arXiv subjects
Publications and source records attributed to Federico Barbacovi.
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.
Homological mirror symmetry predicts that there is a relation between autoequivalence groups of derived categories of coherent sheaves on Calabi-Yau varieties, and the symplectic mapping class groups of symplectic manifolds. In this paper, as an analogue of Dehn twists for closed oriented real surfaces, we study spherical twists for dg-enhanced triangulated categories. We introduce the intersection number and relate it to group-theoretic properties of spherical twists. We show an inequality analogous to a fundamental inequality in the theory of mapping class groups about the behavior of the intersection number via iterations of Dehn twists. We also classify the subgroups generated by two spherical twists using the intersection number. In passing, we prove a structure theorem for finite dimensional dg-modules over the graded dual numbers and use this to describe the autoequivalence group. As an application, we compute the center of autoequivalence groups of derived categories of K3 surfaces.
E. Segal proved that any autoequivalence of an enhanced triangulated category can be realised as a spherical twist. However, when exhibiting an autoequivalence as a spherical twist one has various choices for the source category of the spherical functor. We describe a construction that realises the composition of two spherical twists as the twist around a single spherical functor whose source category semiorthogonally decomposes into the source categories for the spherical functors we started with. We give a description of the cotwist for this spherical functor and prove, in the special case when our starting twists are around spherical objects, that the cotwist is the Serre functor (up to a shift). We finish with an explicit treatment for the case of P-objects.
Flops are birational transformations which, conjecturally, induce derived equivalences. In many cases an equivalence can be produced as pull-push via a resolution of the birational transformation; when this happens, we have a non-trivial autoequivalence of either sides of the flop known as the \emph{flop-flop autoequivalence}. We prove that such autoequivalence can be realised as the inverse of a spherical twist around a conservative, spherical functor in a natural way. More precisely, we prove that a natural, conservative spherical functor exists in a more general framework and that the flop-flop autoequivalence fits into this picture. We also give an explicit description of the source category of the spherical functor for standard flops (local model and family case) and Mukai flops. We conclude with some speculation about Grassmannian flops and the Abuaf flop.
In arXiv:2007.14415 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.
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.
Given a categorical dynamical system, i.e. a triangulated category together with an endofunctor, one can try to understand the complexity of the system by computing the entropy of the endofunctor. Computing the entropy of the composition of two endofunctors is hard, and in general the result doesn't have to be related to the entropy of the single pieces. In this paper we compute the entropy of the composition of two spherical twists around spherical objects, showing that it depends on the dimension of the graded vector space of morphisms between them. As a consequence of these computations we produce new counterexamples to Kikuta-Takahashi's conjecture. In particular, we describe the first counterexamples in odd dimension and examples for the $d$-Calabi-Yau Ginzburg dg algebra associated to the $A_2$ quiver.