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 ∈ Aut D(B) and the co-twist F ∈ Aut D(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.

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 in abelian case. 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-HilbðC 3 Þ. This correspondence appears to be related to Reid's recipe. IntroductionThe study of the McKay correspondence began with an observation by John McKay in [11] that there exists a bijective correspondence between irreducible representations of a finite subgroup G L SL 2 ðCÞ and exceptional divisors of the minimal resolution Y of C 2 =G. Gonzales-Sprinberg and Verdier in [6] gave a geometric construction of this correspondence. The aim of this paper is to give a generalization of this construction for dimension three. Our approach is via the derived McKay equivalence of [1] and for G abelian it appears to give a categorification of 'Reid's recipe' from [13].The original construction of [6] is as follows. Denote by K G ðC 2 Þ the Grothendieck ring of G-equivariant coherent sheaves on C 2 and by KðY Þ the Grothendieck ring of Y . Let M A CohðY Â C 2 Þ be the structure sheaf of the reduced fiber product Y Â C 2 =G C 2 . Define the transform Y : K G ðC 2 Þ ! KðY Þ bywhere p Y and p C 2 are the projections from Y Â C 2 to Y and C 2 .Denote by ExcðY Þ the set of irreducible exceptional divisors on Y . Let r be an irreducible representation of G. Since M is flat over Y its pushforward to Y is a vector bundle and L r :¼ YðO C 2 n r 4 Þ is its r-eigensheaf. In general L r is a vector bundle of dimension dimðrÞ and is called a tautological or GSp-V sheaf. It is proven in [6] that: Brought to you by | Lund University Libraries Authenticated Download Date | 5/25/15 6:09 PM Brought to you by | Lund University Libraries Authenticated Download Date | 5/25/15 6:09 PM2. The inverse transform and the dual family 2.1. Notation. In this section, let G be an arbitrary finite subgroup of SL n ðCÞ and Y a smooth n-dimensional separable scheme of finite type over C. We equip Y with the trivial G-action. Then G acts naturally on Y Â C n and we can consider the bounded derived category of G-equivariant coherent sheaves D G ðY Â C n Þ.We denote by V giv the representation of G induced by its inclusion into SL n ðCÞ and by R the symmetric algebra SðV 4 giv Þ. We identify C n with the a‰ne G-scheme Spec R. We also call a G-equivariant sheaf a G-sheaf for short (cf.[1], Section 4).

We show that the adjunction counits of a Fourier-Mukai transform Φ : D(X 1 ) → 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 Φ. We also give another description of these maps, better suited to computing cones if the kernel of Φ is a pushforward from a closed subscheme Z ⊂ X 1 × 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.On the other hand, the identity functor Id is the Fourier-Mukai transform D(X 1 ) → D(X 1 ) with kernelConsider now the left adjunction counitIn general, morphisms between Fourier-Mukai kernels map neither injectively nor surjectively to natural transformations between the Fourier-Mukai transforms. Thus there is no a priori reason for (1.4) to come 1 arXiv:1004.3052v3 [math.AG]

For any finite subgroup G ⊂ 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 Reid's recipe and extends earlier work from [CL09] and [Log10] which dealt only with the case when C 3 /G has one isolated singularity. ∼

Abstract. We introduce a relative version of the spherical objects of Seidel and Thomas [ST01]. Define an object E in the derived category D(Z × 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) orthogonal over Z. We prove that such an object is spherical over Z if and only if it possesses 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.

Given a differentially graded (DG)-category ${{\mathcal{A}}}$, we introduce the bar category of modules ${\overline{\textbf{{Mod}}}-{\mathcal{A}}}$. It is a DG enhancement of the derived category $D({{\mathcal{A}}})$ of ${{\mathcal{A}}}$, which is isomorphic to the category of DG ${{\mathcal{A}}}$-modules with ${A_{\infty }}$-morphisms between them. However, it is defined intrinsically in the language of DG categories and requires no complex machinery or sign conventions of ${A_{\infty }}$-categories. We define for these bar categories Tensor and Hom bifunctors, dualisation functors, and a convolution of twisted complexes. The intended application is to working with DG-bimodules as enhancements of exact functors between triangulated categories. As a demonstration, we develop a homotopy adjunction theory for tensor functors between derived categories of DG categories. It allows us to show in an enhanced setting that given a functor $F$ with left and right adjoints $L$ and $R$, the functorial complex $FR \xrightarrow{F{\operatorname{act}}{R}} FRFR \xrightarrow{FR{\operatorname{tr}} - {\operatorname{tr}}{FR}} FR \xrightarrow{{\operatorname{tr}}} {\operatorname{Id}}$ lifts to a canonical twisted complex whose convolution is the square of the spherical twist of $F$. We then write down four induced functorial Postnikov systems computing this convolution.

We prove two conjectures from Cautis and Logvinenko (2009) [CL09] which describe the geometrical McKay correspondence for a finite, abelian subgroup of SL 3 (C). We do it by studying the relation between the derived category mechanics of computing a certain Fourier-Mukai transform and a piece of toric combinatorics known as 'Reid's recipe', effectively providing a categorification of the latter.

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