[Paper Review] Invariant hypersurfaces of endomorphisms of projective varieties

De‐Qi Zhang|arXiv (Cornell University)|Oct 22, 2013

Algebraic Geometry and Number Theory22 references3 citations

TL;DR

This paper establishes sharp upper bounds on the number of $f^{-1}$-stable prime divisors under polarized endomorphisms of projective varieties with log canonical singularities. It proves that such divisors are rationally chain connected and provides optimal bounds: $c \leq n + \rho(X)$, with equality implying $X$ is rationally connected or fibred over an elliptic curve, and $K_X + \sum V_i \sim_\mathbb{Q} 0$ when $c = n + \rho(X)$. The results generalize to $\mathbb{Q}$-factorial, Kawamata log terminal varieties and are optimal in the case of $\mathbb{P}^n$. The key contribution is a classification of invariant hypersurfaces under endomorphisms via birational geometry and dynamical systems techniques.

ABSTRACT

We consider surjective endomorphisms f of degree > 1 on projective manifolds X of Picard number one and their f^{-1}-stable hypersurfaces V, and show that V is rationally chain connected. Also given is an optimal upper bound for the number of f^{-1}-stable prime divisors on (not necessarily smooth) projective varieties.

Motivation & Objective

• To determine the maximum number of $f^{-1}$-stable prime divisors on a projective variety under a polarized endomorphism of degree $>1$.
• To establish conditions under which such invariant divisors are rationally chain connected, generalizing rational connectedness to singular varieties.
• To classify the structure of the variety when the number of stable divisors reaches the theoretical upper bound.
• To prove that the upper bound $n + \rho(X)$ is sharp and characterize the equality case, including the case of $\mathbb{P}^n$ with coordinate hyperplanes.
• To extend results from smooth varieties to $\mathbb{Q}$-factorial, Kawamata log terminal varieties with minimal singularities.

Proposed method

• Use of the Néron-Severi group and Picard number $\rho(X)$ to bound the number of $f^{-1}$-stable prime divisors via linear equivalence and numerical conditions.
• Application of the inversion of adjunction and log canonical singularity theory to handle singular varieties and ensure the validity of the bounds.
• Employment of induction on fibrations and normalization maps to reduce the problem to lower-dimensional cases.
• Use of $\mathbb{Q}$-linear equivalence and canonical divisor conditions ($K_X + \sum V_i \sim_\mathbb{Q} 0$) to characterize equality cases.
• Leveraging the dynamics of $f$ via pullback actions on $\operatorname{Pic}(X)$ and $N^1(X)$, particularly $f^*H \sim qH$, to derive invariance and finiteness.
• Analysis of the branch locus and pullbacks via normalization maps to show that certain divisors $\Delta$ vanish, leading to $K_X + \sum V_i \sim_\mathbb{Q} 0$.

Experimental results

Research questions

• RQ1 What is the maximum number of $f^{-1}$-stable prime divisors on a projective variety with a polarized endomorphism of degree $>1$?
• RQ2 Under what conditions are the irreducible components of such invariant divisors rationally chain connected?
• RQ3 When does the number of $f^{-1}$-stable divisors reach the upper bound $n + \rho(X)$, and what structure does the variety admit in that case?
• RQ4 Can the bound be achieved only in the case $X = \mathbb{P}^n$ with coordinate hyperplanes and $f$ given by $[X_1,\dots,X_{n+1}] \mapsto [X_1^q,\dots,X_{n+1}^q]$?
• RQ5 What happens when the number of stable divisors is at least $n + \rho(X) - 1$ or $n + \rho(X) - 2$? Does this force rational connectedness or a fibration over an elliptic curve?

Key findings

• The number $c$ of $f^{-1}$-stable prime divisors on a projective variety $X$ of dimension $n$ with polarized endomorphism $f$ satisfies $c \leq n + \rho(X)$, and this bound is optimal.
• Equality $c = n + \rho(X)$ holds if and only if $K_X + \sum V_i \sim_\mathbb{Q} 0$, $(f^t)^* = q^t \cdot \mathrm{id}$ on $\mathrm{Pic}(X)$, and $f$ is étale outside the singular locus and the union of the $V_i$.
• If $c \geq n + \rho(X) - 1$, then $X$ is rationally connected.
• If $c \geq n + \rho(X) - 2$, then either $X$ is rationally connected or there exists a fibration $X \to E$ onto an elliptic curve $E$ such that $f^k$ descends to an endomorphism of degree $q$ on $E$, and fibers are rationally connected.
• Each irreducible component $V_i$ and its normalization are rationally chain connected under the conditions of Theorem 1.2, especially when $-K_X \sim_\mathbb{Q} rH$ and $X$ has log canonical singularities.
• The bound $c \leq n + \rho(X)$ is sharp and is achieved precisely when $X = \mathbb{P}^n$, $V_i$ are coordinate hyperplanes, and $f$ is the $q$-th power map.