Spherical DG-functors

Rina Anno ; Timothy Logvinenko

Journal of the European Mathematical Society, Tome 19 (2017) no. 9, pp. 2577-2656

Résumé

For two DG-categories A and B we define the notion of a spherical Morita quasi-functor A→B. We construct its associated autoequivalences: the twist T∈AutD(B) and the cotwist F∈AutD(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.

Détail

DOI : 10.4171/jems/724

Classification : 14-XX, 18-XX
Keywords: Algebraic geometry, derived categories, DG-categories, autoequivalences, Fourier–Mukai transforms, spherical functors, braid group actions