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Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

   
 

 

The derived category of a GIT quotient


Author: Daniel Halpern-Leistner
Journal: J. Amer. Math. Soc. 28 (2015), 871-912
MSC (2010): Primary 14F05, 14L25, 14L30; Secondary 19L47
DOI: https://doi.org/10.1090/S0894-0347-2014-00815-8
Published electronically: October 31, 2014
MathSciNet review: 3327537
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Abstract: 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 derived equivalences between birational varieties. We provide a criterion under which two different GIT quotients are derived equivalent, and apply it to prove that any two generic GIT quotients of an equivariantly Calabi-Yau projective-over-affine manifold by a torus are derived equivalent.


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Additional Information

Daniel Halpern-Leistner
Affiliation: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
Email: danhl@math.columbia.edu

DOI: https://doi.org/10.1090/S0894-0347-2014-00815-8
Received by editor(s): December 31, 2012
Received by editor(s) in revised form: March 7, 2014, and May 19, 2014
Published electronically: October 31, 2014
Dedicated: Dedicated to Ernst Halpern, who inspired my scientific pursuits
Article copyright: © Copyright 2014 Daniel Halpern-Leistner