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.

arXiv commentNoncommutative deformations have been removed from the second version of the preprint and will be treated in greater detail in a subsequent paper