## The derived category of a GIT quotient

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- by Daniel Halpern-Leistner
- J. Amer. Math. Soc.
**28**(2015), 871-912 - DOI: https://doi.org/10.1090/S0894-0347-2014-00815-8
- Published electronically: October 31, 2014

## 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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## Bibliographic Information

**Daniel Halpern-Leistner**- Affiliation: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
- MR Author ID: 1101864
- Email: danhl@math.columbia.edu
- 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
- © Copyright 2014 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
- MathSciNet review: 3327537

Dedicated: Dedicated to Ernst Halpern, who inspired my scientific pursuits