×

Kernels for Grassmann flops. (English. French summary) Zbl 1468.14034

Summary: We develop a generalization of the \(Q\)-construction of the first author et al. [“Kernels from compactifications”, Preprint, arXiv:1710.01418] for Grassmann flips. This generalization provides a canonical idempotent kernel on the derived category of the associated global quotient stack. The idempotent kernel, after restriction, induces a semi-orthogonal decomposition which compares the flipped varieties. Furthermore its image, after restriction to the geometric invariant theory semistable locus, “opens” a canonical “window” in the derived category of the quotient stack. We check this window coincides with the set of representations used by M. M. Kapranov [Invent. Math. 92, No. 3, 479–508 (1988; Zbl 0651.18008)] to form a full exceptional collection on Grassmannians.

MSC:

14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14E05 Rational and birational maps
14L24 Geometric invariant theory

Citations:

Zbl 0651.18008
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ballard, M. R.; Diemer, C.; Favero, D., Kernels from compactifications (Oct. 2017)
[2] Drinfeld, V., On algebraic spaces with an action of \(\mathbb{G}_m\) (Aug. 2013)
[3] Bondal, A.; Orlov, D., Semiorthogonal decomposition for algebraic varieties (1995)
[4] Kawamata, Y., D-equivalence and k-equivalence, J. Differ. Geom., 61, 1, 147-171 (2002) · Zbl 1056.14021
[5] Donovan, W.; Segal, E., Window shifts, flop equivalences and Grassmannian twists, Compos. Math., 150, 6, 942-978 (2014) · Zbl 1354.14028
[6] Kapranov, M. M., On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math., 92, 479 (1988) · Zbl 0651.18008
[7] Buchweitz, R.-O.; Leuschke, G. J.; Van den Bergh, M., Non-commutative desingularization of determinantal varieties, II: arbitrary minors, Int. Math. Res. Not. IMRN, 9, 2748-2812 (2016) · Zbl 1434.14001
[8] Halpern-Leistner, D., The derived category of a GIT quotient, J. Am. Math. Soc., 28, 3, 871-912 (2015) · Zbl 1354.14029
[9] Ballard, M.; Favero, D.; Katzarkov, L., Variation of geometric invariant theory quotients and derived categories, J. Reine Angew. Math., 746, 235-303 (2019) · Zbl 1432.14015
[10] Špenko, Š.; Van den Bergh, M., Non-commutative resolutions of quotient singularities for reductive groups, Invent. Math., 210, 1, 3-67 (2017) · Zbl 1375.13007
[11] Mumford, D., Geometric Invariant Theory, Ergebnisse der Mathematik und Ihrer Grenzgebiete, Neue Folge, vol. 34 (1965), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0147.39304
[12] Ballard, M.; Favero, D.; Katzarkov, L., A category of kernels for equivariant factorizations and its implications for Hodge theory, Publ. Math. l’IHÉS, 120, 1, 1-111 (2014) · Zbl 1401.14086
[13] Kraft, H.; Procesi, C., A primer of classical invariant theory: preliminary version (1996), éditeur inconnu
[14] Weyl, H., The Classical Groups, Their Invariants and Representations. Vol. 45 (1946), Princeton University Press
[15] Krause, H., Localization Theory for Triangulated Categories, London Mathematical Society Lecture Note Series (2010) · Zbl 1232.18012
[16] Weyman, J., Cohomology of Vector Bundles and Syzygies, Cambridge Tracts in Mathematics (2003), Cambridge University Press · Zbl 1075.13007
[17] Procesi, C., Lie Groups: An Approach Through Invariants and Representations, Universitext (2007), Springer Science+Buisness Media LLC
[18] Orlov, D. O., Derived categories of coherent sheaves on abelian varieties and equivalences between them, Izv. Ross. Akad. Nauk, Ser. Mat., 66, 3, 131-158 (2002) · Zbl 1031.18007
[19] Demazure, M., A very simple proof of Bott’s theorem, Invent. Math., 33, 3, 271-272 (1976) · Zbl 0383.14017
[20] Kuznetsov, A., Exceptional collections for Grassmannians of isotropic lines, Proc. Lond. Math. Soc. (3), 97, 1, 155-182 (2008) · Zbl 1168.14032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.