Groupes et algébres de lie

(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)

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Equivariant cohomology, Koszul duality, and the localization theorem

Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Progress in Mathematics

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.

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The McKay correspondence as an equivalence of derived categories

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.

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Braid group actions on derived categories of coherent sheaves

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

Equivariant Sheaves and Functors

The paper contains general results on the uniqueness of a DG enhancement for trian- gulated categories. As a consequence we obtain such uniqueness for the unbounded categories of quasi-coherent sheaves, for the triangulated categories of perfect complexes, and for the bounded de- rived categories of coherent sheaves on quasi-projective schemes. If a scheme is projective then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded de- rived categories of coherent sheaves. These results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product.

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Uniqueness of enhancement for triangulated categories

The paper contains general results on the uniqueness of a DG enhancement for triangulated categories. As a consequence we obtain such uniqueness for the unbounded categories of quasi-coherent sheaves, for the triangulated categories of perfect complexes, and for the bounded derived categories of coherent sheaves on quasi-projective schemes. If a scheme is projective then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded derived categories of coherent sheaves. These results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product.

/pdf/uniqueness-of-enhancement-for-triangulated-categories-6if2z4pweo.pdf

We study the motivic Grothendieck group of algebraic vari- eties from the point of view of stable birational geometry. In particular, we obtain a counterexample to a conjecture of M. Kapranov on the rationality of motivic zeta-functions.

Motivic measures and stable birational geometry

We consider the abelian group PT generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semiorthogonal decompositions of corresponding triangulated categories. We introduce an operation of “multiplication” • on the collection of DG categories, which makes this abelian group into a commutative ring. A few applications are considered: representability of “standard” functors between derived categories of coherent sheaves on smooth projective varieties and a construction of an interesting motivic measure.

Valery A. Lunts

Papers

Equivariant Sheaves and Functors

Uniqueness of enhancement for triangulated categories

Uniqueness of enhancement for triangulated categories

Motivic measures and stable birational geometry

Grothendieck ring of pretriangulated categories