A note on varieties of non-negative Kodaira dimension with polarized self maps

Technical Summary: Varieties of Non-Negative Kodaira Dimension with Polarized Self Maps

Problem Statement
The paper investigates the geometric structure of smooth projective varieties $X$ defined over a field $k$ that possess a non-negative Kodaira dimension and admit a polarized self-map. Specifically, the study focuses on triples $(X, \phi, L)$ where $L$ is an ample line bundle and $\phi: X \to X$ is a self-map satisfying $\phi^*L \cong L^{\otimes d}$ for an integer $d > 1$. While the case of characteristic 0 has been partially addressed (notably by Nakayama and Zhang in [2]), the behavior of such varieties in positive characteristic, and over non-algebraically closed fields, remains a subject requiring further clarification and generalization.

Methodology
The author employs a combination of algebraic geometry, étale fundamental group theory, and deformation theory to establish structural results. The methodology proceeds through the following logical steps:

1. Reduction to Abelian Fundamental Groups: The proof begins by utilizing the hypothesis that the étale fundamental group $\pi_1^{\text{et}}(X)$ contains a normal abelian subgroup of finite index. Lemma 2.1 demonstrates that under this condition, one can construct a finite étale cover $Z \to X$ such that the fundamental group of $Z$ is abelian, and the polarized map lifts to $Z$.
2. Analysis of the Albanese Map: With the assumption that $\pi_1^{\text{et}}(X)$ is abelian, the paper analyzes the Albanese morphism $\text{alb}_X: X \to \text{Alb}_X$. The core of the argument involves proving that this map is finite.
   • Case $k = \mathbb{F}_p$: The proof leverages the existence of periodic points for the map $\phi$ (guaranteed by the finiteness of the field). By lifting the periodic point to the Albanese variety, the author constructs a commutative diagram involving the induced map on the Albanese variety. Using results on the injectivity of the pullback on global differential forms (Igusa) and the properties of relative differentials, it is shown that the induced map on the Albanese variety is an isogeny. A contradiction argument involving the kernel of the map on fundamental groups is used to prove that the fibers of the Albanese map are finite.
   • General Characteristic $p > 0$: The result for $k = \mathbb{F}_p$ is extended to any algebraically closed field of characteristic $p$ via a specialization argument. The variety and map are defined over a finitely generated subfield, allowing reduction to a discrete valuation ring (DVR) with a residue field of lower transcendence degree. An induction on the transcendence degree reduces the general case to the finite field case.
3. Lifting to Characteristic Zero: In Section 3, the paper addresses the condition on the fundamental group. Lemma 3.1 proves that if $X$ admits a lift to characteristic 0, the fundamental group necessarily contains a normal abelian subgroup of finite index. This allows for an alternative proof of the main theorem in the lifting case that relies on elementary results (specifically, the structure of the generic fiber) rather than deep theorems like those of Hacon–Patakfalvi–Zhang.
4. Non-Algebraically Closed Fields: Section 4 extends the results to fields $k$ that are not necessarily algebraically closed. By utilizing the theory of torsors and the splitting of the fundamental group exact sequence, the author constructs a form $A'$ of an abelian variety defined over $k$ that maps finitely and étalely to $X$, provided $X(k) \neq \emptyset$.

Key Contributions and Results

• Theorem 1.1 (Main Result): For a smooth projective variety $X$ of non-negative Kodaira dimension over an algebraically closed field $k$ of characteristic $p > 0$, if $X$ admits a separable polarized self-map and its étale fundamental group contains a normal abelian subgroup of finite index, then there exists an abelian variety $A$ with a finite étale map to $X$.
• Generalization to Non-Closed Fields (Theorems 4.3 & 4.4): The paper establishes that over any field $k$ (characteristic 0 or $p > 0$), if $X$ has a $k$-rational point and satisfies the fundamental group conditions (virtually abelian), then $X$ is a finite free quotient of an abelian variety defined over $k$.
• Lifting Implication (Lemma 3.1 & 3.2): The paper proves that the "lifting to characteristic 0" hypothesis implies the necessary fundamental group structure, thereby providing a proof of the main structural theorem for such varieties that avoids reliance on more advanced machinery used in characteristic 0 literature.
• Extension to Int-Amplified Maps: The author notes that the arguments can be adapted to "int-amplified" self-maps (where $f^*L \otimes L^{-1}$ is ample), provided the triviality of the first Chern class of the canonical bundle can be established in that context.

Significance and Claims
The paper claims to provide a positive characteristic analogue to the results of Nakayama and Zhang ([2, Theorem 4.2]) regarding the structure of varieties with polarized endomorphisms. The significance lies in:

1. Bridging Characteristic Gaps: It explicitly addresses the positive characteristic case, which was previously suggested to hold only under additional hypotheses.
2. Fundamental Group Conditions: It clarifies that the existence of a normal abelian subgroup of finite index in the fundamental group is a sufficient condition for the variety to be a finite étale quotient of an abelian variety.
3. Field Independence: It generalizes these structural results to non-algebraically closed fields, provided a rational point exists.
4. Elementary Proofs: In the case where the variety lifts to characteristic 0, the paper offers a proof utilizing more elementary nature results (Lemma 3.2), contrasting with the heavy machinery often required in this domain.

The author modestly frames the work as a "note" that proves specific structural theorems and provides analogues for existing results, rather than solving the broader dynamical conjectures (such as the Bogomolov or Mordell–Lang conjectures) mentioned in the introduction as context.