Geometric invariant theory for graded unipotent groups and applications

Abstract

Let U be a graded unipotent group over the complex numbers, in the sense that it has an extensionÛ 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 ofÛ 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 ofÛ by a suitable (rational) character, and provided an additional condition is satisfied which is the analogue of the condition in classical geometric invariant theory (GIT) that there should be no strictly semistable points for the action, we show that theÛ -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 byÛ 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.

Variation of non-reductive geometric invariant theory

“…This is a linear algebraic group but is not reductive, so Mumford's classical GIT cannot be used to construct compactifications of the orbit space J reg k (1, n)/Diff k (1) (cf. [BDHK,BDHK18]). This matrix group is parametrised along its first row with free parameters α 1 ∈ C * , α 2 , .…”

“…, n − 1 on the Lie algebra Lie(U k ) of U k . In Bérczi and Kirwan [BK] and Bérczi, Doran, Hawes and Kirwan [BDHK,BDHK18] we study actions of non-reductive groups of this type in a more general context. The action of λ ∈ C * on k-jets is thus described by…”

“…This paper introduces a third compactification coming from the recent development of non-reductive GIT. As we have seen, the reparametrisation group Diff k (1) is not reductive, but it is a linear algebraic group with internally graded unipotent radical in the sense of [BDHK,BDHK18], and hence the construction and results of these papers apply. We use the fibrewise completion…”

Stratifying Quotient Stacks and Moduli Stacks

“…When H is non-reductive, then the graded algebra m 0 H 0 (X, L ⊗m ) H is not necessarily finitely generated and in general the attractive properties of Mumford's GIT fail [2]. However when the unipotent radical U of H = U ⋊ R is graded in the sense described above by a central 1-PS λ : G m → R of the Levi subgroup R, then after twisting the linearisation by an appropriate rational character, so that it becomes 'graded' itself in the sense of [5], some of the desirable properties of classical GIT still hold [3,4]. More precisely, we first quotient by the linear action of the graded unipotent group U := U ⋊ λ(G m ) using the results of [3,4] described in §2.3, then we quotient by the residual action of the reductive group H/ U ∼ = R/λ(G m ).…”

Towards the Green–Griffiths–Lang conjecture via equivariant localisation

“…This bundle gives a better reflection of the geometry of entire curves, since it only takes care of the image of such curves and not of the way they are parameterised. However, it also comes with a technical difficulty, namely, the reparameterisation group is nonreductive, and the classical geometric invariant theory of Mumford is not applicable to describe the invariants and the quotient ; for details see .…”