Orthogonally spherical objects and spherical fibrations

Abstract

We introduce a relative version of the spherical objects of Seidel and Thomas. Define an object E in the derived category D(Z x X) to be spherical over Z if the corresponding functor from D(Z) to D(X) gives rise to autoequivalences of D(Z) and D(X) in a certain natural way. Most known examples come from subschemes of X fibred over Z. This categorifies to the notion of an object of D(Z x X) orthogonal over Z. We prove that such an object is spherical over Z if and only if it has certain cohomological properties similar to those in the original definition of a spherical object. We then interpret this geometrically in the case when our objects are actual flat fibrations in X over Z.

29 pages; v2: A missing assumption reinstated in Prop. 3.7, some notation cleaned up. The final version to appear in Adv. in Math

References in corpus (4)

• Noncommutative Counterparts of the Springer Resolution
• On adjunctions for Fourier-Mukai transforms
• Stability conditions on K3 surfaces
• Autoequivalences of toric surfaces

Cited by in corpus (1)

• ${\mathbb{P}}^n$-functors