# Spherical DG-functors

### Rina Anno

University of Pittsburgh, USA### Timothy Logvinenko

Cardiff University, UK

## Abstract

For two DG-categories A and B we define the notion of a *spherical* Morita quasi-functor $\mathcal A \to \mathcal B$. We construct its associated autoequivalences: the *twist* $T \in \mathrm {Aut} \mathcal D(\mathcal B)$ and the *cotwist* $F \in \mathrm {Aut} \mathcal D(\mathcal A)$. We give sufficiency criteria for a quasi-functor to be spherical and for the twists associated to a collection of spherical quasi-functors to braid. Using the framework of DG-enhanced triangulated categories, we translate all of the above to Fourier–Mukai transforms between the derived categories of algebraic varieties. This is a broad generalisation of the results on spherical objects in [ST01] and on spherical functors in [Ann07]. In fact, this paper replaces [Ann07], which has a fatal gap in the proof of its main theorem. Though conceptually correct, the proof was impossible to fix within the framework of triangulated categories.

## Cite this article

Rina Anno, Timothy Logvinenko, Spherical DG-functors. J. Eur. Math. Soc. 19 (2017), no. 9, pp. 2577–2656

DOI 10.4171/JEMS/724