We describe three algorithms to determine the stable, semistable, and torus-polystable loci of the GIT quotient of a projective variety by a reductive group. The algorithms are efficient when the group is semisimple. By using an implementation of our algorithms for simple groups, we provide several applications to the moduli theory of algebraic varieties, including the K-moduli of algebraic varieties, the moduli of algebraic curves and the Mukai models of the moduli space of curves for low genus. We also discuss a number of potential improvements and some natural open problems arising from this work.
Let $G= GL_n(\mathbb{C})$, the general linear group over the complex numbers, and let $B$ be the set of invertible upper triangular matrices in $G$. Let $\mathfrak{b}=\text{Lie}(B)$. For $μ:T^*(\mathfrak{b}\times \mathbb{C}^n)\rightarrow \mathfrak{b}^*$, where $\mathfrak{b}^*\cong \mathfrak{g}/\mathfrak{u}$ and $\mathfrak{u}$ being strictly upper triangular matrices in $\mathfrak{g}=\text{Lie}(G)$, we prove that the Hamiltonian reduction $μ^{-1}(0)^{rss}/\!\!/B$ of the extended regular semisimple locus $\mathfrak{b}^{rss}$ of the Borel subalgebra is smooth, affine, reduced, and scheme-theoreti…
In this paper we classify all the quivers and corresponding dimension vectors having a smooth space of semisimple representation classes. The result is that these quiver settings can be reduced via some specific reduction steps to 3 simple types.
Let $G$ be a connected semisimple algebraic group over an algebraically closed field $k$. In 1965 Steinberg proved that if $G$ is simply connected, then in $G$ there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary $G$ such a cross-section exists if and only if the universal covering isogeny $τ\colon \tG\to G$ is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for ${\rm char}\,k=0$, the converse to Steinberg's theorem holds. The existence of a cross-section in $G$ implies, at least fo…
Let $U$ be a maximal unipotent subgroup of a connected semisimple group $G$ and $U'$ the derived group of $U$. If $X$ is an affine $G$-variety, then the algebra of $U'$-invariants, $k[X]^U'$, is finitely generated and the quotient morphism $π: X \to X//U'$ is well-defined. In this article, we study properties of such quotient morphisms, e.g. the property that all the fibres of $π$ are equidimensional. We also establish an analogue of the Hilbert-Mumford criterion for the null-cones with respect to $U'$-invariants.
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]).
For an affine spherical homogeneous space G/H of a connected semisimple algebraic group G, we consider the factorization morphism by the action on G/H of a maximal unipotent subgroup of G. We prove that this morphism is equidimensional if and only if the weight semigroup of G/H satisfies some simple condition.
We study deformations of orbit closures for the action of a connected semisimple group $G$ on its Lie algebra $\mathfrak{g}$, especially when $G$ is the special linear group. The tools we use are on the one hand the invariant Hilbert scheme and on the other hand the sheets of $\mathfrak{g}$. We show that when $G$ is the special linear group, the connected components of the invariant Hilbert schemes we get are the geometric quotients of the sheets of $\mathfrak{g}$. These quotients were constructed by Katsylo for a general semisimple Lie algebra $\mathfrak{g}$; in our case, they happen to be af…
Based on the work of Conrad-Gabber-Prasad, the paper deals with the geometry of particular pseudo-semisimple groups, namely those which can be written as quotient of Weil restriction of semisimple groups. We establish that these groups are retract rational when their are split, and give results on their Picard groups.
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.
For a split semisimple Chevalley group scheme G with Lie algebra g over an arbitrary base scheme S, we consider the quotient of g by the adjoint action of G. We study in detail the structure of g over S. Given a maximal torus T with Lie algebra t and associated Weyl group W, we show that the Chevalley morphism t/W -> g/G is an isomorphism except for the group Sp_{2n} over a base with 2-torsion. In this case this morphism is only dominant and we compute it explicitly. We compute the adjoint quotient in some other classical cases, yielding examples where the formation of the quotient g -> g/G co…
In this paper we consider the orbifold curve, which is a quotient of an elliptic curve $\mathcal{E}$ by a cyclic group of order 4. We develop a systematic way to obtain a Givental-type reconstruction of Gromov-Witten theory of the orbifold curve via the product of the Gromov-Witten theories of a point. This is done by employing mirror symmetry and certain results in FJRW theory. In particular, we present the particular Givental's action giving the CY/LG correspondence between the Gromov-Witten theory of the orbifold curve $\mathcal{E} / \mathbb{Z}_4$ and FJRW theory of the pair defined by the …
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.
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 $m\in \mathbb{Z}^6_{>0}$.
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.
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.
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…
We study the Newton stratification of the adjoint quotient of a connected split reductive group G with simply connected derived group over the field F of formal Laurent series in one variable over the field of complex numbers. Our main result describes the centralizer of a regular semisimple element in G(F) whose image in the adjoint quotient lies in a certain generic subset of a given Newton stratum. Other noteworthy results include analogues of some results of Springer on regular elements of finite reflection groups, as well as a geometric construction of a well known homomorphism from the f…
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 $C$ 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…
In this paper, we find a composition series of GKZ-systems with semisimple successive quotients. We also study the composition series of the corresponding perverse sheaves and compare these two composition series under the Riemann-Hilbert correspondence.
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.
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.
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.
Given an action of a reductive group on a normal variety, we construct all invariant open subsets admitting a good quotient with a quasiprojective or a divisorial quotient space. Our approach extends known constructions like Mumford's Geometric Invariant Theory. We obtain several new Hilbert-Mumford type theorems, and we extend a projectivity criterion of Bialynicki-Birula and Swiecicka for varieties with semisimple group action from the smooth to the singular case.
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…
We consider the action of a semisimple subgroup $\hat G$ of a semisimple complex group $G$ on the flag variety $X=G/B$, and the linearizations of this action by line bundles $\mathcal L$ on $X$. The main result is an explicit description of the associated unstable locus in dependence of $\mathcal L$, as well as a combinatorial formula for its (co)dimension. We observe that the codimension is equal to 1 on the regular boundary of the $\hat G$-ample cone, and grows towards the interior in steps by 1, in a way that the line bundles with unstable locus of codimension $q$ form a convex polyhedral c…
We study the semi-decomposable invariants of a split semisimple group and their extension to a split reductive group by using the torsion in the codimension $2$ Chow groups of a product of Severi-Brauer varieties. In particular, for any $n\geq 2$ we completely determine the degree $3$ invariants of a split semisimple group, the quotient of $(\operatorname{\mathbf{SL}}_{2})^{n}$ by its maximal central subgroup, as well as of the corresponding split reductive group. We also provide an example illustrating that a modification of our method can be applied to find the semi-decomposable invariants o…
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.
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.
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…
In this paper we classify and give Kazhdan-Lusztig type character formulas for equivariantly irreducible representations of Lie algebras of reductive algebraic groups over a field of large positive characteristic. The equivariance is with respect to a group whose connected component is a torus. Character computation is done in two steps. First, we treat the case of distinguished $p$-characters: those that are not contained in a proper Levi. Here we essentially show that the category of equivariant modules we consider is a cell quotient of an affine parabolic category $\mathcal{O}$. For this, w…
In this paper, we consider the diagonal action of a connected semisimple group of adjoint type on its wonderful compactification. We show that the semi-stable locus is a union of the $G$-stable pieces and we calculate the geometric quotient.
We consider actions of complex algebraic groups $\mathbf{G}$ on complex algebraic varieties $\mathbf{X}$, coming from actions of real forms $G$ of $\mathbf{G}$ and $X$ of $\mathbf{X}$. We explore the links between the real points of the complex GIT quotient $\mathbf{X /\!\!/ G}$ and the real GIT quotient $X /\!\!/ G$ defined by Richardson and Slodowy. We prove that some type of real points of $\mathbf{X /\!\!/ G}$ can be lifted to a quotient of the form $X /\!\!/ G$ maybe after changing the real forms, and we link the number of possible lifts to a co-homology set. We apply then the results to …
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.
Let $G'$ be a complex semisimple group, and let $G \subseteq G'$ be a semisimple subgroup. We show that the branching cone of the pair $(G, G')$, which (asymptotically) parametrizes all pairs $(W, V)$ of irreducible finite-dimensional $G$-representations $W$ which occur as subrepresentations of a finite-dimensional irreducible $G'$-representation $V$, can be identified with the pseudo-effective cone, $\overline{\mbox{Eff}}(Y)$, of some GIT quotient $Y$ of the flag variety of the group $G \times G'$. Moreover, we prove that the quotient $Y$ is a Mori dream space. As a consequence, the global Ok…
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}…
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.
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.
