相关论文: Towards non-reductive geometric invariant theory

In this paper we deal with a Hamiltonian action of a reductive algebraic group $G$ on an irreducible normal affine Poisson variety $X$. We study the invariant moment map $\psi_{G,X}:X\to \g$, that is, the composition of the moment map…

Given an algebraic torus action on a normal projective variety with finitely generated total coordinate ring, we study the GIT-equivalence for not necessarily ample linearized divisors, and we provide a combinatorial description of the…

For a subgroup $H$ of a reductive group $G$, let $\mathfrak m\subset \mathfrak g^*$ be the cotangent space of $eH\in G/H$. The linear action $(H:\mathfrak m)$ is the coisotropy representation. It is known that the complexity and rank of…

We give another proof that a reductive algebraic group is geometrically reductive. We show that a quotient of the semi-stable locus (by a linear action of a reductive algebraic group on a projective scheme) exists, and from this Haboush's…

An example of a finite dimensional factorizable ribbon Hopf C-algebra is given by a quotient H=u_q(g) of the quantized universal enveloping algebra U_q(g) at a root of unity q of odd degree. The mapping class group M_{g,1} of a surface of…

Let $V$ be a linear representation of a connected complex reductive group $G$. Given a choice of character $\theta$ of $G$, Geometric Invariant Theory defines a locus $V^{ss}_\theta(G) \subseteq V$ of semistable points. We give necessary,…

Many geometric learning problems require invariants on heterogeneous product spaces, i.e., products of distinct spaces carrying different group actions, where standard techniques do not directly apply. We show that, when a group $G$ acts…

Classical invariant theory establishes a systematic correspondence between algebraic and smooth invariants for compact and reductive Lie groups. However, the extension of these results to non-compact and non-reductive regimes remains a…

To any Hamiltonian action of a reductive algebraic group $G$ on a smooth irreducible symplectic variety $X$ we associate certain combinatorial invariants: Cartan space, Weyl group, weight and root lattices. For cotangent bundles our…

We present a proof of Thue-Siegel-Roth's Theorem (and its more recent variants, such as those of Lang for number fields and that "with moving targets" of Vojta) as an application of Geometric Invariant Theory (GIT). Roth's Theorem is…

We define a version of stable maps into the classifying stack $B\mathrm{GL}_N$, and develop a corresponding notion of $K$-theoretic Gromov-Witten invariants. In this setting, the evaluation morphisms are not of finite type; the definition…

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…

Let G be an affine algebraic group and let X be an affine algebraic variety. An action $G\times X \to X$ is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant $f\in K[X]^G$ such that f(Y) =0.…

This is the first in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology $QH_G(X)$ of a smooth complex projective variety X with the action of a connected complex reductive…

Given an affine scheme X with an action of a reductive group G and a G-linearized coherent sheaf M, we construct the ``invariant Quot scheme'' that parametrizes the quotients of M whose space of global sections is a direct sum of simple…

Let $X=G/\Gamma$ be the quotient of a semisimple Lie group $G$ by its non-cocompact arithmetic lattice. Let $H$ be a reductive algebraic subgroup of $G$ acting on $X$. We give several equivalent algebraic conditions on $H$ for the existence…

We presented a Hilbert-Mumford criterion for polystablility associated with an action of a real reductive Lie group $G$ on a real submanifold $X$ of a Kahler manifold $Z$. Suppose the action of a compact Lie group with Lie algebra…

Let $X$ be a projective variety with a torus action, which for simplicity we assume to have dimension 1. If $X$ is a smooth complex variety, then the geometric invariant theory quotient $X//G$ can be identifed with the symplectic reduction…

This is a survey of results that extend notions of the classical invariant theory of linear actions by finite groups on $k[x_1, \dots, x_n]$ to the setting of finite group or Hopf algebra $H$ actions on an Artin-Schelter regular algebra…

We study de Rham cohomology for various differential calculi on finite groups G up to order 8. These include the permutation group S_3, the dihedral group D_4 and the quaternion group Q. Poincare' duality holds in every case, and under some…