ID: math/0110137

The Decomposition Theorem and the Intersection Cohomology of Quotients in Algebraic Geometry

October 12, 2001

View on ArXiv

Similar papers 5

Descent theory for semiorthogonal decompositions

June 13, 2012

83% Match
Alexey Elagin
Algebraic Geometry

In this paper a method of constructing a semiorthogonal decomposition of the derived category of $G$-equivariant sheaves on a variety $X$ is described, provided that the derived category of sheaves on $X$ admits a semiorthogonal decomposition, whose components are preserved by the action of the group $G$ on $X$. Using this method, semiorthogonal decompositions of equivariant derived categories were obtained for projective bundles and for blow-ups with a smooth center, and als...

Find SimilarView on arXiv

Secant varieties and degrees of invariants

November 29, 2018

83% Match
Valdemar V. Tsanov
Representation Theory
Algebraic Geometry
Symplectic Geometry

The ring of invariant polynomials ${\mathbb C}[V]^G$ over a given finite dimensional representation space $V$ of a complex reductive group $G$ is known, by a famous theorem of Hilbert, to be finitely generated. The general proof being nonconstructive, the generators and their degrees have remained a subject of interest. In this article we determine certain divisors of the degrees of the generators. Also, for irreducible representations, we provide lower bounds for the degrees...

Find SimilarView on arXiv

Smoothness of non-reductive fixed point sets and cohomology of non-reductive GIT quotients

May 7, 2021

83% Match
Eloise Hamilton
Algebraic Geometry

We establish a method for calculating the Poincar\'e 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 enable...

Find SimilarView on arXiv

Abelianization and Quantum Lefschetz for Orbifold Quasimap $I$-Functions

September 24, 2021

83% Match
Rachel Webb
Algebraic Geometry

Let $Y$ be a complete intersection in an affine variety $X$, with action by a complex reductive group $G$. Let $T \subset G$ be a maximal torus. A character $\theta$ of $G$ defines GIT quotients $Y//_\theta G$ and $X//_\theta T$. We prove formulas relating the small quasimap I-function of $Y//_\theta G$ to that of $X//_\theta T$. When $X$ is a vector space, this provides a completely explicit formula for the small $I$-function of $Y//_\theta G$.

Find SimilarView on arXiv

Geometric Invariant Theory based on Weil Divisors

January 19, 2003

83% Match
Juergen Hausen
Algebraic Geometry

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.

Find SimilarView on arXiv

Group Actions on Riemann-Roch Space

April 4, 2019

83% Match
Angel Carocca, Daniela Vásquez
Algebraic Geometry

Let $ \; G \; $ be a group acting on a compact Riemann surface $ \; {\mathcal X} \; $ and $ \; D \; $ be a $ \; G$-invariant divisor on $\; {\mathcal X}. \; $ The action of $ \; G \; $ on $ \; {\mathcal X} \; $ induces a linear representation $ \; L_G(D) \; $ of $ \; G \; $ on the Riemann-Roch space associated to $ \; D.$ In this paper we give some results on the decomposition of $ \; L_G(D) \; $ as sum of complex irreducible representations of $ \; G, \; $ for $ \; D \; $ ...

Find SimilarView on arXiv

Algebraic cycles and completions of equivariant K-theory

February 22, 2007

83% Match
Dan Edidin, William Graham
Algebraic Geometry
K-Theory and Homology

Let $G$ be a complex, linear algebraic group acting on an algebraic space $X$. The purpose of this paper is to prove a Riemann-Roch theorem (Theorem 5.3) which gives a description of the completion of the equivariant Grothendieck group $G_0(G,X)$ at any maximal ideal of the representation ring $R(G) \otimes \C$ in terms of equivariant cycles. The main new technique for proving this theorem is our non-abelian completion theorem (Theorem 4.3) for equivariant $K$-theory. Theorem...

Find SimilarView on arXiv

On the localization theorem in equivariant cohomology

November 7, 1997

83% Match
Michel Brion, Michèle Vergne
Differential Geometry

We present a simple proof of a precise version of the localization theorem in equivariant cohomology. As an application, we describe the cohomology algebra of any compact symplectic variety with a multiplicity-free action of a compact Lie group. This applies in particular to smooth, projective spherical varieties.

Find SimilarView on arXiv

The derived category of a GIT quotient

March 1, 2012

83% Match
Daniel Halpern-Leistner
Algebraic Geometry
K-Theory and Homology
Representation Theory

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 ...

Find SimilarView on arXiv

Graded linearisations

March 15, 2017

83% Match
Gergely Bérczi, Brent Doran, Frances Kirwan
Algebraic Geometry

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 Mumford's GIT can be extended effectively to suitable actions of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive ...

Find SimilarView on arXiv