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 Mosc. Math. J., 2015, Volume 15, Number 2, Pages 187–203 (Mi mmj555)

Stability conditions for Slodowy slices and real variations of stability

Rina Annoa, Roman Bezrukavnikovbc, Ivan Mirkovićd

a Department of Mathematics, University of Pittsburg, Pittsburgh, PA 15260, USA
b Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts ave., Cambridge, MA 02139, USA
c National Research University Higher School of Economics, International Laboratory of Representation Theory and Mathematical Physics, 20 Myasnitskaya st., Moscow 101000, Russia
d Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA

Abstract: The paper provides new examples of an explicit submanifold in Bridgeland stabilities space of a local Calabi–Yau.
More precisely, let $X$ be the standard resolution of a transversal slice to an adjoint nilpotent orbit of a simple Lie algebra over $\mathbb C$. An action of the affine braid group on the derived category $D^b(Coh(X))$ and a collection of $t$-structures on this category permuted by the action have been constructed earlier by the last two authors and S. Riche. In this note we show that the $t$-structures come from points in a certain connected submanifold in the space of Bridgeland stability conditions. The submanifold is a covering of a submanifold in the dual space to the Grothendieck group, and the affine braid group acts by deck transformations.
We also propose a new variant of definition of stabilities on a triangulated category, which we call a “real variation of stability conditions” and discuss its relation to Bridgeland's definition. The main theorem provides an illustration of such a relation. We state a conjecture by the second author and A. Okounkov on examples of this structure arising from symplectic resolutions of singularities and its relation to equivariant quantum cohomology. We verify this conjecture in our examples.

Key words and phrases: stability conditions on triangulated categories, Slodowy slices, symplectic resolutions, (affine) braid group actions, quantization in positive characteristic, quantum cohomology.

DOI: https://doi.org/10.17323/1609-4514-2015-15-2-187-203

Full text: http://www.mathjournals.org/.../2015-015-002-002.html
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MSC: 14F05, 14N35, 17B08, 17B50
Received: April 9, 2014
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Citation: Rina Anno, Roman Bezrukavnikov, Ivan Mirković, “Stability conditions for Slodowy slices and real variations of stability”, Mosc. Math. J., 15:2 (2015), 187–203

Citation in format AMSBIB
\Bibitem{AnnBezMir15} \by Rina~Anno, Roman~Bezrukavnikov, Ivan~Mirkovi{\'c} \paper Stability conditions for Slodowy slices and real variations of stability \jour Mosc. Math.~J. \yr 2015 \vol 15 \issue 2 \pages 187--203 \mathnet{http://mi.mathnet.ru/mmj555} \crossref{https://doi.org/10.17323/1609-4514-2015-15-2-187-203} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3427420} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000361607300002} 

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This publication is cited in the following articles:
1. V. Nandakumar, “Stability conditions for Gelfand–Kirillov subquotients of category $\mathcal O$”, Int. Math. Res. Notices, 2017, no. 19, 5800–5832
2. A. Bayer, A. Craw, Z. Zhang, “Nef divisors for moduli spaces of complexes with compact support”, Sel. Math.-New Ser., 23:2 (2017), 1507–1561
3. A. Okounkov, “Enumerative geometry and geometric representation theory”, Algebraic Geometry (Salt Lake City 2015), v. 1, Proceedings of Symposia in Pure Mathematics, 97, no. 1, eds. T. DeFernex, B. Hassett, M. Mustata, M. Olsson, M. Popa, R. Thomas, Amer. Math. Soc., 2018, 419–457
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