EQUIVARIANT DERIVED CATEGORIES FOR TOROIDAL GROUP IMBEDDINGS
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Springer Science+Business Media New York (2020)
EQUIVARIANT DERIVED CATEGORIES FOR TOROIDAL GROUP IMBEDDINGS ROY JOSHUA∗ Department of Mathematics Ohio State University Columbus, Ohio, 43210, USA [email protected]
Abstract. Let X denote a projective variety over an algebraically closed field on which a linear algebraic group acts with finitely many orbits. Then, a conjecture of Soergel and Lunts in the setting of Koszul duality and Langlands’ philosophy, postulates that the equivariant derived category of bounded complexes with constructible equivariant cohomology sheaves on X is equivalent to a full subcategory of the derived category of modules over a graded ring defined as a suitable graded Ext. Only special cases of this conjecture have been proven so far. The purpose of this paper is to provide a proof of this conjecture for all projective toroidal imbeddings of complex reductive groups. In fact, we show that the methods used by Lunts for a proof in the case of toric varieties can be extended with suitable modifications to handle the toroidal imbedding case. Since every equivariant imbedding of a complex reductive group is dominated by a toroidal imbedding, the class of varieties for which our proof applies is quite large. We also show that, in general, there exist a countable number of obstructions for this conjecture to be true and that half of these vanish when the odd dimensional equivariant intersection cohomology sheaves on the orbit closures vanish. This last vanishing condition had been proven to be true in many cases of spherical varieties by Michel Brion and the author in prior work.
1. 2. 3. 4. 5. 6.
Table of contents Introduction and statement of results Toroidal group imbeddings and the proof of Theorem 1.3 Proof of Theorem 1.2 A general obstruction theory and conclusions: proof of Theorem 1.5 Comparison of equivariant derived categories Appendix 1. Introduction
This paper concerns a variant of a conjecture attributed to Soergel and Lunts (see [So98], [So01], [Lu95]) for the action of linear algebraic groups on projective varieties with finitely many orbits and over any algebraically closed field k. Let X DOI: 10.1007/S00031-020-09610-3 Supported by the NSF. Received May 26, 2019. Accepted March 14, 2020. Corresponding Author: Roy Joshua, e-mail: [email protected] ∗
ROY JOSHUA
denote such a projective variety provided with the action of a linear algebraic group G. Recall that a sheaf F on X is equivariant, if it satisfies the following condition: let µ, pr2 : G×X → X denote the group-action and projection to the second factor, respectively. Then there is given an isomorphism φ : µ∗ (F ) → pr∗2 (F ) satisfying a co-cycle condition on further pull-back to G × G × X, and which reduces to the identity on pull-back to X by the degeneracy map x 7→ (e, x). The conjecture asserts that the equivariant derived category DbG,c (X) of complexes of sheaves with bounded, equivariant and constructible cohomology sheaves is equivalent to a full subcategory of the derived category of differential
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