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 of D(B) and the co-twist F of D(A). We give powerful 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.
We propose a three dimensional generalization of the geometric McKay correspondence described by Gonzales-Sprinberg and Verdier in dimension two. We work it out in detail when G is abelian and C^3/G has a single isolated singularity. More precisely,
we show that the Bridgeland-King-Reid derived category equivalence induces a natural geometric correspondence between irreducible representations of G and subschemes of the exceptional set of G-Hilb (C^3). This correspondence appears to be related to Reids recipe.
For any finite subgroup G in SL3(C), work of Bridgeland-King-Reid constructs an equivalence between the G-equivariant derived category of C^3 and the derived category of the crepant resolution Y = G-Hilb(C^3) of C^3/G. When G is abelian we show that
this equivalence gives a natural correspondence between irreducible representations of G and certain sheaves on exceptional subvarieties of Y, thereby extending the McKay correspondence from two to three dimensions. This categorifies Reids recipe and extends earlier work from [CL09] and [Log10] which dealt only with the case when C^3/G has one isolated singularity.
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) i
n 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.
We show that the adjunction counits of a Fourier-Mukai transform $Phi$ from $D(X_1)$ to $D(X_2)$ arise from maps of the kernels of the corresponding Fourier-Mukai transforms. In a very general setting of proper separable schemes of finite type over a
field we write down these maps of kernels explicitly -- facilitating the computation of the twist (the cone of an adjunction counit) of $Phi$. We also give another description of these maps, better suited to computing cones if the kernel of $Phi$ is a pushforward from a closed subscheme $Z$ of $X_1 times X_2$. Moreover, we show that we can replace the condition of properness of the ambient spaces $X_1$ and $X_2$ by that of $Z$ being proper over them and still have this description apply as is. This can be used, for instance, to compute spherical twists on non-proper varieties directly and in full generality.
We prove two existing conjectures which describe the geometrical McKay correspondence for a finite abelian G in SL3(C) such that C^3/G has a single isolated singularity. We do it by studying the relation between the derived category mechanics of comp
uting a certain Fourier-Mukai transform and a piece of toric combinatorics known as `Reids recipe, effectively providing a categorification of the latter.
