Spherical functors provide a formal way to package autoequivalences of enhanced triangulated categories. Moreover, E. Segal proved that any autoequivalence of an enhanced triangulated category can be realized as a spherical twist.When exhibiting an autoequivalence as a spherical twist, one has various choices for the source category of the spherical functor. We prove that, in the DG setting, given two spherical twists there is a natural way to produce a new spherical functor whose twist is the composition of the spherical twists we started with, and whose source category semiorthogonally decomposes into the source categories for the two spherical functors we began with.

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.↑

In topological dynamics, the Gromov-Yomdin theorem states that the topological entropy of a holomorphic automorphism f of a smooth projective variety is equal to the logarithm of the spectral radius of the induced map f * . In order to establish a categorical analogue of the Gromov-Yomdin theorem, one first needs to find a categorical analogue of a holomorphic automorphism. In this paper, we propose a categorical analogue of a holomorphic automorphism and prove that the Gromov-Yomdin type theorem holds for them.

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. We also discuss the cases of Grassmannian flops, and the Abuaf flop. Contents 14 2.3. Mukai flops 19 2.4. Other examples 31 Appendix A. Computations 34 A.1. Standard flops (local model) 34 A.2. Standard flops (family case) 37 A.3. Mukai flops 48 References 64

In this short note, we observe that the Serre functor on the residual category of a complete intersection can be easily described in the framework of hybrid models. Using this description, we recover some recent results of Kuznetsov and Perry.