A geometric construction for invariant jet differentials

TL;DR

This paper introduces a geometric approach to understanding invariant jet differentials, crucial for complex hyperbolicity, by analyzing the action of a non-reductive reparametrization group on holomorphic jet germs.

Contribution

It provides a new geometric description of the Demailly-Semple algebra and proves finite generation of invariants under a non-reductive group action.

Findings

Invariants form a finitely generated algebra.

Provides a geometric interpretation of the Demailly-Semple algebra.

Extends understanding of jet differential invariants in complex geometry.

Abstract

Motivated by Demailly's strategy towards the Kobayashi hyperbolicity conjecture, we study the action on the k-jets of germs of holomorphic discs in a complex manifold X of the reparametrization group of k-jets of germs of biholomorphisms of the source. This reparametrization group is a subgroup of the general linear group GL(k) which is not reductive, but nonetheless we show that its invariants for any linear action which extends to GL(k) form a finitely generated algebra, and give a new geometric description of the Demailly-Semple algebra of invariant jet differentials.