Abstract
We give a new proof of the derived equivalence of a pair of varieties connected by the flop of type C2 in the list of Kanemitsu (2018), which is originally due to Segal (Bull. Lond. Math. Soc., 48 (3) 533–538, 2016). We also prove the derived equivalence of a pair of varieties connected by the flop of type {A}_{4}^{G} in the same list. The latter proof follows that of the derived equivalence of Calabi–Yau 3-folds in Grassmannians Gr(2,5) and Gr(3,5) by Kapustka and Rampazzo (Commun. Num. Theor. Phys., 13 (4) 725–761 2019) closely.
Highlights
Let G be a semisimple Lie group and B a Borel subgroup of G
The pull-back to F of the hyperplane classes h and H will be denoted by the same symbol
When G is the simple Lie group of type G2, Ueda [24] used sequence of mutations of semiorthogonal decompositions of Db(V) obtained by applying Orlov’s theorem [20] to the diagram Eq 1.1 to prove the derived equivalence of V− and V+. This sequence of mutations in turn follows that of Kuznetsov [18] closely
Summary
Let G be a semisimple Lie group and B a Borel subgroup of G. For distinct maximal parabolic subgroups P and Q of G containing B, three homogeneous spaces G/P , G/Q, and G/(P ∩ Q) form the following diagram:. We write the hyperplane classes of P and Q as h and H respectively. The pull-back to F of the hyperplane classes h and H will be denoted by the same symbol
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