We study the Borel moment map $μ_B:T^*(\mathfrak{b}\times \mathbb{C}^n)\rightarrow \mathfrak{b}^*$, given by $(r,s,i,j)\mapsto [r,s]+ij$, and describe our algorithm to construct the geometric invariant theory (GIT) quotients $μ_B^{-1}(0)/\!\!/_{\det}B$ and $μ_B^{-1}(0)/\!\!/_{\det^{-1}}B$, and the affine quotient $μ_B^{-1}(0)/\!\!/B$. We also provide an insight of the singular locus of $2^n$ irreducible components of $μ_B$. Finally, analogous to the Hilbert--Chow morphism, we discuss that the GIT quotient for the Borel setting is a resolution of singularities.
Let X be a locally symmetric variety. Let EBS(X) and TorE(X) denote its excentric Borel-Serre and excentric toroidal compactifications, resp. We determine their least common modification and use it to prove a conjecture of Goresky and Tai concerning canonical extensions of homogeneous vector bundles. In the process, we see that EBS(X) and TorE(X) are homotopy equivalent.
Consider an algebraic action of a connected complex reductive algebraic group on a complex polarized projective variety. In this paper, we first introduce the nilpotent quotient, the quotient of the polarized projective variety by a maximal unipotent subgroup. Then, we introduce and investigate three induced actions: one by the reductive group, one by a Borel subgroup, and one by a maximal torus, respectively. Our main result is that there are natural correspondences among quotients of these three actions. In the end, we mention a possible application to the moduli spaces of parabolic bundles …
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 characterize the tempered part of the automorphic Langlands category D-mod(Bun_G) using the geometry of the big cell in the affine Grassmannian. We deduce that, for $G$ non-abelian, tempered D-modules have no de Rham cohomology with compact supports. The latter fact boils down to a concrete statement, which we prove using the Ran space and some explicit t-structure estimates: for $G$ non-abelian and $Σ$ a smooth affine curve, the Borel-Moore homology of the indscheme $Maps(Σ,G)$ vanishes.
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 compute motivic Donaldson-Thomas invariants for crepant resolutions of quotients of affine three-space by even dihedral groups in terms of an affine type D root system, using double dimensional reduction and the representation theory of affine type D quivers.
We study the geometry of the stratification induced by an affine hyperplane arrangement H on the quotient of a complex affine space by the action of a discrete group preserving H. We give conditions ensuring normality or normality in codimension 1 of strata. As an application, we provide the list of those categorical quotients of closures of Jordan classes and of sheets in all complex simple algebraic groups that are normal. In the simply connected case, we show that normality of such a quotient is equivalent to its smoothness.
We consider two linear reductive algebraic groups $ G $ and $ G' $ over $ C $. Take a finite dimensional rational representation $ W $ of $ G \times G' $. Let $ Y = W // G := Spec C[W]^G $ and $ X = W // G' := \Spec C[W]^{G'} $ be the affine quotients. The quotient space $ X $ (respectively $ Y $) naturally inherits the action of $ G $ (respectively $ G' $). In this note, we study the interrelation between the orbit structures of $ X / G $ and $ Y / G' $. In a good situation, we can embed $ Y / G' $ into $ X / G $, and the embedding map preserves important properties such as the closure relati…
We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of ${\rm GL}_n(\mathbf{C})$, especially of the Borel subgroup $B$ and of the standard unipotent subgroup $U$ of the latter on the nilpotent cone of complex nilpotent matrices. We obtain generic normal forms of the orbits and describe generating (semi-) invariants for the Borel semi-invariant ring as well as for the $U$-invariant ring. The latter is described in more detail in terms of algebraic quotients by a special toric variety closely related. The study of a GIT-quotient for the Borel-action is initiated.
Let L be k((ε)), where k is an algebraic closure of a finite field with q elements and εis an indeterminate, and let σbe the Frobenius automorphism. Let G be a split connected reductive group over the fixed field of σin L, and let I be the Iwahori subgroup of G(L) associated to a given Borel subgroup of G. Let W be the extended affine Weyl group of G. Given x in W and b in G(L), we have some subgroup of G(L) that acts on the affine Deligne-Lusztig variety X_x(b) = {gI in G(L)/I : g^{-1}bσ(g) is in IxI} and hence a representation of this subgroup on the Borel-Moore homology of the variety. We i…
In this paper we study actions of reductive groups on affine spaces. We prove that there is a fan structure on the space of characters of the group, which parameterizes the possible invariant quotients. In the second half of the paper we study various geometrical properties of the quotients: we compute their Chow ring and the cohomology ring (in case we work over the complex numbers), both with rational coefficients. Actually, the cycle map turns out to be an isomorphism. Further, we construct families of such quotients over reduced base schemes, and compute again the Chow and the cohomology r…
Working over a ground field of characteristic zero, this paper studies the quotient morphism $π:X\to Y$ for an affine $\mathbb{G}_a$-variety $X$ with affine quotient $Y$. It is shown that the degree modules associated to the $\mathbb{G}_a$-action give a uniquely determined sequence of dominant $\mathbb{G}_a$-equivariant morphisms, $X=X_r\to X_{r-1}\to\cdots\to X_1\to X_0=Y$, where $X_i$ is an affine $\mathbb{G}_a$-variety and $X_{i+1}\to X_i$ is birational for each $i\ge 1$. This is the canonical factorization of $π$. We give an algorithm for finding the degree modules associated to the given …
For a connected reductive group G and a finite-dimensional G-module V, we study the invariant Hilbert scheme that parameterizes closed G-stable subschemes of V affording a fixed, multiplicity-finite representation of G in their coordinate ring. We construct an action on this invariant Hilbert scheme of a maximal torus T of G, together with an open T-stable subscheme admitting a good quotient. The fibers of the quotient map classify affine G-schemes having a prescribed categorical quotient by a maximal unipotent subgroup of G. We show that V contains only finitely many multiplicity-free G-subva…
The $L^2$-cohomology of a locally symmetric variety is known to have the topological interpretation as the intersection homology of its Baily-Borel Satake compactification. In this article, we observe that even without the Hermitian hypothesis, the $L^p$-cohomology of an arithmetic quotient, for $p$ finite and sufficiently large, is isomorphic to the ordinary cohomology of its reductive Borel-Serre compactification. We use this to generalize a theorem of Mumford concerning homogeneous vector bundles, their invariant Chern forms and the canonical extensions of the bundles; here, though, we are …
Following Steinberg, we construct an adjoint quotient for the Vinberg semi-group and a section to this quotient. Then, after Ngô, we show the existence of a regular centralizer on it and use it to compute the affine Springer fibers for groups.
We introduce a notion of $ξ$-stability on the affine grassmannian $\xx$ for the classical groups, this is the local version of the $ξ$-stability on the moduli space of Higgs bundles on a curve introduced by Chaudouard and Laumon. We prove that the quotient $\xx^ξ/T$ of the stable part $\xx^ξ$ by the maximal torus $T$ exists as an ind-$k$-scheme, and we introduce a reduction process analogous to the Harder-Narasimhan reduction for vector bundles. For the group $\mathrm{SL}_{d}$, we calculate the Poincaré series of the quotient $\xx^ξ/T$.
Let G be the (special) affine group, semidirect product of SL_n and C^n. In this paper we study the representation theory of G and in particular the question of rationality for V/G where V is a generically free G-representation. We show that the answer to this question is positive if the dimension of V is sufficiently large and V is indecomposable. We have a more precise theorem if V is a two-step extension 0 -> S -> V -> Q -> 0 with S, Q completely reducible.
Nagata's famous counterexample to Hilbert's fourteenth problem shows that the ring of invariants of an algebraic group action on an affine algebraic variety is not always finitely generated. In some sense, however, invariant rings are not far from affine. Indeed, invariant rings are always quasi-affine, and there always exist finite separating sets. In this paper, we give a new method for finding a quasi-affine variety on which the ring of regular functions is equal to a given invariant ring, and we give a criterion to recognize separating algebras. The method and criterion are used on some kn…
Let $G=SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the simple root $α_{4n+2}$. In this article, we prove projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle.
In this work we show that the homogeneous space of an affine algebraic group $G$ by a one-dimensional unipotent subgroup $H$ is affine if and only if the subgroup is not contained in any reductive subgroup of $G$.
Let X be an affine T-variety. We study two different quotients for the action of T on X: the toric Chow quotient X/_CT and the toric Hilbert scheme H. We introduce a notion of the main component H_0 of H which parameterizes general T-orbit closures in X and their flat limits. The main component U_0 of the universal family U over H is a preimage of H_0. We define an analogue of a universal family W_X over the main component of the X/_CT. We show that the toric Chow morphism restricted on the main components lifts to a birational projective morphism from U_0 to W_X. The variety W_X also provides…
Affine varieties of dimension greater than two can be explored their structures with the help of fibrations by the affine line or plane and quotient morphisms by $\mathbb{G}_a$-actions. We consider $\mathbb{G}_a$-actions on affine threefolds and discuss the structure and the singularities of the quotient surface as well as the singular fibers of the quotient morphism. Relative $\mathbb{G}_a$ and $\mathbb{G}_m$-actions on $\mathbb{A}^4$ or on an affine 4-fold having $\mathbb{A}^3$-fibrations over $\mathbb{A}^1$ are discussed when the actions leave one variable invariant or the fibration invaria…
We study quotients of quasi-affine schemes by unipotent groups over fields of characteristic 0. To do this, we introduce a notion of stability which allows us to characterize exactly when a principal bundle quotient exists and, together with a cohomological vanishing criterion, to characterize whether or not the resulting quasi-affine quotient scheme is affine. We completely analyze the case of G_a-invariant hypersurfaces in a linear G_a-representation W; here the above characterizations admit simple geometric and algebraic interpretations. As an application, we produce arbitrary dimensional f…
Let X be an affine toric variety. The total coordinates on X provide a canonical presentation of X as a quotient of a vector space by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient presentation.
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…
We study the equivariant K-group of the affine flag manifold with respect to the Borel group action. We prove that the structure sheaf of the (infinite-dimensional) Schubert variety in the K-group is represented by a unique polynomial, which we call the affine Grothendieck polynomial.
We study Hilbert's fourteenth problem from a geometric point of view. Nagata's celebrated counterexample demonstrates that for an arbitrary group action on a variety the ring of invariant functions need not be isomorphic to the ring of functions of an affine variety. Nevertheless one can prove that such a ring of invariants is always isomorphic to the ring of functions on a quasi-affine variety. Conversely, for a given quasi-affine variety V there exists always an action of the additive group on some affine variety W such that the ring of functions of V is isomorphic to the ring of invariant f…
We calculate the Borel-Moore homology of affine Springer fibers of type $A$ associated to some regular semisimple nil elliptic elements. As a result, we obtain bigraded $\mf{S}_{n}$-modules whose bigraded Frobenius series are generalization of the symmetric functions introduced by Haglund, Haiman, Loehr, Remmel, and Ulyanov.
Let X be a smooth projective algebraic curve of genus g minus $r\geq 1$ points defined over an algebraically closed field k of characteristic $p\geq 0$. The structure of the largest prime to p quotient of the étale fundamental group is well known by transcendental methods : it is isomorphic to the largest prime to p quotient of a free pro-finite group on 2g+r-1 generators. We show that, with purely algebraic means, we can prove the corresponding result for the largest pro-solvable quotient of these groups.
We introduce and study smooth compactifications of the moduli space of n labeled points with weights in projective space, which have normal crossings boundary and are defined as GIT quotients of the weighted Fulton-MacPherson compactification. We show that the GIT quotient of a wonderful compactification is also a wonderful compactification under certain hypotheses. We also study a weighted version of the configuration spaces parametrizing n points in affine space up to translation and homothety. In dimension one, the above compactifications are isomorphic to Hassett's moduli space of rational…
We prove that the actions mentioned in the title are translations. We show also that for certain $G_a$-actions on affine fourfolds the quotient of the action is automatically affine and describe the geometric structure of such quotients.
The aim of this article is to make a first step towards the classification of complex normal affine $\mathbb G_a$-threefolds $X$. We consider the case where the restriction of the quotient morphism $π\colon X\to S$ to $π^{-1}(S_*)$, where $S_*$ denotes the complement of some regular closed point in $S$, is a principal $\mathbb G_a$-bundle. The variety $\mathrm{SL}_2$ will be of special interest and a source of many examples. It has a natural right $\mathbb G_a$-action such that the quotient morphism $\mathrm{SL}_2\to\mathbb A^2$ restricts to a principal $\mathbb G_a$-bundle over the punctured …
We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. We propose a general algebraic framework to find the solutions and to compute the structure of the quotient ring R/I from the null space of a Macaulay-type matrix. The affine dense, affine sparse, homogeneous and multi-homogeneous cases are treated. In the presented framework, the concept of a border basis is generalized by relaxing the conditions on the set of basis elements. This allows for algorithms to adapt the choice of basis…
We show that although the fundamental group of the complement of an algebraic affine plane curve is not easy to compute, it possesses a more accessible quotient, which we call the Orevkov invariant.
We show that an algebraic stack with affine stabilizer groups satisfies the resolution property if and only if it is a quotient of a quasi-affine scheme by the action of the general linear group, or equivalently, if there exists a vector bundle whose associated frame bundle has quasi-affine total space. This generalizes a result of B. Totaro to non-normal and non-noetherian schemes and algebraic stacks. Also, we show that the vector bundle induces such a quotient structure if and only if it is a tensor generator in the category of quasi-coherent sheaves.
Let A denote the ring of differential operators on the affine line with its two usual generators t and d/dt given degrees +1 and -1 respectively. Let X be the stack having coarse moduli space the affine line Spec k[z] and isotropy groups Z/2 at each integer point. Then the category of graded A-modules is equivalent to the category of quasi-coherent sheaves on X. Version 2: corrected typos and deleted appendix at referee's suggestion.
In this paper, we show that any 3-dimensional normal affine quasihomogeneous SL(2)-variety can be described as a categorical quotient of a 4-dimensional affine hypersurface. Moreover, we show that the Cox ring of an arbitrary 3-dimensional normal affine quasihomogeneous SL(2)-variety has a unique defining equation. This allows us to construct SL(2)-equivariant flips by different GIT-quotients of hypersurfaces. Using the theory of spherical varieties, we describe SL(2)-flips by means of 2-dimensional colored cones.
We study the Cox realization of an affine variety, i.e., a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi quasitorus. The realization is described explicitly for the quotient space of a linear action of a finite group. A universal property of this realization is proved, and some results on the divisor theory of an abstract semigroup emerging in this context are given. We show that each automorphism of an affine variety can be lifted to an automorphism of the Co…
The GIT chamber decomposition arising from a subtorus action on a quasiprojective toric variety is a polyhedral complex. Denote by Sigma the fan that is the cone over the polyhedral complex. In this paper we show that the toric variety defined by the fan Sigma is the normalization of the toric Chow quotient of a closely related affine toric variety by a complementary torus.
We show that all toric noncommutative crepant resolutions (NCCRs) of affine GIT quotients of "weakly symmetric" unimodular torus representations are derived equivalent. This yields evidence for a non-commutative extension of a well known conjecture by Bondal and Orlov stating that all crepant resolutions of a Gorenstein singularity are derived equivalent. We prove our result by showing that all toric NCCRs of the affine GIT quotient are derived equivalent to a fixed Deligne-Mumford GIT quotient stack associated to a generic character of the torus. This extends a result by Halpern-Leistner and …
We study the category of perfect cdg-modules over a curved algebra, and in particular the category of B-branes in an affine Landau-Ginzburg model. We construct an explicit chain map from the Hochschild complex of the category to the closed state space of the model, and prove that this is a quasi-isomorphism from the Borel-Moore Hochschild complex. Using the lowest-order term of our map we derive Kapustin and Li's formula for the correlator of an open-string state over a disc.
We study the derived critical locus of a function $f:[X/G]\to \mathbb{A}_{\mathbb{K}}^1$ on the quotient stack of a smooth affine scheme $X$ by the action of a smooth affine group scheme $G$. It is shown that $\mathrm{dCrit}(f) \simeq [Z/G]$ is a derived quotient stack for a derived affine scheme $Z$, whose dg-algebra of functions is described explicitly. Our results generalize the classical BV formalism in finite dimensions from Lie algebra to group actions.
