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\(G_2\)-Grassmannians and derived equivalences. (English) Zbl 1440.14096

Summary: We prove the derived equivalence of a pair of non-compact Calabi-Yau 7-folds, which are the total spaces of certain rank 2 bundles on \(G_2\)-Grassmannians. The proof follows that of the derived equivalence of Calabi-Yau 3-folds in \(G_2\)-Grassmannians by A. Kuznetsov [J. Math. Soc. Japan 70, No. 3, 1007–1013 (2018; Zbl 1423.14128)] closely.

MSC:

14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14J32 Calabi-Yau manifolds (algebro-geometric aspects)

Citations:

Zbl 1423.14128
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References:

[1] Addington, N., Donovan, W., Segal, E.: The Pfaffian-Grassmannian equivalence revisited. Algebr. Geom. 2(3), 332-364 (2015) · Zbl 1322.14037
[2] Ballard, M., Favero, D., Katzarkov, L.: Variation of geometric invariant theory quotients and derived categories, arXiv:1203.6643 · Zbl 1432.14015
[3] Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors, and reconstructions. Izv. Akad. Nauk SSSR Ser. Mat. 53(6), 1188-1205 (1989). 1337
[4] Bondal, A., Orlov, D.: Derived categories of coherent sheaves. In: Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, pp. 47-56 (2002) · Zbl 0996.18007
[5] Baranovsky, V., Pecharich, J.: On equivalences of derived and singular categories. Cent. Eur. J. Math. 8(1), 1-14 (2010) · Zbl 1191.14004
[6] Donovan, W., Segal, E.: Window shifts, flop equivalences and Grassmannian twists. Compos. Math 150(6), 942-978 (2014) · Zbl 1354.14028
[7] Herbst, M., Hori, K., Page, D.: Phases of N=2 theories in 1+1 dimensions with boundary, arXiv:0803.2045
[8] Hirano, Y.: Equivalences of derived factorization categories of gauged Landau-Ginzburg models, arXiv:1506.00177 · Zbl 1386.14075
[9] Halpern-Leistner, D.: The derived category of a GIT quotient. J. Am. Math. Soc. 28(3), 871-912 (2015) · Zbl 1354.14029
[10] Halpern-Leistner, D., Shipman, I.: Autoequivalences of derived categories via geometric invariant theory. Adv. Math. 303, 1264-1299 (2016) · Zbl 1371.14023
[11] Ito, A., Miura, M., Okawa, S., Ueda, K.: Calabi-Yau complete intersections in \[G_2\] G2-grassmannians, arXiv:1606.04076
[12] Ito, A., Miura, M., Okawa, S., Ueda, K.: The class of the affine line is a zero divisor in the Grothendieck ring: via \[G_2\] G2-Grassmannians, arXiv:1606.04210 · Zbl 1420.14019
[13] Ito, A., Miura, M., Okawa, S., Ueda, K.: The class of the affine line is a zero divisor in the Grothendieck ring: via K3 surfaces of degree 12 and abelian varieties, arXiv:1612.08497 · Zbl 1420.14019
[14] Isik, M.U.: Equivalence of the derived category of avariety with a singularity category, Int. Math. Res. Not. IMRN no. 12, 2787-2808 (2013) · Zbl 1312.14052
[15] Kawamata, \[Y.: D\] D-equivalence and \[KK\]-equivalence. J. Differ. Geom. 61(1), 147-171 (2002) · Zbl 1056.14021
[16] Kapustin, A., Katzarkov, L., Orlov, D., Yotov, M.: Homological mirror symmetry for manifolds of general type. Cent. Eur. J. Math. 7(4), 571-605 (2009) · Zbl 1200.53079
[17] Kuznetsov, A.: Derived equivalence of Ito-Miura-Okawa-Ueda Calabi-Yau 3-folds, arXiv:1611.08386 · Zbl 1423.14128
[18] Kuznetsov, A.G.: Hyperplane sections and derived categories. Izv. Ross. Akad. Nauk Ser. Mat. 70(3), 23-128 (2006) · Zbl 1133.14016
[19] Orlov, D.O.: Projective bundles, monoidal transformations, and derived categories of coherent sheaves. Izv. Ross. Akad. Nauk Ser. Mat. 56(4), 852-862 (1992) · Zbl 0798.14007
[20] Segal, E.: Equivalence between GIT quotients of Landau-Ginzburg B-models. Commun. Math. Phys 304(2), 411-432 (2011) · Zbl 1216.81122
[21] Segal, E.: A new 5-fold flop and derived equivalence. Bull. Lond. Math. Soc. 48(3), 533-538 (2016) · Zbl 1342.14028
[22] Shipman, I.: A geometric approach to Orlov’s theorem. Compos. Math. 148(5), 1365-1389 (2012) · Zbl 1253.14019
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