×

zbMATH — the first resource for mathematics

Minimal Lefschetz decompositions of the derived categories for Grassmannians. (English. Russian original) Zbl 1287.14007
Izv. Math. 77, No. 5, 1044-1065 (2013); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 77, No. 5, 203-224(2013).
The author uses Kapranov’s exceptional collection on the derived category of bounded coherent sheaves on the Grassmannian \(\mathrm{Gr}(k,n)\) of \(k\) dimensional subspaces in an \(n\) dimensional vector space, to construct two different Lefschetz decompositions of the respective derived category.
Lefschetz decompositions are important in the context of homological projective duality, as introduced by Kuznetsov, and are special semiorthogonal decompositions that behave well (they induce a Lefschetz type property) with respect to taking linear sections (and, in particular, hyperplane sections). Such decompositions were already known for \(\mathrm{Gr}(2,n)\), the Grassmannian of 2 dimensional subpaces in an \(n\) dimensional vector space, and in other sporadic examples.
The author proves that one of the collections is full and, for \(k\) and \(n\) coprime, that it is also minimal. For \(k\) and \(n\) not coprime, the other collection is smaller and the conjecture is that it is both minimal and full. If \(k=2\) then both collections coincide to the one constructed in an earlier work by Kuznetsov.
While proving the statements in the paper, the author constructs a new class of exact Schur-Weyl type complexes of vector bundles on the Grassmannian which are interesting by themselves.

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
18E30 Derived categories, triangulated categories (MSC2010)
Full Text: DOI
References:
[1] А. И. Бондал, “Представления ассоциативных алгебр и когерентные пучки”, Изв. АН СССР. Сер. матем., 53:1 (1989), 25 – 44 · Zbl 0692.18002 · doi:10.1070/IM1990v034n01ABEH000583 · mi.mathnet.ru
[2] A. I. Bondal, “Representation of associative algebras and coherent sheaves”, Math. USSR-Izv., 34:1 (1990), 23 – 42 · Zbl 0692.18002 · doi:10.1070/IM1990v034n01ABEH000583
[3] A. L. Gorodentsev, A. N. Rudakov, “Exceptional vector bundles on projective spaces”, Duke Math. J., 54:1 (1987), 115 – 130 · Zbl 0646.14014 · doi:10.1215/S0012-7094-87-05409-3
[4] А. А. Бейлинсон, “Когерентные пучки на \(\mathbf P^n\) и проблемы линейной алгебры”, Функц. анализ и его прил., 12:3 (1978), 68 – 69 · Zbl 0402.14006 · mi.mathnet.ru
[5] A. A. Beilinson, “Coherent sheaves on \(\mathbf P^n\) and problems of linear algebra”, Funct. Anal. Appl., 12:3 (1978), 214 – 216 · Zbl 0424.14003 · doi:10.1007/BF01681436
[6] А. И. Бондал, М. М. Капранов, “Представимые функторы, функторы Серра и перестройки”, Изв. АН СССР. Сер. матем., 53:6 (1989), 1183 – 1205 · Zbl 0703.14011 · doi:10.1070/IM1990v035n03ABEH000716 · mi.mathnet.ru
[7] A. I. Bondal, M. M. Kapranov, “Representable functors, Serre functors, and mutations”, Math. USSR-Izv., 35:3 (1990), 519 – 541 · Zbl 0703.14011 · doi:10.1070/IM1990v035n03ABEH000716
[8] A. Kuznetsov, “Lefschetz decompositions and categorical resolutions of singularities”, Selecta Math. (N.S.), 13:4 (2008), 661 – 696 · Zbl 1156.18006 · doi:10.1007/s00029-008-0052-1 · arxiv:math/0609240
[9] M. M. Kapranov, “On the derived categories of coherent sheaves on some homogeneous spaces”, Invent. Math., 92:3 (1988), 479 – 508 · Zbl 0651.18008 · doi:10.1007/BF01393744 · eudml:143579
[10] A. Kuznetsov, “Exceptional collections for Grassmannians of isotropic lines”, Proc. Lond. Math. Soc. (3), 97:1 (2008), 155 – 182 · Zbl 1168.14032 · doi:10.1112/plms/pdm056 · arxiv:math/0512013
[11] A. Kuznetsov, “Homological projective duality”, Publ. Math. Inst. Hautes Eťudes Sci., 105:1 (2007), 157 – 220 · Zbl 1131.14017 · doi:10.1007/s10240-007-0006-8 · numdam:PMIHES_2007__105__157_0 · eudml:104222 · arxiv:math/0507292
[12] M. Demazure, “A very simple proof of Bott/s theorem”, Invent. Math., 33:3 (1976), 271 – 272 · Zbl 0383.14017 · doi:10.1007/BF01404206 · eudml:142384
[13] V. Gasharov, “A short proof of the Littlewood – Richardson rule”, European J. Combin., 19:4 (1998), 451 – 453 · Zbl 0918.05100 · doi:10.1006/eujc.1998.0212
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.