It is expected that flops do not change the derived category [BO, Kaw02]. This is proved for 3-folds by Bridgeland [Bri02], but open in higher dimensions.
Ito-Miura-Okawa-Ueda (IMOU) varieties [IMOU] are pairs of Calabi–Yau 3-folds which are non-birational and derived-equivalent. They are obtained as complete intersections in G2- Grassmannians defined by globally generated homogeneous vector bundles, and deformation- equivalent to the Pfaffian-Grassmannian pairs of Calabi–Yau 3-folds. The derived equivalence is first proved by Kuznetsov [Kuz18] using mutation of semiorthogonal decomposition. By taking the total spaces of the dual of the vector bundles defining the IMOU varieties, one obtains a pair of non-compact Calabi–Yau 7-folds related by a flop. Ueda [Ued19] proved the derived equivalence under this flop using mutation of semiorthogonal decomposition closely related to that of Kuznetsov. This derived equivalence of 7-folds implies the derived equivalence of IMOU varieties.
In the talk, we will discuss a new proof of the derived equivalence for a certain 5-fold flop, first proved by Segal [Seg16]. Our proof is based on mutation of semiorthogonal decompo- sition, and very different from Segal’s original proof based on a tilting object. Yet another proof based on a tilting object, together with its relation with Segal’s proof, is given by Hara [Hara]. We also discuss a proof of the derived equivalence for the flop of type AG4 in the sense of [Kan]. The proof is based on mutation of semiorthogonal decomposition again, and follows [KR] closely.
References
[BO] A. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties, arXiv:alg- geom/9506012
[Bri02] T. Bridgeland, Flops and derived categories, Inventiones Mathematicae. 147(3), 613-632 (2002). [Hara] W. Hara, On derived equivalence for Abuaf flop: mutation of non-commutative crepant resolutions and
spherical twists, arXiv:1706.04417.
[IMOU] A. Ito, M. Miura, S. Okawa, K. Ueda, Calabi–Yau complete intersections in G2-Grassmannians,
arXiv:1606.04076
[KR] M. Kapustka and M. Rampazzo Torelli problem for Calabi-Yau threefolds with GLSM description,
arXiv:1711.10231.
[Kaw02] Y. Kawamata, D-equivalence and K-equivalence, Journal of Differential Geometry. 61(1), 147-171
(2002).
[Kan] A. Kanemitsu, Mukai pairs and simple K-equivalence, arXiv:1812.05392.
[Kuz18] A. Kuznetsov, Derived equivalence of Ito–Miura–Okawa–Ueda Calabi–Yau 3-folds, Journal of the
Mathematical Society of Japan. 70(3), 1007-1013 (2018).
[Seg16] E. Segal, A new 5-fold flop and derived equivalence, Bulletin of the London Mathematical Society. 48,
533-538 (2016).
[Ued19] K. Ueda, G2-Grassmannians and derived equivalences, Manuscripta Mathematica. 159(3-4), 549-559
(2019).