2015/11/02 by Agnieszka Bodzenta, Bodzenta, Agnieszka, Alexey Bondal +1
Mathematics · #14E05 #14E30 #18E30 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1511.00665
openalex publication_date 2015/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study derived categories of Gorenstein varieties X and X+ connected by a flop. We assume that the flopping contractions f: X → Y, f+: X+ → Y have fibers of dimension bounded by 1 and Y has canonical hypersurface singularities of multiplicity 2. We consider the fiber product W=X ×Y X+ with projections p: W → X, q: W → X+ and prove that the flop functors F = Rq_* Lp^*: Db(X) → Db(X+), F+= Rp_*Lq^*: Db(X+) → Db(X) are equivalences, inverse to those constructed by M. Van den Bergh. The composite F+ ∘ F: Db(X) → Db(X) is a non-trivial auto-equivalence. When variety Y is affine, we present F+∘ F as the spherical cotwist associated to a spherical functor Ψ. The functor Ψis constructed by deriving the inclusion of the null-category Af of sheaves F in \Coh (X) with Rf_*(F)=0 into Coh (X). We construct a spherical pair (Db(X),Db(X+)) in the quotient Db(W)/Kb, where Kb 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 L1f^*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.