Proof of Miyanishi's conjecture on endomorphisms of varieties

1. Problem Statement

The paper addresses a long-standing conjecture in algebraic geometry regarding the nature of endomorphisms on algebraic varieties. Specifically, it investigates whether a birational endomorphism that is "almost" injective must be an automorphism.

• Context: For normal projective varieties, endomorphisms impose strong geometric restrictions. However, for non-complete (quasi-projective) varieties, the behavior is less understood.
• The Conjecture (Miyanishi, 1990s): Let $X$ be a complex variety and $\phi: X \to X$ an endomorphism. If $\phi$ is injective outside a closed subset of codimension $\ge 2$, then $\phi$ is an isomorphism (automorphism).
• Previous Status:
  • Ax (1968): Proved that any globally injective endomorphism of a reduced separated scheme of finite type is bijective.
  • Nowak: Proved that for $k=\mathbb{C}$ (char 0), bijective endomorphisms are automorphisms.
  • Limitations: Global injectivity is a strong condition. If injectivity is only assumed on a dense open set, the map may not be an automorphism (e.g., $\phi(x,y) = (x, xy)$ on $\mathbb{A}^2$). Miyanishi conjectured that if the non-injective locus has codimension $\ge 2$, the map becomes an automorphism.
  • Partial Results: The conjecture was known to hold for $Q$-factorial varieties, affine varieties, complete varieties, and when $\dim X \le 2$.

2. Methodology

The author proves a generalized version of the conjecture (Theorem A) that holds over any field $k$ (not just $\mathbb{C}$) and for quasi-projective varieties. The proof relies on two main technical pillars:

A. Reduction to Finitely Generated Fields

The author first reduces the problem to the case where the base field $k$ is finitely generated over its prime field. This allows the application of arithmetic geometry tools and finiteness theorems regarding class groups.

B. Analysis of Class Groups and Picard Groups

The core of the proof involves comparing the structure of the Class Group ($\text{Cl}(X)$) and the Picard Group ($\text{Pic}(X)$) of the variety before and after the morphism.

• Finiteness (Lemma 2.1): The author establishes that for a variety over a finitely generated field, the $\mathbb{Q}$-vector space $\text{Cl}(X) \otimes \mathbb{Q}$ is finite-dimensional. Furthermore, if the variety is birational to a geometrically normal proper variety, $\text{Cl}(X)$ is a finitely generated abelian group.
• Dimensional Drop Argument (Lemma 2.2): This is the critical step. The author considers the quotient space $\text{Cl}(X) / \text{Pic}(X) \otimes \mathbb{Q}$.
  • Let $\phi: Y \to X$ be a birational morphism between normal quasi-projective varieties with exceptional locus of codimension $\ge 2$.
  • The author constructs a surjective map on the quotient groups induced by $\phi$.
  • Key Insight: If $\phi$ is not an open immersion (i.e., it contracts something), the dimension of the quotient group $\text{Cl}(Y) / \text{Pic}(Y) \otimes \mathbb{Q}$ is strictly less than that of $\text{Cl}(X) / \text{Pic}(X) \otimes \mathbb{Q}$.
  • Contradiction: Since $\phi$ is an endomorphism ($Y=X$), the dimension cannot strictly decrease. Therefore, the assumption that $\phi$ is not an open immersion must be false. Thus, $\phi$ must be an open immersion.

C. Final Step: Bijectivity

Once it is established that $\phi$ is an open immersion (an isomorphism onto its image), the author invokes Ax's Theorem (which states that an injective endomorphism of a variety of finite type is surjective). Since $\phi$ is an open immersion and surjective, it is an isomorphism.

3. Key Contributions

1. Proof of Miyanishi's Conjecture: The paper provides a complete proof of Conjecture 1.1 for all quasi-projective varieties over any field.
2. Generalization to Arbitrary Characteristic: Unlike previous results restricted to characteristic 0, this proof works for fields of arbitrary characteristic (with the necessary caveat that in positive characteristic, the map must be assumed birational, as the Frobenius map is injective but not birational).
3. Finiteness Result on Class Groups: The paper establishes a significant independent result (Lemma 2.1) regarding the finite dimensionality of $\text{Cl}(X) \otimes \mathbb{Q}$ and the finite generation of $\text{Cl}(X)$ for varieties over finitely generated fields.
4. Refinement of Hypotheses: The author clarifies that the quasi-projectivity assumption is only used to ensure the existence of a Cartier divisor separating two points (Remark 2.3), suggesting the result may hold for a broader class of varieties satisfying this separation property.

4. Main Results

• 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 $\ge 2$. Then $\phi$ is an isomorphism.
• Corollary: This confirms Miyanishi's conjecture for complex varieties, removing the need for $X$ to be affine, complete, or $Q$-factorial.

5. Significance

• Resolution of an Open Problem: This paper settles a conjecture that had remained open for decades, unifying various partial results known for specific classes of varieties.
• Methodological Innovation: The proof introduces a novel approach using the dimension of the quotient of the Class Group by the Picard Group to detect non-trivial exceptional loci. This "dimensional drop" argument provides a new tool for studying birational endomorphisms.
• Foundational Impact: By proving the result over arbitrary fields and establishing finiteness properties of class groups, the paper strengthens the bridge between the geometry of varieties and arithmetic properties of their divisor class groups. It clarifies the relationship between injectivity, birationality, and automorphism in the context of non-complete varieties.