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Register a new userProjective completions of graded unipotent quotients
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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.
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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 $U$.
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.
Let $S$ be a Riemann surface obtained by deleting a finite number of points, called cusps, from a compact Riemann surface. Let $rho: pi_1(S)to Sl(n, mathbb{C})$ be a semisimple linear representation of $pi_1(S)$ which is unipotent near the cusps. We investigate various cohomologies associated to $rho$ of $bar S$ with degenerating coefficients $L_{rho}$ (considered as a local system -- a flat vector bundle, a Higgs bundle, or a $mathcal{D}$-module, depending on the context): the v{C}ech cohomology of $j_*L_{rho}$, the $L^2$-cohomology, the $L^2$-Dolbeault cohomology, and the $L^2$-Higgs cohomology, and the relationships between them. This paper is meant to be a part of the general program of studying cohomologies with degenerating coefficients on quasiprojective varieties and their Kahlerian generalizations. The general aim here is not restricted to the case of curves nor to the one of representations that are unipotent near the divisor. The purpose of this note therefore is to illuminate at this particular case where many of the (analytic and geometric) difficulties of the general case are not present what differences will appear when we consider unipotent harmonic bundles instead of Variations of Hodge Structures where the results are known.
In this paper we prove that the etale sheafification of the functor arising from the quotient of an algebraic supergroup by a closed subsupergroup is representable by a smooth superscheme.
When the action of a reductive group on a projective variety has a suitable linearisation, Mumfords geometric invariant theory (GIT) can be used to construct and study an associated quotient variety. In this article we describe how Mumfords GIT can be extended effectively to suitable actions of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action induces a graded linearisation in a natural way. The classical examples of moduli spaces which can be constructed using Mumfords GIT are moduli spaces of stable curves and of (semi)stable bundles over a fixed nonsingular curve. This more general construction can be used to construct moduli spaces of unstable objects, such as unstable curves or unstable bundles (with suitable fixed discrete invariants in each case, related to their singularities or Harder--Narasimhan type).
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