Flops and spherical functors

Abstract

We study derived categories of Gorenstein varieties X and X^+ connected by a flop. We assume that the flopping contractions f: X \to Y, f^+: X^+ \to Y have fibers of dimension bounded by 1 and Y has canonical hypersurface singularities of multiplicity 2. We consider the fiber product W=X \times_Y X^+ with projections p: W \to X, q: W \to X^+ and prove that the flop functors F = Rq_* Lp^*: D^b(X) \to D^b(X^+), F^+= Rp_*Lq^*: D^b(X^+) \to D^b(X) are equivalences, inverse to those constructed by M. Van den Bergh. The composite F^+ \circ F: D^b(X) \to D^b(X) is a non-trivial auto-equivalence. When variety Y is affine, we present F^+\circ F as the spherical cotwist associated to a spherical functor Ψ. The functor Ψis constructed by deriving the inclusion of the null-category A_f of sheaves F in \Coh (X) with Rf_*(F)=0 into Coh (X). We construct a spherical pair (D^b(X),D^b(X^+)) in the quotient D^b(W)/K^b, where K^b is the common kernel of the derived push-forwards for the projections to X and X^+, thus implementing in geometric terms a schober for the flop. A technical innovation of the paper is the L^1f^*f_* vanishing for the Van den Bergh's projective generator. We construct a projective generator in the null-category and prove that its endomorphism algebra is the contraction algebra.