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On Endomorphisms of Projective Algebraic Varieties

Sami al-Asaad

TL;DR

This work analyzes endomorphisms of projective varieties through three intertwined lenses: iterated images, dynamical Stein factorizations, and algebraic endomorphism structures. It proves that iterated images stabilize and correspond to finite morphisms to normalizations, and that iterates’ Stein factorizations exhibit stability with the contraction data forming a monothetic algebraic group. It develops a precise description of algebraic endomorphisms via the semigroup S(f) and its monothetic subgroup G, showing that large iterates eventually lie in G and that S(f) decomposes into a finite initial block and an algebraic component. The results yield equivalences characterizing when endomorphisms (and their iterates) are algebraic or automorphic and provide a concrete geometric realization through varieties with monothetic subgroup actions. Together, these findings illuminate the algebraic dynamics of endomorphisms and establish structural constraints on their iterates and factorization behavior.

Abstract

We study the algebraic dynamics of endomorphisms of projective varieties. First, we characterize their iterated images, i.e. the intersection of the images of their iterates. Next, we explore the Stein factorizations of the iterates, proving some stability phenomena they exhibit. Finally, we study endomorphisms whose iterates lie in a finite union of connected components of the endomorphism scheme, thereby completing a result of Brion.

On Endomorphisms of Projective Algebraic Varieties

TL;DR

This work analyzes endomorphisms of projective varieties through three intertwined lenses: iterated images, dynamical Stein factorizations, and algebraic endomorphism structures. It proves that iterated images stabilize and correspond to finite morphisms to normalizations, and that iterates’ Stein factorizations exhibit stability with the contraction data forming a monothetic algebraic group. It develops a precise description of algebraic endomorphisms via the semigroup S(f) and its monothetic subgroup G, showing that large iterates eventually lie in G and that S(f) decomposes into a finite initial block and an algebraic component. The results yield equivalences characterizing when endomorphisms (and their iterates) are algebraic or automorphic and provide a concrete geometric realization through varieties with monothetic subgroup actions. Together, these findings illuminate the algebraic dynamics of endomorphisms and establish structural constraints on their iterates and factorization behavior.

Abstract

We study the algebraic dynamics of endomorphisms of projective varieties. First, we characterize their iterated images, i.e. the intersection of the images of their iterates. Next, we explore the Stein factorizations of the iterates, proving some stability phenomena they exhibit. Finally, we study endomorphisms whose iterates lie in a finite union of connected components of the endomorphism scheme, thereby completing a result of Brion.

Paper Structure

This paper contains 15 sections, 18 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction of Main Results
  2. Iterated Images
  3. Dynamical Stein Factorizations
  4. Algebraic Endomorphisms
  5. Proofs of Statements About Iterated Images
  6. Proof of Theorem \ref{['itimage']}
  7. Proof of Proposition \ref{['pinchprop']}
  8. Proofs of Statements About Dynamical Stein Factorizations
  9. Proof of Proposition \ref{['stability']}
  10. Proof of Proposition \ref{['steinfactoriteratedimage']}
  11. Proofs of Statements About Algebraic Endomorphisms
  12. Proof of Theorem \ref{['mainthm0']}
  13. Proof of Theorem \ref{['mainthm1']}
  14. Proof of Proposition \ref{['mainprop2']}
  15. Proof of Theorem \ref{['mainthm2']}

Key Result

Lemma 1

Let $Y$ be the iterated image of the dynamical system $(X,f)$.

Figures (1)

  • Figure 1: An infinite zigzag of affine lines

Theorems & Definitions (50)

  • Lemma 1
  • Theorem 1
  • Proposition 1
  • Definition 1
  • Proposition 2
  • Proposition 3
  • Definition 2
  • Definition 3
  • Remark 1
  • Theorem 2
  • ...and 40 more