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 in geometric terms. Namely, as the varieties involved are birational, they map to a common scheme, and the fibre product with respect to these maps gives a Fourier Mukai kernel which often induces the derived equivalence. When this happens, we have a non trivial autoequivalence of either sides of the flop known as the "flop-flop" autoequivalence.We investigate this autoequivalence, and we prove that can be realized as the inverse of a spherical twist around a spherical functor whose source category arises naturally from the geometry. Precisely, we consider the derived category of the fibre product and we prove that a suitable Verdier quotient of this category admits a four periodic SOD inducing the mentioned autoequivalence. This picture also implements a perverse Schober for the flop.To give a feeling of what the source category we found looks like, we study in detail standard flops (both in the local model and in the family version), and Mukai flops. In particular, we show that in these cases the category respects the known decomposition associated to the flop-flop autoequivalence, by which we mean that it is the gluing of (some of the possible) source categories of the single spherical functors. To conclude, we provide further examples of where our construction can be used, such as Grassmannian flops, and the Abuaf flop. ContentsFEDERICO BARBACOVI 3.7. Relative dimension one 36 4. Examples 38 4.1. Standard flops (local model) 38 4.2. Standard flops in families 43 4.3. Mukai flops 49 4.4. Other examples 58 Appendix A. Computations 60 A.1. Standard flops (local model) 60 A.2. Standard flops (family case) 64 A.3. Mukai flops 71 References 85 9We will call them bounded complexes from now on to be short.↑

On some examples of spherical functors related to flops

“…Let T be a triangulated category and let T ∈ T . Assume that T has a DG enhancement Φ : Moreover, let us recall the main theorem of [Bar20b]. Consider a diagram as…”

“…The first example in which we apply the general theory we described in [Bar20b] is that of standard flops. Let X − = X + = Tot(O P n (−1) ⊕n+1 ), and consider…”

“…In [Bar20b] considering a roof as below 1 X X − X + q p we proved that if p * O X ≃ O X − , q * O X ≃ O X − , and the flop functors…”

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In [Bar20b] we proved that the "flop-flop" autoequivalence can be realized as the spherical twist around a spherical functor whose source category arises naturally from the geometry.In this companion paper we study in detail some examples so to explicitly describe what the source category looks like. In some cases we are able to prove that the source category respects the known decomposition of the flop-flop autoequivalence, and therefore we tie up our geometric description with formal results which appear in the literature about gluing and splitting of spherical twists around spherical functors.The examples we treat completely are standard flops (both in the local model and in the family version), and Mukai flops.

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“…Similar statements hold for the subcategory of perfect complexes and for that of complexes with bounded and coherent cohomologies, but for the latter we need to assume that p and q are proper and of finite Tor dimension. We refer the interested reader to [Bar20b] for precise statements and for an explanation of why we pass from D b (−) to D qc (−). This paper will be a companion to ibidem in the sense that we will be concerned in describing explicitly the source category Ker p * ⊂ D qc ( X)/K in some examples.…”