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

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.