Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGeometric invariant theory for graded unipotent groups and applications
89
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic 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.
rate research
Read More
Let $U$ be a unipotent group which is graded in the sense that it has an extension $H$ by the multiplicative group of the complex numbers such that all the weights of the adjoint action on the Lie algebra of $U$ are strictly positive. We study embeddings of $H$ in a general linear group $G$ which possess Grosshans-like properties. More precisely, suppose $H$ acts on a projective variety $X$ and its action extends to an action of $G$ which is linear with respect to an ample line bundle on $X$. Then, provided that we are willing to twist the linearisation of the action of $H$ by a suitable (rational) character of $H$, we find that the $H$-invariants form a finitely generated algebra and hence define a projective variety $X/!/H$; moreover the natural morphism from the semistable locus in $X$ to $X/!/H$ is surjective, and semistable points in $X$ are identified in $X/!/H$ if and only if the closures of their $H$-orbits meet in the semistable locus. A similar result applies when we replace $X$ by its product with the projective line; this gives us a projective completion of a geometric quotient of a $U$-invariant open subset of $X$ by the action of the unipotent group $U$.
The aim of this paper is to show that classical geometric invariant theory (GIT) has an effective analogue for linear actions of a non-reductive algebraic group $H$ with graded unipotent radical on a projective scheme $X$. Here the linear action of $H$ is required to extend to a semi-direct product $hat{H} = H rtimes mathbb{G}_m$ with a multiplicative one-parameter group which acts on the Lie algebra of the unipotent radical $U$ of $H$ with all weights strictly positive, and which centralises a Levi subgroup $R cong H/U$ of $H$. We show that $X$ has an $H$-invariant open subscheme (the hat-stable locus) which has a geometric quotient by the $H$-action. This geometric quotient has a projective completion which is a categorical quotient (indeed, a good quotient) by $hat{H}$ of an open subscheme of a blow-up of the product of $X$ with the affine line; with additional blow-ups a projective completion which is itself a geometric quotient can be obtained. Furthermore the hat-stable locus of $X$ and the corresponding open subsets of the blow-ups of the product of $X$ with the affine line can be described effectively using Hilbert-Mumford-like criteria combined with the explicit blow-up constructions. Applications include the construction of moduli spaces of sheaves and Higgs bundles of fixed Harder--Narasimhan type over a fixed nonsingular projective scheme, and of moduli spaces of unstable projective curves of fixed singularity. More recently, cohomology theory for reductive GIT quotients were extended by the first and fourth author to the non-reductive situation studied in this paper, and this was used to prove the Green--Griffiths--Lang and Kobayashi hyperbolicity conjectures for generic projective hypersurfaces with polynomial bounds on their degree.
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.
We combine recently developed intersection theory for non-reductive geometric invariant theoretic quotients with equivariant localisation to prove a formula for Thom polynomials of Morin singularities. These formulas use only toric combinatorics of certain partition polyhedra, and our new approach circumvents the poorly understood Borel geometry of existing models.
The wall-and-chamber structure of the dependence of the reductive GIT quotient on the choice of linearisation is well known. In this article, we first give a brief survey of recent results in non-reductive GIT, which apply when the unipotent radical is graded. We then examine the dependence of these non-reductive quotients on the linearisation and an additional parameter, the choice of one-parameter subgroup grading the unipotent radical, and arrive at a picture similar to the reductive one.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
