Equivariant Sheaves and Functors

Book•

Equivariant Sheaves and Functors

28 Jun 1994-

TL;DR: In this paper, the DG-modules and equivariant cohomology of toric varieties have been studied, and the derived category D G (X) and functors have been defined.

read less

Abstract: Derived category D G (X) and functors.- DG-modules and equivariant cohomology.- Equivariant cohomology of toric varieties.

...read moreread less

Citations

PDF

Open Access

More filters

Journal Article•DOI•

Equivariant cohomology, Koszul duality, and the localization theorem

[...]

Mark Goresky¹, Robert E. Kottwitz², Robert MacPherson¹•Institutions (2)

Institute for Advanced Study¹, University of Chicago²

17 Dec 1997-Inventiones Mathematicae

TL;DR: In this paper, the authors considered the action of a compact Lie group K on a space X and gave a description of equivariant homology and intersection homology in terms of Equivariant geometric cycles.

...read moreread less

Abstract: (1.1) This paper concerns three aspects of the action of a compact group K on a space X . The ®rst is concrete and the others are rather abstract. (1) Equivariantly formal spaces. These have the property that their cohomology may be computed from the structure of the zero and one dimensional orbits of the action of a maximal torus in K. (2) Koszul duality. This enables one to translate facts about equivariant cohomology into facts about its ordinary cohomology, and back. (3) Equivariant derived category. Many of the results in this paper apply not only to equivariant cohomology, but also to equivariant intersection cohomology. The equivariant derived category provides a framework in both of these may be considered simultaneously, as examples of ``equivariant sheaves''. We treat singular spaces on an equal footing with nonsingular ones. Along the way, we give a description of equivariant homology and equivariant intersection homology in terms of equivariant geometric cycles. Most of the themes in this paper have been considered by other authors in some context. In Sect. 1.7 we sketch the precursors that we know about. For most of the constructions in this paper, we consider an action of a compact connected Lie group K on a space X , however for the purposes of the introduction we will take K S1 to be a torus. Invent. math. 131, 25±83 (1998)

...read moreread less

797 citations

Cites background or methods from "Equivariant Sheaves and Functors"

...Similarly, if A = (AX, Ā, β) ∈ D K(X) is an element of the equivariant derived category, then ([BL] §2....
[...]
...([BL] §2....
[...]
...11, or by the sheaf-theoretic construction of [BL] §5....
[...]
...In this section we recall the construction [BL] and some of the basic properties of the equivariant derived category of sheaves of vectorspaces over the real numbers R (although these constructions work more generally for sheaves of modules over any ring of finite cohomological dimension....
[...]
...[BL] §12....
[...]

Journal Article•DOI•

The McKay correspondence as an equivalence of derived categories

[...]

Tom Bridgeland¹, Alastair King², Miles Reid³•Institutions (3)

University of Edinburgh¹, University of Bath², University of Warwick³

22 Mar 2001-Journal of the American Mathematical Society

TL;DR: Gonzalez-Sprinberg and Verdier as discussed by the authors interpreted the McKay correspondence as an isomorphism on K theory, observing that the representation of G is equal to the G-equivariant K theory of C2.

...read moreread less

Abstract: The classical McKay correspondence relates representations of a finite subgroup G ⊂ SL(2,C) to the cohomology of the well-known minimal resolution of the Kleinian singularity C2/G. Gonzalez-Sprinberg and Verdier [10] interpreted the McKay correspondence as an isomorphism on K theory, observing that the representation ring of G is equal to the G-equivariant K theory of C2. More precisely, they identify a basis of the K theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of G.

...read moreread less

678 citations

Journal Article•DOI•

Braid group actions on derived categories of coherent sheaves

[...]

Paul Seidel, Richard P. Thomas

15 May 2001-Duke Mathematical Journal

TL;DR: In this paper, the authors give a construction of braid group actions on coherent sheaves on a variety of manifolds and show that these actions are always faithful when the manifold is smooth.

...read moreread less

Abstract: This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is M. Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dim $X\geq 2$, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.

...read moreread less

663 citations

Journal Article•DOI•

Geometric Langlands duality and representations of algebraic groups over commutative rings

[...]

Ivan Mirković¹, Kari Vilonen²•Institutions (2)

University of Massachusetts Amherst¹, Northwestern University²

01 Jul 2007-Annals of Mathematics

TL;DR: In this article, the basic relationship between G and G is discussed, and a canonical construction of G, starting from G, is presented, which leads to a rather explicit construction of a Hopf algebra by Tannakian formalism.

...read moreread less

Abstract: As such, it can be viewed as a first step in the geometric Langlands program. The connected complex reductive groups have a combinatorial classification by their root data. In the root datum the roots and the co-roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs. If G is a reductive group, we write G for its companion and call it the dual group G. The notion of the dual group itself does not appear in Satake's paper, but was introduced by Langlands, together with its various elaborations, in [LI], [L2] and is a cornerstone of the Langlands program. It also appeared later in physics [MO], [GNO]. In this paper we discuss the basic relationship between G and G. We begin with a reductive G and consider the affine Grassmannian Qx, the Grassmannian for the loop group of G. For technical reasons we work with formal algebraic loops. The affine Grassmannian is an infinite dimen sional complex space. We consider a certain category of sheaves, the spherical perverse sheaves, on ?r. These sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a Hopf algebra, by what has come to be known as Tannakian formalism. The resulting Hopf algebra turns out to be the ring of functions on G. In this interpretation, the spherical perverse sheaves on the affine Grassman nian correspond to finite dimensional complex representations of G. Thus, instead of defining G in terms of the classification of reductive groups, we pro vide a canonical construction of G, starting from G. We can carry out our construction over the integers. The spherical perverse sheaves are then those with integral coefficients, but the Grassmannian remains a complex algebraic object.

...read moreread less

554 citations

Posted Content•

Braid group actions on derived categories of coherent sheaves

[...]

Paul Seidel, Richard P. Thomas

07 Jan 2000-arXiv: Algebraic Geometry

TL;DR: In this article, the authors give a construction of braid group actions on coherent sheaves on a variety of manifolds and show that these actions are always faithful when the manifold is an elliptic curve.

...read moreread less

Abstract: This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when $\dim X \geq 2$, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities that lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.

...read moreread less

495 citations

Cites background from "Equivariant Sheaves and Functors"

...categories C, D is a pair (F, νF) consisting of a functor F : C −→ D and a natural isomorphism νF : F ◦[ 1 ]C ∼= [1]D◦F (here [1]C,[1]D are the translation functors) with the property that exact triangles in C are mapped to exact triangles in D. The appropriate equivalence relation between such functors is ‘graded natural isomorphism’ which means natural isomorphism compatible with the maps νF [4, section 1]. Ignoring set-theoretic ......
[...]
...Hom∗(E, F) ⊗C E // F // TE(F) [ 1 ] hh...
[...]
...The grading is such that if one ignores the differential, TE(F) = F ⊕ (hom(E, F) ⊗ E)[ 1 ]....
[...]
...categories C, D is a pair (F, νF) consisting of a functor F : C −→ D and a natural isomorphism νF : F ◦[1]C ∼= [1]D◦F (here [ 1 ]C,[1]D are the translation functors) with the property that exact triangles in C are mapped to exact triangles in D. The appropriate equivalence relation between such functors is ‘graded natural isomorphism’ which means natural isomorphism compatible with the maps νF [4, section 1]. Ignoring set-theoretic ......
[...]
...By a curve in (D,�) we mean a subset c ⊂ D \ ∂D which can be represented as the image of a smooth embedding γ : [0; 1 ] −→ D such that γ−1(�) = {0;1}....
[...]

Collapse

Equivariant Sheaves and Functors

Citations

Cites background or methods from "Equivariant Sheaves and Functors"

Cites background from "Equivariant Sheaves and Functors"

Related Papers (5)