Finite Generation and Structure of Invariant Jets under Non-Reductive Reparametrization

TL;DR

This paper proves the finite generation of invariant jet differentials under a non-reductive group action, advancing the algebraic understanding crucial for complex hyperbolicity and the Green–Griffiths–Demailly program.

Contribution

It establishes the finite generation of the algebra of invariants for non-reductive reparametrization groups acting on jet spaces, a key step in complex hyperbolicity research.

Findings

Finite generation of the invariant jet algebra for all dimensions and jet orders.

The invariant algebra's projective spectrum matches the Demailly–Semple tower.

Provides a uniform algebraic framework for non-reductive group actions.

Abstract

We study invariant jet differentials in the framework of complex hyperbolicity, focusing on the algebra of invariants for the non--reductive reparametrization group $G_k = \mathbb{C}^{\ast} \ltimes U_k$. The paper develops a uniform, representation--theoretic, and graded--algebraic strategy for the $\rho$--action of $G_k$ on $J_k\mathbb{C}^n$, establishing in particular the finite generation of the invariant jet algebra central to the Green--Griffiths--Demailly program. Specifically, we prove that the $\mathbb{C}^{\ast}$--graded algebra of unipotent invariants $\mathbb{C}[J_k\mathbb{C}^n]^{U_k}$ is finitely generated for all $n,k$; equivalently, the fiber ring of invariant jet differentials is a finitely generated positively graded $\mathbb{C}$--algebra, so that its projective spectrum $\operatorname{Proj}\,\mathbb{C}[J_k\mathbb{C}^n]^{U_k}$ exists and coincides with the Demailly--Semple tower.