Bosio generalized the construction by Meersseman of a family of non-algebraic compact complex manifolds of any dimension. We establish a link between Bosio's construction and GIT quotients. We show that his generalization parallels exactly the extension from Mumford's GIT to the more general GIT developed by Bialynicki-Birula and Swiecicka. This gives new insights into the relationship between the two non-algebraic families, from which we obtain new results on their geometry.
We consider Fano manifolds M that admit a collection of finite automorphism groups G_1, ..., G_k, such that the quotients M/G_i are smooth Fano manifolds possessing a Kaehler-Einstein metric. Under some numerical and smoothness assumptions on the ramification divisors, we prove that M admits a Kaehler-Einstein metric too.
The results known for Green-Lazarsfeld sets and solvable or nilpotent quotients of Kaehler groups are extended to the class of (compact Kaehler) geometric orbifolds with finite and integral multiplicities. The proofs are by reduction to the known case of compact Kaehler manifolds.
We investigate algebraicity properties of quotients of complex spaces by complex reductive Lie groups G. We obtain a projectivity result for compact momentum map quotients of algebraic G-varieties. Furthermore, we prove equivariant versions of Kodaira's Embedding Theorem and Chow's Theorem relative to an analytic Hilbert quotient. Combining these results we derive an equivariant algebraisation theorem for complex spaces with projective quotient.
Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is f…
We show that any set of quotients with fixed Chern classes of a given coherent sheaf on a compact Kaehler manifold is bounded in a sense which we define. The result is proved by adapting Grothendieck's boundedness criterium expressed via the Hilbert polynomial to the Kaehler set-up. As a consequence we obtain the compactness of the connected components of the Douady space of a compact Kaehler manifold.
The aim of this paper is to study the pro-algebraic fundamental group of a compact Kaehler manifold. Following work by Simpson, the structure of this group's pro-reductive quotient is already well understood. We show that Hodge-theoretic methods can also be used to establish that the pro-unipotent radical is quadratically presented. This generalises both Deligne et al.'s result on the de Rham fundamental group, and Goldman and Millson's result on deforming representations of Kaehler groups, and can be regarded as a consequence of formality of the schematic homotopy type. New examples are given…
We show that the moduli space of stable n-pointed rational curves can be flatly degenerated to a projective toric variety. We arrive at this by showing that the Chow quotients of the Grassmannians admit toric degenerations, which in turn, follows from a theorem that we prove for toric degenerations of more general Chow quotients. Along the way, we also argued that GIT quotient of a flat family is again flat. In particular, all GIT quotients of flag varieties by maximal tori can be flatly degenerated to projective toric varieties ([9] and [12]).
We construct small desingularizations of moduli spaces of semistable quiver representations for indivisible dimension vectors using deformations of stabilites and a dimension estimate for nullcones. We apply this construction to several classes of GIT quotients.
We study how to use a suitably ample locally free sheaf over a proper Deligne-Mumford stack to furnish an embedding of the stack into a geometric invariant theory (GIT) quotient stack constructed from a finite-dimensional linear representation of the general linear group.
Let X be a compact Kaehler manifold of complex dimension n. Let G be a connected solvable subgroup of the automorphism group Aut(X), and let N(G) be the normal subgroup of G of elements of null entropy. One of the goals of this paper is to show that G/N(G) is a free abelian group of rank r(G) less than or equal to n-1 and that the rank estimate is optimal. This gives an alternative proof of the conjecture of Tits type. In addition, we show some non-obvious implications on the structure of solvable automorphism groups of compact Kaehler manifolds. Furthermore, we also show that if the rank r(G)…
We study the GIT quotients for the diagonal action of the algebraic group $SL_3(\mathbb{C})$ on the $n$-fold product of $\mathbb{P}^2(\mathbb{C})$: in particular we determine a strategy in order to determine the (intersection) Poincaré polynomial of any quotient variety. In the special case $n=6$ we determine an explicit formula for the (intersection) Betti numbers of a quotient variety, depending only on the combinatorics of the weights of the polarization $m\in \mathbb{Z}^6_{>0}$.
We describe the GIT compactification for the moduli space of smooth quintic surfaces in projective space. In particular, we show that a normal quintic surface with at worst an isolated double point or a minimal elliptic singularity is stable. We also describe the boundary of the GIT quotient, and we discuss the stability of the non-normal surfaces.
The aim of this work is to study the quotients for the diagonal action of SL_3(C) on the product of n-fold of \mathbb{P}^2(C): we are interested in describing how the quotient changes when we vary the polarization (i.e. the choice of an ample linearized line bundle). We illustrate the different techniques for the construction of a quotient, in particular the numerical criterion for semi-stability and the ``elementary transformations'' which are resolutions of precisely described singularities (case n=6).
We establish a method for calculating the Poincaré series of moduli spaces constructed as quotients of smooth varieties by suitable non-reductive group actions; examples of such moduli spaces include moduli spaces of unstable vector or Higgs bundles on a smooth projective curve, with a Harder-Narasimhan type of length two. To do so, we first prove a result concerning the smoothness of fixed point sets for suitable non-reductive group actions on smooth varieties. This enables us to prove that quotients of smooth varieties by such non-reductive group actions, which can be constructed using Non-R…
A 3-fold terminal quotient singularity X=C^3/G admits the economic resolution Y-> X, which is "close to being crepant". This paper proves that the economic resolution Y is isomorphic to a distinguished component of a moduli space of certain G-equivariant objects using the the King stability condition $θ$ introduced by Kędzierski.
Let $G=SL(n, \mathbb{C}),$ and $T$ be a maximal torus of $G,$ where $n$ is a positive even integer. In this article, we study the GIT quotients of the Schubert varieties in the Grassmannian $G_{2,n}.$ We prove that the GIT quotients of the Richardson varieties in the minimal dimensional Schubert variety admitting stable points in $G_{2,n}$ are projective spaces. Further, we prove that the GIT quotients of certain Richardson varieties in $G_{2,n}$ are projective toric varieties. Also, we prove that the GIT quotients of the Schubert varieties in $G_{2,n}$ have at most finite set of singular poin…
Let $C$ be a hyperelliptic curve of genus $g \geq 3$. We give a new description of the theta map for moduli spaces of rank 2 semistable vector bundles with trivial determinant. In orther to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb{P}^1)^{2g}//\operatorname{PGL(2)}$. Then, we use recent results of Kumar to identify the restriction of the theta map to these GIT quotients with some explicit osculating projection. As a corollary of this construction, we obtain a birational equivalence between the ramification locus of th…
We suggest to endow Mumford's GIT quotient scheme with a stack structure, by replacing Proj(-) of the invariant ring with its stack theoretic analogue. We analyse the stacks resulting in this way from classically studied invariant rings, and in particular for binary forms of low degree. Our viewpoint is that the stack structure carries interesting geometric information that is intrinsically present in the invariant ring, but lost when passing to its Proj(-).
We study GIT quotients parametrizing n-pointed conics that generalize the GIT quotients $(\mathbb{P}^1)^n//SL2$. Our main result is that $\overline{M}_{0,n}$ admits a morphism to each such GIT quotient, analogous to the well-known result of Kapranov for the simpler $(\mathbb{P}^1)^n$ quotients. Moreover, these morphisms factor through Hassett's moduli spaces of weighted pointed rational curves, where the weight data comes from the GIT linearization data.
In this paper we consider the Brauer groups of algebraic stacks and GIT quotients: the only algebraic stacks we consider in this paper are quotient stacks [X/G], where X is a smooth scheme of finite type over a field k, and G is a linear algebraic group over k and acting on X, as well as various moduli stacks of principal G-bundles on a smooth projective curve X, associated to a reductive group G. We also consider the Brauer groups of the corresponding coarse moduli spaces, which most often identify with the corresponding GIT-quotients. One conclusion that we seem to draw then is that the Brau…
We give explicit equations for the Chow and Hilbert quotients of a projective scheme X by the action of an algebraic torus T in an auxiliary toric variety. As a consequence we provide GIT descriptions of these canonical quotients, and obtain other GIT quotients of X by variation of GIT quotient. We apply these results to find equations for the moduli space \bar{M}_{0,n} of stable genus zero n-pointed curves as a subvariety of a smooth toric variety defined via tropical methods.
We study the Fulton-Macpherson operational Chow rings of good moduli spaces of properly stable, smooth, Artin stacks. Such spaces are étale locally isomorphic to geometric invariant theory quotients of affine schemes, and are therefore natural extensions of GIT quotients. Our main result is that, with rational coefficients, every operational class can be represented by a so-called topologically strong cycle on the corresponding stack. Moreover, this cycle is unique modulo rational equivalence on the stack. Using out methods, we prove that if $X$ is the good moduli space of a properly stable, s…
Let $X$ be flat scheme over $\mathbb{Z}$ such that its base change, $X_p$, to $\bar{\mathbb{F}}_p$ is Frobenius split for all primes $p$. Let $G$ be a reductive group scheme over $\mathbb{Z}$ acting on $X$. In this paper, we prove a result on the $N_p$ property for line bundles on GIT quotients of $X_{\mathbb{C}}$ for the action of $G_{\mathbb{C}}$. We apply our result to the special cases of (1) an action of a finite group on the projective space and (2) the action of a maximal torus on the flag variety of type $A_n$.
The aim of this note is to announce some results on the GIT problem for the Hilbert and Chow scheme of curves of degree d and genus g in P^{d-g}, whose full details will appear in a subsequent paper. In particular, we extend the previous results of L. Caporaso up to d>4(2g-2) and we observe that this is sharp. In the range 2(2g-2)<d<7/2(2g-2), we get a complete new description of the GIT quotient. As a corollary, we get a new compactification of the universal Jacobian over the moduli space of pseudo-stable curves.
Motivated by constructing moduli spaces of unstable objects, we use new ideas in non-reductive GIT to construct quotients by parabolic group actions. For moduli problems with semistable moduli spaces constructed by reductive GIT, we consider associated instability (or HKKN) stratifications, which are often closed related to Harder-Narasimhan stratifications, and construct quotients of the unstable strata under various stabiliser assumptions by further developing ideas of non-reductive GIT. Our approach is to construct parabolic quotients in stages, in order for the required stabiliser assumpti…
Let $k$ be an algebraically closed field. Consider a reductive group $G$ over $k$. Let $X$ be a projective variety over $k$ with a $G$-action and let $L$ be a very ample $G$-linearized line bundle on $X$. Suppose that $L$ descends to the GIT quotient of $X$ by $G$. If $L$ satisfies the property $N_p$ one can ask if its descent also has $N_p$ property. In this article, we show this is the case under certain conditions. We then apply our results to some cases of interest. As a consequence of our results, we show that if $G$ is a finite group and $L$ satisfies $N_p$ property and its descent satis…
We study the GIT compactifications of pairs formed by a hypersurface and a hyperplane. We provide a general setting to characterize all polarizations which give rise to different GIT quotients. Furthermore, we describe a finite set of one-parameter subgroups sufficient to determine the stability of any GIT quotient. We characterize all maximal orbits of non stable and strictly semistable pairs, as well as minimal closed orbits of strictly semistable pairs. Our construction gives natural compactifications of the space of log smooth pairs for Fano and Calabi-Yau hypersurfaces.
The main goal of this paper is to study varieties with the best possible Mori theoretic properties (measured by the existence of a certain decomposition of the cone of effective divisors). We call such a variety a Mori Dream Space. There turn out to be many examples, including quasi-smooth projective toric (or more generally, spherical) varieties, many GIT quotients, and log Fano 3-folds. We characterize Mori dream spaces as GIT quotients of affine varieties by a torus in a manner generalizing Cox's construction of toric varieties as quotients of affine space. Via the quotient description, the…
We study the deformations of the minimally elliptic surface singularity $N_{16}$. A standard argument reduces the study of the deformations of $N_{16}$ to the study of the moduli space of pairs $(C,L)$ consisting of a plane quintic curve and a line. We construct this moduli space in two ways: via the periods of $K3$ surfaces and by using geometric invariant theory (GIT). The GIT construction depends on the choice of the linearization. In particular, for one choice of linearization we recover the space constructed via $K3$ surfaces and for another we obtain the full deformation space of $N_{16}…
Let T be an algebraic torus acting on a smooth variety V. We study the relationship between the various GIT quotients of V and the symplectic quotient of the cotangent bundle of V.
If G is a reductive group which acts on a linearized smooth scheme $X$ then we show that under suitable standard conditions the derived category of coherent sheaves of the corresponding GIT quotient stack $X^{ss}/G$ has a semi-orthogonal decomposition consisting of derived categories of coherent sheaves of rings on the categorical quotient $X^{ss}/\!/G$ which are locally of finite global dimension. One of the components of the decomposition is a certain non-commutative resolution of $X^{ss}/\!/G$ constructed earlier by the authors. As a concrete example we obtain in the case of odd Pfaffians a…
We construct new compactifications with good properties of moduli spaces of maps from nonsingular marked curves to a large class of GIT quotients. This generalizes from a unified perspective many particular examples considered earlier in the literature.
Let L be a homogeneous ample line bundle on any flag variety G/P and let T be a maximal torus of G. We prove a general necessary and sufficient condition for L to descend as a line bundle on the GIT quotient of G/P by T. We use this result to explicitly determine exactly which L descend to the GIT quotient for any simple complex algebraic group G and any parabolic subgroup P.
In this paper we study heights on quotient varieties in the sense of Geometric Invariant Theory (GIT). We generalise a construction of Burnol and we generalise diverse lower bounds of the height of semi-stable points due to Bost, Zhang, Gasbarri and Chen. In order to prove Burnol's formula for the height on the quotient we develop a Kempf-Ness theory in the setting of Berkovich analytic spaces, completing the former work of Burnol.
S. Kondo has constructed a ball quotient compactification for the moduli space of non-hyperelliptic genus four curves. In this paper, we show that this space essentially coincides with a GIT quotient of the Chow variety of canonically embedded genus four curves. More specifically, we give an explicit description of this GIT quotient, and show that the birational map from this space to Kondo's space is resolved by the blow-up of a single point. This provides a modular interpretation of the points in the boundary of Kondo's space. Connections with the slope nine space in the Hassett-Keel program…
In this article we review the question of constructing geometric quotients of actions of linear algebraic groups on irreducible varieties over algebraically closed fields of characteristic zero, in the spirit of Mumford's geometric invariant theory (GIT). The article surveys some recent work on geometric invariant theory and quotients of varieties by linear algebraic group actions, as well as background material on linear algebraic groups, Mumford's GIT and some of the challenges that the non-reductive setting presents. The earlier work of two of the authors in the setting of unipotent group a…
We define filtered ADHM data and connect a notion of filtered quiver representations to Grothendieck--Springer resolutions. We also provide current developments and give a list of research problems to further study filtered ADHM equation.
We discuss the basic properties of various versions of two variable elliptic genus with special attention to the equivariant elliptic genus. The main applications are to the elliptic genera attached to non-compact GITs, including the elliptic genera of Witten's phases on $N=2$ theories.
We introduce and study the GIT CONE of $\bar{M}_{0,n}$, which is generated by the pullbacks of the natural ample line bundles on the GIT quotients $(\mathbb P^1)^n//SL(2)$. We give an explicit formula for these line bundles and prove a number of basic results about the GIT cone. As one application, we prove unconditionally that the log canonical models of $\bar{M}_{0,n}$ with a symmetric boundary divisor coincide with the moduli spaces of weighted curves or with the symmetric GIT quotient, extending the result of Matt Simpson arXiv:0709.4037. (Cf. also a different proof by Fedorchuk and Smyth …
Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result which, for example, settles the case of a finitely generated algebra of invariants. As an application, we provide a combinatorial GIT-type construction of categorial quotients for actions on, e.g. complete varieties with finitely generated Cox ring via lifting to the characteristic space.
Given a quasiprojective algebraic variety with a reductive group action, we describe a relationship between its equivariant derived category and the derived category of its geometric invariant theory quotient. This generalizes classical descriptions of the category of coherent sheaves on projective space and categorifies several results in the theory of Hamiltonian group actions on projective manifolds. This perspective generalizes and provides new insight into examples of derived equivalences between birational varieties. We provide a criterion under which two different GIT quotients are deri…
We define the category of B-branes in a (not necessarily affine) Landau-Ginzburg B-model, incorporating the notion of R-charge. Our definition is a direct generalization of the category of perfect complexes. We then consider pairs of Landau-Ginzburg B-models that arise as different GIT quotients of a vector space by a one-dimensional torus, and show that for each such pair the two categories of B-branes are quasi-equivalent. In fact we produce a whole set of quasi-equivalences indexed by the integers, and show that the resulting auto-equivalences are all spherical twists.
Recent results in geometric invariant theory (GIT) for non-reductive linear algebraic group actions allow us to stratify quotient stacks of the form [X/H], where X is a projective scheme and H is a linear algebraic group with internally graded unipotent radical acting linearly on X, in such a way that each stratum [S/H] has a geometric quotient S/H. This leads to stratifications of moduli stacks (for example, sheaves over a projective scheme) such that each stratum has a coarse moduli space.
In this paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying the theory of Variation of Geometric Invariant Theory Quotients ([3]), we show that they are related by a sequence of GIT wall-crossing flips.
We study linear actions of algebraic groups on smooth projective varieties X. A guiding goal for us is to understand the cohomology of "quotients" under such actions, by generalizing (from reductive to non-reductive group actions) existing methods involving Mumford's geometric invariant theory (GIT). We concentrate on actions of unipotent groups H, and define sets of stable points X^s and semistable points X^{ss}, often explicitly computable via the methods of reductive GIT, which reduce to the standard definitions due to Mumford in the case of reductive actions. We compare these with definiti…
