Variation of non-reductive geometric invariant theory

Abstract

The Green-Griffiths-Lang and Kobayashi hyperbolicity conjectures for generic hypersurfaces of polynomial degree are proved using intersection theory for non-reductive geometric invariant theoretic quotients and recent work of Riedl and Yang.is the associated bundle whose structure group is Diff k (n).Let J reg k X denote the bundle of k-jets of germs of parametrised curves f : C → X in X which are regular in the sense that they have nonzero first derivative f ′ 0. After fixing local coordinates near p ∈ X, the fibre J reg k X p can be identified with J reg k (1, n) and J reg k X = Diff X × Diff k (n) J reg k (1, n).

Graded linearisations

“…Remark 2.4. When H is a linear algebraic group with T -graded unipotent radical U and L is a T -graded linearisation for an action of H on a projective variety X with respect to an ample line bundle L on X, then an analogous picture to that of Remark 2.2 holds [5]. Thus the geometric quotient X ŝ,Y,L T ,L /H has a projective completion which is a geometric quotient by Ĥ of an open subset of a Ĥ-equivariant blow-up of the product of X with the toric variety Y .…”

Hyperbolicity and specialness of symmetric powers

“…Remark 7.7. -As explained in [BK19], the coefficient 5n+3 comes from Darondeau's improvements [Dar16] for the pole order of slanted vector fields on the universal hypersurface. It seems to us by reading [Dar16] that we should actually expect the slightly better value 5n − 2.…”

“…The recent work of Bérczi and Kirwan [BK19] gives new effective degrees for which a generic hypersurface has enough jet differentials to ensure the degeneracy of entire curves. This improvement of [DMR10] yields the following result.…”

Moduli Spaces of Unstable Objects: Sheaves of Harder-Narasimhan Length 2

“…Remark 3.1.10. A final important property of this generalisation of reductive GIT, which we will not use here, is that Variation of GIT as in [DH98] [Tha96], can be shown to have an avatar in this setting [BJK18].…”