×

zbMATH — the first resource for mathematics

\(t\)-structures on some local Calabi–Yau varieties. (English) Zbl 1069.14044
Author’s abstract: Let \(Z\) be a smooth Fano variety satisfying the condition that the rank of the Grothendieck group of \(Z\) is one more than the dimension of \(Z\). Let \(\omega_Z\) denote the total space of the canonical line bundle of \(Z\), considered as a non-compact Calabi-Yau variety. We use the theory of exceptional collections to describe \(t\)-structures on the derived category of coherent sheaves on \(\omega_Z\). The combinatorics of these \(t\)-structures is determined by a natural action of an affine braid group, closely related to the well-known action of the Artin braid group on the set of exceptional collections on \(Z\).

MSC:
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J45 Fano varieties
18E30 Derived categories, triangulated categories (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Beilinson, A.A., Coherent sheaves on \(\mathbb{P}^n\) and problems of linear algebra, Funktsional. anal. i prilozhen., Funct. anal. appl., 12, 3, 68-69, (1978), English translation in · Zbl 0402.14006
[2] Beilinson, A.A.; Bernstein, J.; Deligne, P., Faisceaux pervers, Astérisque, 100, (1983) · Zbl 0536.14011
[3] Beilinson, A.A.; Ginzburg, V.; Soergel, W., Koszul duality patterns in representation theory, J. amer. math. soc., 9, 2, 473-527, (1996) · Zbl 0864.17006
[4] Birman, J.S., Braids, links and mapping class groups, Ann. of math. stud., vol. 82, (1974), Princeton Univ. Press
[5] Bondal, A.I., Representation of associative algebras and coherent sheaves, Izv. akad. nauk SSSR ser. mat., Math. USSR-izv., 34, 1, 23-44, (1990), English translation in · Zbl 0692.18002
[6] Bondal, A.I.; Polishchuk, A.E., Homological properties of associative algebras: the method of helices, Izv. ross. akad. nauk ser. mat., Russian acad. sci. izv. math., 42, 2, 219-260, (1994), English translation in · Zbl 0847.16010
[7] Brenner, S.; Butler, M.C.R., Generalizations of the bernstein – gelfand – ponomarev reflection functors, (), 103-169 · Zbl 0446.16031
[8] Bridgeland, T., Stability conditions on triangulated categories, Preprint · Zbl 1137.18008
[9] T. Bridgeland, Stability conditions on \(\mathcal{O}_{\mathbb{P}^2}(- 3)\), in preparation
[10] Cassels, J.W.S., The markoff chain, Ann. of math., 50, 676-685, (1949) · Zbl 0035.31701
[11] Chow, W.-L., On the algebraical braid group, Ann. of math., 49, 654-658, (1948) · Zbl 0033.01002
[12] L. Costa, R.M. Miró-Roig, Tilting bundles, helix theory and Castelnuovo-Mumford regularity, Preprint
[13] Feng, B.; Hanany, A.; He, Y.-H.; Iqbal, A., Quiver theories, soliton spectra and picard – lefschetz transformations, J. high energy phys., 2, 056, (2003), Also
[14] Franco, S.; Hanany, A.; He, Y.-H., A trio of dualities: walls, trees and cascades, Proc. 36th internat. symp. ahrenshoop on the theory of elementary particles, fortschr. phys., 52, 6-7, 540-547, (2004), Also · Zbl 1052.81586
[15] Gelfand, S.I.; Manin, Yu.I, Methods of homological algebra, (1996), Springer · Zbl 0855.18001
[16] Gorodentsev, A.L., Surgeries of exceptional bundles on \(\mathbb{P}^n\), Izv. akad. nauk SSSR ser. mat., Math. USSR-izv., 32, 1, 1-13, (1989), English translation in · Zbl 0664.14010
[17] Gorodentsev, A.L.; Rudakov, A.N., Exceptional vector bundles on projective spaces, Duke math. J., 54, 1, 115-130, (1987) · Zbl 0646.14014
[18] Hille, L., Consistent algebras and special tilting sequences, Math. Z., 220, 2, 189-205, (1995) · Zbl 0841.14013
[19] Happel, D.; Reiten, I.; Smalø, S.O., Tilting in abelian categories and quasitilted algebras, Mem. amer. math. soc., 120, 575, (1996) · Zbl 0849.16011
[20] Kapranov, M.M., The derived category of coherent sheaves on a quadric, Funktsional. anal. i prilozhen., Funct. anal. appl., 20, 2, 67, (1986), English translation in · Zbl 0607.18004
[21] Karpov, B.V.; Nogin, D.Yu., Three-block exceptional sets on del Pezzo surfaces, Izv. ross. akad. nauk ser. mat., Russian acad. sci. izv. math., 62, 3, 429-463, (1998), English translation in · Zbl 0949.14026
[22] Kent, R.P.; Peifer, D., A geometric and algebraic description of annular braid groups, Internat. J. algebra comput., 12, 1 & 2, 85-97, (2002) · Zbl 1010.20024
[23] Orlov, D.O., An exceptional set of vector bundles on the variety \(V_5\), Vestnik moskov. univ. ser. I mat. mekh., Moscow univ. math. bull., 46, 5, 48-50, (1991), English translation in · Zbl 0784.14010
[24] Rankin, R.A., Modular forms and functions, (1977), Cambridge University Press · Zbl 0376.10020
[25] Rickard, J., Morita theory for derived categories, J. London math. soc., 39, 3, 436-456, (1989) · Zbl 0642.16034
[26] Rudakov, A.N., Helices and vector bundles: seminaire rudakov, London math. soc. lecture note ser., vol. 148, (1990), Cambridge University Press Cambridge · Zbl 0727.00022
[27] Rudakov, A.N., Exceptional vector bundles on a quadric, Izv. akad. nauk SSSR ser. mat., Math. USSR-izv., 33, 1, 115-138, (1989), 896. English translation in · Zbl 0708.14012
[28] Seidel, P.; Thomas, R.P., Braid group actions on derived categories of coherent sheaves, Duke math. J., 108, 1, 37-108, (2001) · Zbl 1092.14025
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.