Mukai flops and ℙ-twists

Associated to a Mukai flop {X\dashrightarrow X^{\prime}} is on the one hand a sequence of equivalences {D^{b}(X)\to D^{b}(X^{\prime})} , due to Kawamata and Namikawa, and on the other hand a sequence of autoequivalences of {D^{b}(X)} , due to Huybrechts and Thomas. We work out a complete picture of the relationship between the two. We do the same for standard flops, relating Bondal and Orlov’s derived equivalences to spherical twists, extending a well-known story for the Atiyah flop to higher dimensions.

Window shifts, flop equivalences and Grassmannian twists

We introduce a new class of autoequivalences that act on the derived categories of certain vector bundles over Grassmannians. These autoequivalences arise from Grassmannian flops: they generalize Seidel–Thomas spherical twists, which can be seen as arising from standard flops. We first give a simple algebraic construction, which is well suited to explicit computations. We then give a geometric construction using spherical functors which we prove is equivalent.