Twists and braids for general 3-fold flops
Will Donovan, Michael Wemyss
2019
Journal of the European Mathematical Society (Print)
Given a quasi-projective 3-fold X with only Gorenstein terminal singularities, we prove that the flop functors beginning at X satisfy higher degree braid relations, with the combinatorics controlled by a real hyperplane arrangement H. This leads to a general theory, incorporating known special cases with degree 3 braid relations, in which we show that higher degree relations can occur even for two smooth rational curves meeting at a point. This theory yields an action of the fundamental group
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... the complexified complement π 1 (C n \H C ) on the derived category of X, for any such 3-fold that admits individually floppable curves. We also construct such an action in the more general case where individual curves may flop analytically, but not algebraically, and furthermore we lift the action to a form of affine pure braid group under the additional assumption that X is Q-factorial. Along the way, we produce two new types of derived autoequivalences. One uses commutative deformations of the scheme-theoretic fibre of a flopping contraction, and the other uses noncommutative deformations of the fibre with reduced scheme structure, generalising constructions of Toda and the authors [T07, DW1] which considered only the case when the flopping locus is irreducible. For type A flops of irreducible curves, we show that the two autoequivalences are related, but that in other cases they are very different, with the noncommutative twist being linked to birational geometry via the Bridgeland-Chen [B02, C02] flop-flop functor. WILL DONOVAN AND MICHAEL WEMYSS Proof. (1) This follows as in [DW1, (7.D), (7.E)], using 5.8 above. (a) This is shown as in [DW1, 7.4(1)]. (b) Exactly as in the proof of [DW1, (7.E)], G RA J ∼ = RHom X (E J , −). Further, setting F is the morita equivalence between mod Λ J and mod A J induced from (4.A). Thus the functor G LA J takes D − (coh X ) to D − (mod A J ), and also takes D b (coh X ) to D b (mod A J ), since each individual functor does: F −1 and α * since they are already exact, RHom Up (V, −) since it is a tilting equivalence [TU, 3.3], and (− ⊗ L Λ Λ J ) since Λ J has finite projective dimension. Since A J is a self-injective algebra, we can now use duality as in [YZ, 1.3]. Denoting restriction of scalars from Λ J to Λ by res, we see But now being self-injective, A J is clearly a dualizing complex for D(Mod A J ), and so since G LA J (x) ∈ D − (mod A J ) for all x ∈ D − (coh X ), applying the dualizing functor RHom AJ (−, A J ) gives the result. (c) As remarked above, G LA J takes D b (coh X ) to D b (mod A J ). The functor G J takes the simple A J -modules to the coherent sheaves E i , hence since A J is a finite dimensional algebra, it follows that G J takes D b (mod A J ) to D b (coh X ). We conclude that the composition G J • G LA J Corollary 5.12. With the global quasi-projective flops setup of 2.2, and notation in (5.I), for all y ∈ D(QcohX), there is a functorial triangle RHomX (E J , y) ⊗ L AJ E J → y → JTwist p (y) → . If further U p is Q-factorial, there is a similar triangle for F Twist p . Proof. By the above, all kernels in (5.K) have right adjoints given by applying (−) R := RHomX ×X (−, π ! 2 OX ). We know that D R J gives JTwist p , and clearly the right adjoint of the identity functor is the identity. Further by 5.9 the FM functor given by Q is G J • G LA J , and it has right adjoint G J • G RA J . Since G J := − ⊗ L AJ E J , it is clear that G J • G RA J ∼ = RHomX (E J , −) ⊗ L AJ E J . 5.5. The Projective Twist is an Equivalence. A formal consequence of the twist definition 5.4 is the following intertwinement lemma. Proposition 5.13. The following diagram is naturally commutative.
doi:10.4171/jems/868
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