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Combinatorial constructions of derived equivalences


Authors: Daniel Halpern-Leistner and Steven V Sam
Journal: J. Amer. Math. Soc. 33 (2020), 735-773
MSC (2010): Primary 14F05, 14L24, 19E08
DOI: https://doi.org/10.1090/jams/940
Published electronically: May 5, 2020
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Abstract: Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of Špenko and Van den Bergh, we construct equivalences between the derived categories of coherent sheaves of its various geometric invariant theory (GIT) quotients for suitably generic stability parameters. These variations of GIT quotient are examples of more complicated wall crossings than the balanced wall crossings studied in recent work on derived categories and variation of GIT quotients.

Our construction is algorithmic and quite explicit, allowing us to: 1) describe a tilting vector bundle which generates the derived category of such a GIT quotient, 2) provide a combinatorial basis for the $ K$-theory of the GIT quotient in terms of the representation theory of $ G$, and 3) show that our derived equivalences satisfy certain relations, leading to a representation of the fundamental groupoid of a ``Kähler moduli space'' on the derived category of such a GIT quotient. Finally, we use graded categories of singularities to construct derived equivalences between all Deligne-Mumford hyperkähler quotients of a symplectic linear representation of a reductive group (at the zero fiber of the algebraic moment map and subject to a certain genericity hypothesis on the representation), and we likewise construct actions of the fundamental groupoid of the corresponding Kähler moduli space.


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Additional Information

Daniel Halpern-Leistner
Affiliation: Department of Mathematics, Columbia University, 2290 Broadway, New York, New York 10027
Address at time of publication: Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853
Email: danielhl@cornell.edu

Steven V Sam
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53703
Address at time of publication: Department of Mathematics, University of California, San Diego, La Jolla, California 92093
Email: ssam@ucsd.edu

DOI: https://doi.org/10.1090/jams/940
Received by editor(s): October 29, 2018
Received by editor(s) in revised form: September 18, 2019
Published electronically: May 5, 2020
Additional Notes: The first author was partially supported by NSF DMS-1303960.
The second author was partially supported by NSF DMS-1500069.
Article copyright: © Copyright 2020 Daniel S. Halpern Leistner and Steven Sam