Spherical functors and the flop-flop autoequivalence

Abstract

Flops are birational transformations which, conjecturally, induce derived equivalences. In many cases an equivalence can be produced as pull-push via a resolution of the birational transformation; when this happens, we have a non-trivial autoequivalence of either sides of the flop known as the \emph{flop-flop autoequivalence}. We prove that such autoequivalence can be realised as the inverse of a spherical twist around a conservative, spherical functor in a natural way. More precisely, we prove that a natural, conservative spherical functor exists in a more general framework and that the flop-flop autoequivalence fits into this picture. We also give an explicit description of the source category of the spherical functor for standard flops (local model and family case) and Mukai flops. We conclude with some speculation about Grassmannian flops and the Abuaf flop.