divisorial contractions, the conjecture also holds; the paper further analyzes threefold flips and shows that, under additional conditions on flips, the conjecture remains valid. The approach hinges on birational geometry techniques, the minimal model program, and the behavior of canonical divisors under resolutions, enabling a reduction to a contradiction via the ampleness of the canonical divisor on the canonical model. The results have implications for the rigidity of endomorphisms on quasi-projective varieties and connect Miyanishi-type questions with modern MMP methods.

Abstract

Miyanishi conjecture claims that for any variety over an algebraically closed field of characteristic zero, any endomorphism of such a variety which is injective outside a closed subset of codimension at least is bijective. We prove Miyanishi conjecture for any quasi-projective variety which is a dense open subset of a -factorial normal projective variety such that codim with the ample canonical divisor or the ample anti-canonical divisor. Also, we observe Miyanishi conjecture without the conditions of its canonical divisor by using minimal model program. In particular, we prove Miyanishi conjecture in the case that has canonical singularities and has the canonical model which is obtained by divisorial contractions.
Paper Structure (5 sections, 15 theorems, 13 equations)

This paper contains 5 sections, 15 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main1']}
  3. Proof of Theorem \ref{['main2']}
  4. Proof of Theorem \ref{['main3']}
  5. Observations on some other cases

Key Result

Theorem 1.1

Let $A$ and $B$ be schemes such that $A$ is of finite type over $B$. For any endomorphism $\varphi$ of $A$ over $B$, if $\varphi$ is injective, then $\varphi$ is bijective.

Theorems & Definitions (24)

  • Theorem 1.1: Ax
  • Conjecture 1.2: Miyanishi conjecture, Open
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 1.6: Kal
  • Lemma 1.7: Kal
  • Lemma 1.8: Das
  • Remark 1.9
  • Theorem 1.10: Mum
  • ...and 14 more