Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-23T18:43:33.130Z Has data issue: false hasContentIssue false
  • English
  • Français

Window shifts, flop equivalences and Grassmannian twists

Part of: Special varieties Abelian categories (Co)homology theory

Published online by Cambridge University Press:  07 April 2014

Will Donovan  and
Ed Segal
Will Donovan
Affiliation:
The Maxwell Institute, School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, UK email [email protected]
Ed Segal
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce a new class of autoequivalences that act on the derived categories of certain vector bundles over Grassmannians. These autoequivalences arise from Grassmannian flops: they generalize Seidel–Thomas spherical twists, which can be seen as arising from standard flops. We first give a simple algebraic construction, which is well suited to explicit computations. We then give a geometric construction using spherical functors which we prove is equivalent.

Keywords

derived categoriesGrassmanniansderived equivalencesspherical functors

MSC classification

Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions 18E30: Derived categories, triangulated categories 14M15: Grassmannians, Schubert varieties, flag manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 6 , June 2014 , pp. 942 - 978
Copyright
© The Author(s) 2014 

References

Addington, N., New derived symmetries of some Hyperkähler varieties, Preprint (2011),arXiv:1112.0487.Google Scholar
Anno, R., Spherical functors, Preprint (2007), arXiv:0711.4409.Google Scholar
Anno, R. and Logvinenko, T., Orthogonally spherical objects and spherical fibrations, Preprint (2010), arXiv:1011.0707.Google Scholar
Beilinson, A., Coherent sheaves on ℙn and problems of linear algebra, Funct. Anal. Appl. 12 (1978), 214–216.CrossRefGoogle Scholar
Ballard, M., Favero, D. and Katzarkov, L., Variation of geometric invariant theory quotients and derived categories, Preprint (2012), arXiv:1203.6643.Google Scholar
Bondal, A. and Orlov, D., Semiorthogonal decomposition for algebraic varieties, Preprint (1995), arXiv:alg-geom/9506012.Google Scholar
Buchweitz, R.-O., Leuschke, G. J. and van den Bergh, M., Non-commutative desingularization of determinantal varieties, II, Preprint (2011), arXiv:1106.1833.Google Scholar
Buchsbaum, D. A. and Rim, D. S., A generalized Koszul complex. II. Depth and multiplicity, Trans. Amer. Math. Soc. 111 (1964), 197–224.Google Scholar
Bridgeland, T., Flops and derived categories, Invent. Math. 147 (2002), 613–632.Google Scholar
Bruns, W. and Vetter, U., Determinantal rings, Lecture Notes in Mathematics, vol. 1327 (Springer, New York, 1988).Google Scholar
Cautis, S., Kamnitzer, J. and Licata, A., Coherent sheaves on quiver varieties and categorification, Math. Ann. 357 (2013), 805–854.Google Scholar
Donovan, W., Grassmannian twists on derived categories of coherent sheaves, PhD thesis, Imperial College London, July 2011,http://www.maths.ed.ac.uk/willdonovan/documents/thesis/twist.pdf.Google Scholar
Donovan, W., Grassmannian twists on the derived category via spherical functors, Proc. Lond. Math. Soc. 107 (2013), 1053–1090.Google Scholar
Donovan, W., Grassmannian twists, derived equivalences and brane transport, Preprint (2013), arXiv:1304.2913 (contribution to the proceedings of String-Math 2012).Google Scholar
Eisenbud, D., Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150 (Springer, Berlin, 1994).Google Scholar
Eagon, J. A. and Northcott, D. G., Ideals defined by matrices and a certain complex associated with them, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 269 (1962), 188–204.Google Scholar
Fonarev, A., On minimal Lefschetz decompositions for Grassmannians, Izv. Math. 77 (2013), 1044–1065.CrossRefGoogle Scholar
Fulton, W. and Harris, J., Representation theory: a first course, Graduate Texts in Mathematics, vol. 129 (Springer, Berlin, 1996).Google Scholar
Halpern-Leistner, D., The derived category of a GIT quotient, Preprint (2012),arXiv:1203.0276.Google Scholar
Halpern-Leistner, D. and Shipman, I., Autoequivalences of derived categories via geometric invariant theory, Preprint (2013), arXiv:1303.5531.Google Scholar
Herbst, M., Hori, K. and Page, D., Phases of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathcal{N}=2$ theories in $1+1$ dimensions with boundary, Preprint (2008), arXiv:0803.2045.Google Scholar
Hori, K. and Tong, D., Aspects of non-Abelian gauge dynamics in two-dimensional N = (2, 2) theories, J. High Energy Phys. 5 (2007), 79; arXiv:hep-th/0609032.CrossRefGoogle Scholar
Horja, R. P., Derived category automorphisms from mirror symmetry, Duke Math. J. 127 (2005), 1–34.Google Scholar
Kapranov, M., On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), 479–508.Google Scholar
Kawamata, Y., D-equivalence and K-equivalence, J. Differential Geom. 61 (2002), 147–171.Google Scholar
Segal, E., Equivalences between GIT quotients of Landau–Ginzburg B-models, Comm. Math. Phys. 304 (2011), 411–432.Google Scholar
Seidel, P. and Thomas, R. P., Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), 37–108.Google Scholar
Thomas, R. P., Notes on GIT and symplectic reduction for bundles and varieties, Preprint (2005), arXiv:math/0512411.Google Scholar
Weyman, J., Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, vol. 149 (Cambridge University Press, Cambridge, 2003).Google Scholar