Related papers: Geometric Invariant Theory via Cox Rings

Given a suitable action on a complex projective variety X of a non-reductive affine algebraic group H, this paper considers how to choose a reductive group G containing H and a projective completion of G x_H X which is a reductive envelope…

Let K be an algebraically closed field of characteristic zero, G_m=(K\{0},*) be its multiplicative group, and G_a=(K,+) be its additive group. Consider a commutative linear algebraic group G=G_m^r\times G_a. We study equivariant…

We define a quasi--projective reduction of a complex algebraic variety $X$ to be a regular map from $X$ to a quasi--projective variety that is universal with respect to regular maps from $X$ to quasi--projective varieties. A toric…

When the action of a reductive group on a projective variety has a suitable linearisation, Mumford's geometric invariant theory (GIT) can be used to construct and study an associated quotient variety. In this article we describe how…

Given a linear action of a group $G$ on a $K$-vector space $V$, we consider the invariant ring $K[V \oplus V^*]^G$, where $V^*$ is the dual space. We are particularly interested in the case where $V =\gfq^n$ and $G$ is the group $U_n$ of…

Consider a finite-dimensional real vector space equipped with a finite group acting unitarily on it. We address the general problem of constructing Euclidean stable embeddings of the quotient space of orbits. Our approach is based on…

Our aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action. By way of application, we give an explicit construction of a…

We investigate Cox rings of symplectic resolutions of quotients of $\mathbb{C}^{2n}$ by finite symplectic group actions. We propose a finite generating set of the Cox ring of a symplectic resolution and prove that under a condition…

Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties,…

For linear actions of real reductive Lie groups we prove the Kempf-Ness Theorem about closed orbits and the Kirwan-Ness Stratification Theorem of the null cone. Since our completely self-contained proof focuses strongly on geometric and…

Let $G/P$ be a generalized flag variety, where $G$ is a complex semisimple connected Lie group and $P\subset G$ a parabolic subgroup. Let also $X\subset G/P$ be a Schubert variety. We consider the canonical embedding of $X$ into a…

An action of a complex reductive group $\mathrm G$ on a smooth projective variety $X$ is regular when all regular unipotent elements in $\mathrm G$ act with finitely many fixed points. Then the complex $\mathrm G$-equivariant cohomology…

Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups…

We develop a method of finding a Cox ring of a crepant resolution of a quotient singularity with a torus action and apply it to examples of symplectic quotient singularities in dimension 4. In addition we obtain a bound on the degrees of…

The equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group. We prove that toric manifolds (i.e. compact smooth toric varieties) are…

Let $G$ be a finite group acting linearly on the vector space $V$ over a field of arbitrary characteristic. The action is called {\em coregular} if the invariant ring is generated by algebraically independent homogeneous invariants and the…

In this manuscript, we define the notion of linearly reductive groups over commutative unital rings and study the Cohen-Macaulay property of the ring of invariants under rational actions of a linearly reductive group. Moreover, we study the…

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…

We provide a Hilbert-Mumford Criterion for actions of reductive groups $G$ on $Q$-factorial complex varieties. The result allows to construct open subsets admitting a good quotient by $G$ from certain maximal open subsets admitting a good…

The article investigates the following question: given a projective variety X acted on by a connected and reductive group G, which is the relationship between the Gromov-Witten invariants of X and those of X//G? In this study we shall also…