Geometric invariant theory for graded unipotent groups and applications

TL;DR

This paper extends geometric invariant theory to actions of graded unipotent groups on projective varieties, establishing conditions for finite generation of invariants and describing quotients similarly to classical GIT.

Contribution

It introduces a GIT framework for graded unipotent groups, proving finite generation of invariants and describing quotients using adapted stability criteria.

Findings

Finite generation of the algebra of invariants under certain conditions.

Construction of geometric quotients as semistable loci.

Extension of Hilbert-Mumford criteria to this setting.

Abstract

Let $U$ be a graded unipotent group over the complex numbers, in the sense that it has an extension $\hat{U}$ by the multiplicative group such that the action of the multiplicative group by conjugation on the Lie algebra of $U$ has all its weights strictly positive. Given any action of $U$ on a projective variety $X$ extending to an action of $\hat{U}$ which is linear with respect to an ample line bundle on $X$, then provided that one is willing to replace the line bundle with a tensor power and to twist the linearisation of the action of $\hat{U}$ by a suitable (rational) character, and provided an additional condition is satisfied which is the analogue of the condition in classical GIT that there should be no strictly semistable points for the action, we show that the $\hat{U}$-invariants form a finitely generated graded algebra; moreover the natural morphism from the semistable subset of $X$ to the enveloping quotient is surjective and expresses the enveloping quotient as a geometric quotient of the semistable subset. Applying this result with $X$ replaced by its product with the projective line gives us a projective variety which is a geometric quotient by $\hat{U}$ of an invariant open subset of the product of $X$ with the affine line and contains as an open subset a geometric quotient of a U-invariant open subset of $X$ by the action of $U$. Furthermore these open subsets of $X$ and its product with the affine line can be described using criteria similar to the Hilbert-Mumford criteria in classical GIT.