Habilthesis.pdf

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Derived Categories

in Algebraic Geometry

Andreas Krug

Habilitation Thesis

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Introduction

Derived categories were rst introduced as the adequate framework for derived functors in homological algebra. Over the last few decades the (bounded) derived category D(X) :=

Db(Coh(X)) of coherent sheaves on a variety or, more generally, an orbifold X was also

recognised as an interesting geometric invariant which controls many other invariants such as cohomology and K-theory. One of the reasons for the recent interest in this category is a conjectured relationship to theoretical physics, more concretely to string theory, via the homological mirror symmetry; see [Kon95]. The derived category is also widely considered as the correct bridge between commutative and non-commutative algebraic geometry. From this point of view, the `spaces' of non-commutative geometry are triangulated categories and the image of a variety X in the non-commutative world is its derived category D(X). The principal problems in the theory of derived categories may be summarised as follows.

P1 Determine under which circumstances two smooth varieties or orbifoldsX and X0 have

equivalent derived categories D(X)∼=D(X0) (FourierMukai partners). P2 Describe the symmetries of the derived category (groups of autoequivalences).

P3 Decompose the derived category into smaller pieces (semi-orthogonal decompositions). P4 Lift interesting structures of other invariants, such as cohomology and K-theory, to the

level of the derived categories (categorication).

One should probably add the study of stability conditions, which has become very popular recently, as a fth central problem. However, stability conditions only play a very minor role in this thesis. Hence, in this introduction, we will focus on the ProblemsP1, P2, P3, and P4.

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One main topic of this Habilitation thesis is the study of the derived category of Hilbert schemes of points on surfaces and generalised Kummer varieties. They are very interesting series of higher dimensional smooth varieties with various connections to combinatorics, rep-resentation theory, and theoretical physics; see [Nak99; Göt02] for an overview of some of these aspects. Their geometry is closely connected to that of the underlying surfaces. Hence, one can use the various results known for surfaces to study the derived categories of these higher dimensional varieties. The hope is that the results obtained for the derived categories of these special series allow to make reasonable conjectures, and ultimately prove them, con-cerning questionsP1,P2, P3, andP4 in higher dimensions. In particular, Hilbert schemes of points on K3 surfaces and generalised Kummer varieties form a large part of the known examples of compact hyperkähler manifolds. Hence, one can view many results of this thesis as a contribution to the study of the derived category of hyperkähler manifolds.

The second, closely related, main theme of this thesis is the McKay correspondence. The derived McKay correspondence is a categorical version of the general philosophy, inspired from physics, that for a nite groupGacting on a smooth varietyM, the geometry of the orbifold

[M/G]and a crepant resolutionY should coincide in an appropriate sense.

InSection 0.1of this introduction, we explain some of the background and make a serious attempt to give an overview of related work of other authors. InSection 0.2, we summarise the results of the author's research papers which constitute this thesis.

Acknowledgements. The author thanks Ciaran Meachan, David Ploog, and Sönke Rol-lenske for useful comments on this introduction. Further acknowledgements are given at the beginning of the chapters, each of which consists of one research paper of the author.

0.1 Background, history, and state of the art

0.1.1 Derived functors and relative Duality

When studying a variety X, one is often interested in vector bundles over this variety (the

same holds for other geometric objects such as manifolds). The category VB(X) of vector bundles, however, does not have very satisfactory properties. While morphisms of varieties induce pull-back functors on the level of vector bundles, they do not induce push-forwards. Furthermore, the category VB(X) is not abelian, which means that kernels and cokernels do not exist in general. The solution of both problems is to pass to the bigger categoryCoh(X) of coherent sheaves on X. One can think of a coherent sheaf as a vector bundle where the

bre dimension is allowed to jump. The category Coh(X) is abelian and, more or less, all geometric functors that one wants to dene can be dened. In particular, for f:X → Y a

proper morphism, there is the push-forwardf∗: Coh(X)→Coh(Y). However, these functors do not preserve short exact sequences in general and this is the reason that one often has to consider derived functors.

Derived categories were introduced by Grothendieck and Verdier in the early 1960s as the adequate framework for these derived functors. The derived category of an abelian category A is dened as D(A) := Kom(A)[qis−1], the category of complexes localised by the class of

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original category A is embedded into the derived category by considering objects in A as complexes concentrated in degree zero. The derived category D(A) is not abelian anymore but a triangulated category instead. This means that there is the autoequivalence[1] :D(A)→

D(A), given by the degree shift of complexes. Furthermore, there is a class of exact (or distinguished) triangles behaving similar to the class of short exact sequences in an abelian category. In particular, under the embedding A → D(A), short exact sequences become distinguished triangles.

When we speak of the derived category of a varietyX (we always work over the eldC),

we usually meanD(X) :=Db(Coh(X))⊂D(Coh(X)), the full subcategory of complexes with bounded cohomology inside the derived category of coherent sheaves. Up to some details, all the geometric functors yield derived functors on the level of the derived categories. For example, forf:X →Y a proper morphism, there is the derived push-forward Rf∗:D(X)→

D(Y) which is an exact functor of triangulated categories, which means that it commutes with the shift functor and preserves exact triangles. Furthermore, it recovers the higher direct image functors as, for every sheaf F ∈ Coh(X) ⊂ D(X) and every i ∈ _Z, we have

Hi₍_Rf

∗F) ∼= Rif∗(F). However, the functor Rf∗ contains more information than all of the functors Rif∗ together. One very convenient consequence is that, given a second proper morphismg:Y →Z, the information of the Leray spectral sequence

E_p,q2 =Rpg∗(Rqf∗F) =⇒ Fp+q =Rp+q(g◦f)∗F can be summarised by the simple formulaRg∗◦Rf∗∼=R(g◦f)∗.

Note that, for a proper morphismf:X→Y, the non-derived push-forwardf∗: Coh(X)→

Coh(Y) does not have a right adjoint in general (if it does,f∗ is already exact). A very deep result, known as Grothendieck duality or GrothendieckVerdier duality, is that this changes if we pass to the derived functors. If X and Y are smooth (which is the case for most of

the varieties that we will consider) the right adjoint is given byf!=Lf∗( )⊗ωf where the

relative canonical bundle isωf =ωX ⊗f∗ωY∨. If we take Y =pt to be the point, the derived

push-forward can be identied with the sheaf cohomology functorH∗(X, )and Grothendieck duality specialises to Serre duality.

The behaviour of smooth projective varieties regarding questions P1, P2, P3 depends strongly on the positivity of the canonical bundle. The reason behind this is that Serre duality is an intrinsic feature of the derived category. Indeed, Serre duality can be reformulated by saying that the Serre functor SX = ( )⊗ωX[dimX] :D(X)→D(X) fulls the property

HomD(X)(E, F)∼=Hom(F,SX(E))∨ for every E, F ∈D(X).

Furthermore, the Serre functor is uniquely determined by this property due to the Yoneda Lemma. Hence, one can say that the derived categoryD(X) sees the canonical bundle ofX.

Problems P1 and P2 are completely solved for ample or anti-ample canonical bundles but tend to become very complicated and deep for varieties with trivial (or close to trivial) canonical bundle. ProblemP3has a simple answer forωX =OX: there are no semi-orthogonal

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0.1.2 FourierMukai partners P1

Originally, the derived category was mainly seen as a tool which allows to deal with derived functors in a natural way and which is necessary for the formulation of relative Serre duality. Starting with the work of Mukai [Muk81] on derived duality for abelian varieties, the derived category began to be recognised as an interesting geometric invariant. Concretely, he proved that, for every abelian variety A, there is an exact equivalence of categories D(A) ∼= D(Ab)

whereAb=Pic0Ais the dual abelian variety. To appreciate this result, one should compare it

to a theorem of Gabriel [Gab62] which says that the category of coherent sheaves determines a variety up to isomorphism:

Coh(X)∼=Coh(Y) ⇐⇒ X ∼=Y .

In contrast, note that A 6∼= Abin general. So it happens that the derived category identies

non-isomorphic varieties but, in all known cases, these varieties, which are then called Fourier Mukai partners (sometimes FM partners for short), are still strongly related. In Mukai's example, the FourierMukai partners are moduli-spaces of line bundles on each other. Fur-thermore, the equivalence Φ :D(A)−∼=→D(Ab) is realised by the universal family, the Poincaré

line bundle P ∈Pic(A×Ab). Namely, we have

Φ =p∗(q∗( )⊗ P)

where p:A×Ab→ Ab and q: A×Ab → A are the projections. Note that, from now on, all

functors are implicitly understood to mean their derived version. This means that p∗ stands for the derived push-forwardRp∗:D(A×Ab)→D(Ab),q∗ stands for Lq∗:D(A)→D(A×Ab),

and so on.

In some regard, the picture that FourierMukai partners are moduli spaces of objects on each other remains true in general due to the following theorem of Orlov.

Theorem 0.1.1 ([Orl03]). Every equivalenceΦ : D(X)−→∼= D(Y) between derived categories of smooth projective varieties is a FourierMukai transform, which means that

Φ =FMQ:=p∗(q∗( )⊗ Q) for someQ ∈D(X×Y).

Thus, we can regard Y as the parameter space of the bres Q_|_X_×{_y_} ∈ D(X) which are objects over X and, conversely, X as the parameter space of the bres Q_|{_x_}×_Y ∈ D(Y). Furthermore, Orlov's result guarantees that derived equivalences induce isomorphisms on the level of cohomology and K-theory. In other words, the derived category controls these other invariants.

FourierMukai partners with (anti-)ample canonical bundle

In general, it is a hard problem to nd the FourierMukai partners of a given smooth projective varietyX. However, for extremal values of the positivity of the canonical bundles, the derived

category still determines the variety.

Theorem 0.1.2 ([BO01]). Let X, Y be smooth projective varieties and let ωX or ω_X−1 be

ample. Then:

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While the actual proof of that theorem is non-trivial, the reason behind its truth is easily summarised. As explained above, the derived category D(X) sees the canonical bundle via its Serre functor. Furthermore, under the assumption that ωX is ample, the variety X is

determined by the canonical ring R(ωX) = ⊕k≥0H0(ω_X⊗k) (and by its anti-canonical ring if

ω−_X1 is ample).

Derived equivalence of abelian varieties

The second class of varieties where a complete description of FourierMukai partners is known are abelian varieties, mainly due to work of Orlov [Orl02]. It turns out that, besides its dualAb,

an abelian variety can have further FourierMukai partners, all of which are abelian varieties too [HN11, Prop. 3.1], and Orlov gives a precise criterion for two abelian varieties to be derived equivalent.

Derived equivalence in low dimension

In the case of elliptic curves, Orlov's criterion for derived equivalence of abelian varieties shows that there are no non-trivial FM partners (note that every elliptic curves is principally polarised which means thatE ∼= ˆE). Together withTheorem 0.1.2this shows that, as mentioned above,

there are no non-trivial FM partners for smooth projective curves at all.

For smooth projective surfaces there are examples of non-trivial FM partners but only for abelian surfaces, K3 surfaces and elliptic surfaces; see [BM01]. This conrms the picture that problemP1is the most interesting for ωX trivial (abelian and K3 surfaces) or fairly close to

trivial (trivial along the bres of an elliptic bration).

In dimension 3, most of the literature concerning problem P1 concentrates on the case of CalabiYau varieties; see for example [C l07], [BC09], [CT16], [HT16]. The main rea-son for the strong interest in this special case is Kontsevich's homological mirror symmetry [Kon95]. Superstring theory only works well if space-time is 10-dimensional. Hence, many string-theorists assume that there is a 3-dimensional (6-dimensional over the real numbers) CalabiYau manifold hidden in every point of ordinary space-time. Drastically oversimpli-ed, Kontsevich's homological mirror symmetry conjecture says that the resulting physical theory only depends on the derived category of the CalabiYau 3-fold, not on the choice of the manifold itself.

Derived equivalences of varieties with trivial canonical bundle

The problem to determine all FM partners of a given smooth projective variety seems to be particularly rich for varieties with trivial canonical bundle. It follows easily from the fact that Serre duality is an intrinsic data of the derived category, that the class of varieties with trivial canonical bundle is stable under derived equivalences: IfD(X)∼=D(Y)holds andωX is trivial,

so isωY.

The three basic subclasses of the class of varieties with trivial canonical bundle are abelian varieties, strict CalabiYau varieties (varieties with trivial canonical bundle and the property that Hi(X,OX) = 0 for i 6= 0,dimX), and hyperkähler varieties, also known as irreducible

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[HN11], the classes of abelian and hyperkähler varieties are stable under derived equivalences. The same is conjectured to hold for strict CalabiYau varieties. This is known to hold in dimension up to four; see [LP15], [Abu15].

0.1.3 Autoequivalences P2

LetXbe a smooth projective variety. We consider the groupAut(D(X))of isomorphism classes of exact autoequivalences of the triangulated category D(X). Concerning Problem P2, the determination of the groupAut(D(X)), there are only very few complete results. To the best of the authors knowledge, the only classes of smooth projective varieties whereAut(D(X))is completely described are those with (anti-)ample canonical bundle [BO01], abelian varieties [Orl02], surfaces of general type whose canonical model has onlyAn-singularities [IU05], toric

surfaces [BP14], some elliptic surfaces [Ueh16], and K3 surfaces of Picard rank 1 [BB17a]. In some other cases, there are at least concrete conjectures predicting how Aut(D(X)) could look like, but in most cases the emphasis of the current research is on constructing new autoequivalences. The most successful construction method is in using twists along certain types of objects or functors as we will explain later.

Standard autoequivalences

There are always two types of autoequivalences which come from autoequivalences of the abelian category Coh(X): Push-forwards (and pull-backs) along automorphisms of X and

tensor productsML := ( )⊗L by line bundles on X. In addition, there is always the shift

autoequivalence[1]along with its powers[m]form∈_Z. The subgroup ofAut(D(X))spanned by these three types of autoequivalences is called the group of standard autoequivalences and denoted by

Autst(D(X)) :=

[1],{ϕ∗}ϕ∈AutX,{ML}L∈PicX

∼

=Z× AutXnPicX).

Theorem 0.1.3 ([BO01]). For ωX ample or anti-ample, the standard autoequivalences are

actually the only autoequivalences of the derived category:

Aut(D(X)) =Autst(D(X)).

In general, however, there are non-standard autoequivalences of the derived category. For example, if A is a principally polarized abelian surface, i.e.A∼=Ab, the FourierMukai

trans-form along the Poincaré bundle FMP ∈Aut(D(A)) sends skyscraper sheaves of points to line bundles and hence cannot be standard.

Spherical objects and twists

An important source of non-standard autoequivalences are twists along spherical objects as introduced by Seidel and Thomas [ST01]. For E, F ∈ D(X), we consider the graded Hom-space

Hom∗(E, F) :=M

i∈Z

Homi(E, F)[−i] where Homi(E, F) :=HomD(X)(E, F[i]).

An objectE∈D(X)is called spherical ifE⊗ωX ∼=E and Hom∗(E, F)∼=C[0]⊕C[−dimX].

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explains the notion. Using the triangulated structure of D(X ×X), one can associate an autoequivalence TE ∈ Aut(D(X)) to every spherical object, the spherical twist. Concretely, TE :=FMQ whereQ ∈D(X×X) is the unique object tting into an exact triangle

EE∨ −→ Oev ∆→ Q →EE∨[1].

The twist autoequivalence satisesTE(E) =E[1−dimX]and acts as the identity on the right

orthogonal complement

E⊥:={F ∈D(X)|Hom∗(E, F)6= 0}.

FordimX > 1 it follows immediately that TE is non-standard (at least if E⊥ is non-empty,

which holds for all known examples but is an open question in general) since a standard autoequivalence could not shift two objects by dierent values.

The motivation for spherical objects and twists came from homological mirror symmetry. Namely, spherical twists are expected to correspond to Dehn twists around Lagrangian spheres on the symplectic side of mirror symmetry.

Let X be a strict CalabiYau variety, which means that ωX is trivial and Hi(OX) = 0

for i 6= 0,dimX. Then every line bundle on X, in particular OX, is a spherical object. We

have TOX = FMI∆[1−dimX]. A second standard example is OC ∈ D(X) for P

1 ∼₌ _C _⊂ _X _a (−2)-curve on a surface.

Autoequivalences on surfaces

For some, but not many, classes of surfaces, there is a complete description of the group of derived autoequivalences. The results of [IU05], [BP14], and [BB17a] can be summarised by saying that for surfaces of general type whose canonical model has onlyAn-singularities,

toric surfaces, and K3 surfaces of Picard rank 1, the groupAut(D(X))is spanned by standard autoequivalences and twist along spherical objects. For the elliptic surfaces studied in [Ueh16], essential the only autoequivalence that has to be added is the FM transform along the relative Poincaré bundle of the bration. In all the cases, there are often braid relations between the generators ofD(X). We will discuss the more general principle behind this inSubsection 0.1.7. For arbitrary K3 surfacesX, there is at least a concrete conjecture of Bridgeland describing

how the groupAut(D(X))looks like. The exact formulation of Bridgeland's conjecture involves the space of stability conditions which is beyond the scope of this introduction. However, the main concrete consequence of the conjecture for the group of autoequivalences is that

Aut(D(X))should again be generated by standard autoequivalences and twists along spherical objects.

Generalisations of spherical objects

Spherical objects and twists have been generalised into various directions. Huybrechts and Thomas [HT06] introduced the notion ofP-objects, mainly in order to construct non-standard

autoequivalences on hyperkähler manifolds. An object E ∈D(X) of the derived category of a smooth projective variety is a Pn-object if E ⊗ωX ∼= E and there is an isomorphism of C-algebras

Hom∗(E, E)∼=H∗(Pn_C,C)∼=C[t]/tn+1 withdegt= 2.

Again, there is an associatedP-twist PE ∈Aut(D(X)). It satisesPE(E)∼=E[2n]andPE|E⊥∼=

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and structure sheaves of centres of Mukai ops, i.e. O_Z ∈ D(X) where dimX = 2n and

Pn∼=Z ⊂X withNZ/X ∼= ΩPn.

Another very important generalisation of spherical objects is the notion of spherical func-tors; see [Rou06], [AL13]. It was inspired by and subsumes the EZ spherical objects of Horja [Hor05] and the fat spherical objects of Toda [Tod07]. Let F = FMQ: D(Z) → D(X) be a FourierMukai transform between the derived categories of smooth projective varieties. It fol-lows by Grothendieck duality thatF has left and right adjointsFL, FR:D(X)→D(Z). Let

η: id → FRF be the unit of adjunction. The rst condition on F to be a spherical functor

is that the endofunctor C:D(Z) → D(Z), called the cotwist of F, that ts into the exact

triangle of functors

id−→η FRF →C →[1] (1) is an autoequivalence. Note that, in general, exact triangles are not functorial. However, in our case all the functors involved are FourierMukai transforms. Hence it makes sense to talk about exact triangles of functors using the triangulated structure ofD(Z ×X), the category where the corresponding FM kernels live. One can still speak about spherical functors in the larger context of dg-enhanced triangulated categories, but this is not necessary for the present thesis. The second condition for F to be a spherical functor is that there is an equivalence FR ∼=CFL relating the right and left adjoints, which can be reformulated in terms of Serre

functors as the condition SXF C ∼= FSZ. For Z = pt a point, the FourierMukai kernel

Q ∈ D(pt×X) ∼= D(X) is a spherical object if and only if F = FMQ:D(pt) → D(X) is a spherical functor. One can think of spherical functors as relative spherical objects which is nicely illustrated by one of the main examples: the pull back along a CalabiYau bration. Analogously to the case of spherical objects, there is a spherical twistTF ∈Aut(D(X))dened

by the exact triangle

F FR ε−→id→TF →F FR[1]

whereεis the counit of adjunction. In many examples of spherical functors, the triangle (1)

splits so that we haveFR_F ∼₌_id_⊕_C_{. In this case we call}_F _{split spherical.}

Segal [Seg16] proved that every autoequivalence Φ ∈ Aut(D(X)) can be realised as a spherical twists Φ = TF along a spherical functor F: C → D(X) where C is a suitable

dg-enhanced triangulated category. However, the category C is usually more complicated than

D(X) itself. Accordingly, in practice, one cannot hope to nd the spherical functor F: C →

D(X) before knowing the autoequivalence Φ. Hence, it is good to have other construction methods of autoequivalences, besides spherical functors, available.

One of the most recent developments in the construction of non-standard autoequivalences is the introduction of P-functors which is a simultaneous generalisation of P-objects and of

split spherical functors due to Addington [Add16] (there is also a similar denition by Cautis [Cau12]). A FourierMukai transform F:D(Z) →D(X) is called a Pn-functor if there exists

an autoequivalenceD∈Aut(D(Z)), called theP-cotwist ofF, such that

FRF ∼=id⊕D⊕D2⊕ · · · ⊕Dn,

we have FR ∼₌ _Dn_FL_{, or equivalently} _S

XF Dn ∼= FSZ, and the monad structure of FRF

fulls a certain condition whose technical formulation we omit. ForZ =ptandD= [−2], the denition of aPn-functor specialises to that of a Pn-object, and the condition on the monad

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spaces but as algebras. Yet again, there is an associatedP-twist PF ∈Aut(D(X))satisfying PF F ∼=F Hn+1[2], PF|(imF)⊥ ∼=id where(imF)⊥= ker(FR) ={A∈D(X)|FR(A) = 0}.

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Note that aP1-functor is the same as a split spherical functor. In this case, theP-twist is the

square of the spherical twist: PF ∼=T2F.

Derived categories of hyperkähler manifolds

Many of the applications ofP-functors involve hyperkähler manifolds. The main example of a Pn−1-functor in [Add16] is the FourierMukai transformF =FMIΞ:D(X) → D(X

[n]₎ _along

the ideal sheaf of the universal familyΞ⊂X×X[n]of lengthnsubschemes, forXa K3 surface

and n ≥ 2. Similarly, for A an abelian surface, the FM transform along the universal ideal

sheaf of the generalised Kummer variety gives a Pn−1-functorD(A)→D(KnA)for n≥2; see

[Mea15].

This ts into the following more general picture: All known examples of compact hyper-kähler manifolds are deformation equivalent to (resolutions of) moduli spaces of sheaves on K3 or abelian surfaces with the Hilbert schemes of points and the generalised Kummer vari-eties being the most prominent representatives of the deformation equivalence classes. It is conjectured (or, at least, considered as an interesting working hypothesis) that every compact hyperkähler manifold can be realised as a moduli space of objects in a 2-dimensional Calabi Yau category (CY2 category), i.e. a triangulated category with the same formal properties as a K3 or abelian surfaces (technically, this means that shift by 2 is the Serre functor of the category). This category can then be viewed as a non-commutative K3 or abelian sur-faces. Furthermore, it is conjectured that the CY2 category and the associated2n-dimensional

hyperkähler are connected by aPn−1-functor withP-cotwistD= [−2].

A case where this works beautifully is that of the variety F(Y) of lines on a smooth cubic fourfold Y ⊂ _P5_{. The variety} _F₍_Y₎ _{is a hyperkähler fourfold deformation equivalent}

to the Hilbert scheme of points on a K3 surface but, in general, not isomorphic to a Hilbert scheme; see [BD85]. However, due to Kuznetsov [Kuz10], there is always a K3 category AY

(a CY2 category with the same Hochschild homology as a K3 surface) contained in D(Y) which is conjectured to be strongly related to the question whether or not Y is rational.

Addington [Add16, Sect. 5] proved that the FourierMukai transformD(Y)→D(F(Y))along the universal family of lines restricts to a split spherical functorAY →D(F(Y))with cotwist

[−2]. Further partial results in the direction of the above conjecture are given by Markman and Mehrotra [MM15], Kuznetsov [Kuz15], and Addington, Donovan, and Meachan [ADM16].

Equivalences induce autoequivalences

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twists introduced above. This questions is particularly interesting in the context of birational geometry; seeSubsection 0.1.5 below.

0.1.4 Semi-orthogonal decompositions and exceptional sequences P3

It is a very common approach throughout various mathematical theories to study the objects of interest by decomposing them until one reaches irreducible pieces. Our objects of interests are the triangulated categories D(X), usually for X a smooth projective variety. There is

indeed a well-dened notion of a direct sum of triangulated categories. However, this only reects the decomposition ofXinto connected components. More precisely, ifD(X) =T1⊕ T2

for triangulated categoriesT₁,T₂, we already haveT_i =D(Xi) withX=X1 `

X2.

It turns out that a much more interesting notion is that of a semi-orthogonal decomposition

D(X) = hT1,T2i. We omit the technical details of the denition, but the main dierence is

that we only requireHom(T₂,T₁)to vanish while, in contrast, for a direct sum decomposition

D(X) = T₁ ⊕ T₂ we also have Hom(T₁,T₂) = 0. However, the notion of a semi-orthogonal decomposition still guarantees that for every objectA∈D(X) there is a unique triangle

A2→A→A1 →A2[1]

withAi ∈ Ti, which makes this into a reasonable notion of a decomposition. In particular, the

two factors of the semi-orthogonal decomposition D(X) = hT1,T2i determine each other as

T1 =T2⊥ andT2 =⊥T1. Of course, there are possibly further semi-orthogonal decompositions

of the triangulated subcategoriesT_i giving rise to semi-orthogonal decompositions ofD(X) of higher length.

Every fully faithful embedding D(Z) ,→ D(X) of derived categories of smooth projective varieties induces a semi-orthogonal decomposition D(X) = hD(Z),T₂i due to a generalised version ofTheorem 0.1.1. ForZ =pta point such an embedding corresponds to an exceptional

object E ∈ D(X) which means that Hom∗(E, E) = C[0]. Many fully faithful embeddings D(Z),→D(X)are constructed by Kuznetsov's homological projective duality [Kuz07]. Irreducible derived categories and the canonical bundle

An example of a triangulated category which is irreducible with regard to the notion of semi-orthogonal decomposition is D(X) for X a smooth projective variety with trivial canonical

bundle. Indeed, Serre duality implies thatHom(T1,T2) = 0 if and only ifHom(T2,T1) = 0 for

triangulated subcategoriesT_i ⊂D(X). Hence, in this case a semi-orthogonal decomposition would already give a direct sum decomposition which cannot happen for X connected. A

more general criterion for irreducibility of D(X) in terms of ωX is given by Kawtani and

Okawa [KO15]. In particular, there are no semi-orthogonal decompositions of D(X) if the base-point locus of ωX is zero-dimensional. It follows that the only smooth projective curve

with a non-trivial semi-orthogonal decomposition is P1 (which was already proven earlier by

Okawa [Oka11]). Furthermore, it rules out semi-orthogonal decompositions on many types of minimal surfaces, such as bielliptic surfaces.

Derived categories of Fano varieties

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ample), every line bundle is an exceptional object by Kodaira's vanishing theorem, realising

D(pt)as a semi-orthogonal factor ofD(X). The abundance of semi-orthogonal decompositions on Fano varieties lead to the question (originally posed by Bondal, see also [BBF16]) whether every smooth projective variety is a Fano visitor: Given a smooth projective varietyY is there

a Fano variety (or at least a Fano orbifold) X, then called Fano host of Y, together with a

fully faithful embedding D(Y) ,→ D(X)? An armative answer would be a very interesting theoretical result. Probably exaggerating its impact, one could say that this would reduce non-commutative geometry to the study of Fano varieties. More concretely, realising a variety as a Fano visitor can be very helpful for understanding the geometry of Y since one can view

it as a moduli space of objects on its Fano host X. Varieties which are known to be Fano

visitors include complete intersections in projective space [Kie+15] and curves [KL15].

Phantom categories

Another very interesting phenomenon, occurring primarily on varieties of general type, is that of a phantom category; see [BBS13], [Böh+15], [GS13], [Gal+15], [AO13], [Cou15], [Lee15]. They are admissible subcategories (which means factors of a semi-orthogonal decomposition) ofD(X)which are non-zero but their two typical categorical invariants, Hochschild homology and Grothendieck group, vanish. Phantom categories are often related to the deformation theory of surfaces of general type. Furthermore, they seem to play an interesting role in homological mirror symmetry; see [FHK14].

Exceptional sequences

Not surprisingly, a case of particular interest of a semi-orthogonal decomposition D(X) = hT₁, . . . ,T_niis when all the factors of the decomposition are given by the derived category of a point (which is equivalent to the derived category ofC-vector spaces). This case corresponds

to a full exceptional sequence ofD(X). An exceptional sequence consists of exceptional objects

E1, . . . , En∈D(X) satisfying the semi-orthogonality assumptionHom∗(Ei, Ej) = 0for i > j.

The sequence is called full if it generates D(X) in the sense that the only full triangulated subcategory of D(X) containing all the Ei is D(X) itself. We call the sequence E1, . . . , En

strong if it satises that Hom∗(Ei, Ej) is concentrated in degree zero (also for i < j). If

E1, . . . , En ∈ D(X) is a full strong exceptional sequence, one obtains an exact equivalence D(X)∼= D(Mod(A)) to the bounded derived category of nitely generated modules over the ring A := End(E) where E := ⊕n

i=1Ei ∈ D(X); see [Bon89]. The ring A is a path

alge-bra of a quiver with relations whose vertices correspond to the members of the exceptional sequence. This phenomenon gives a very interesting connection between algebraic geometry and representation theory. It is a far reaching generalisation of Belinson's description [Be78] of the derived category of the projective space Pn. Later, Kapranov [Kap88] proved that

Grassmannians, ag manifolds, quadrics, and some other rational homogeneous spaces have full strong exceptional collections. This led to the conjecture that the same holds for every rational homogeneous space; see [Böh06] for some more recent work on this conjecture.

The general belief is that varieties whose derived categories have full exceptional sequences are `strongly rational' in some sense. There are concrete conjectural criteria forD(X)to have a full exceptional collection in terms of the the image ofXin the Grothendieck ring of varieties

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0.1.5 Birational geometry and derived categories

Another intriguing aspect of derived categories are connections, some of them conjectured, some of them proven, to birational geometry. Many of them can be summarised under the ho-mological minimal model program. The basic idea is that running the minimal model program for a variety X should correspond to minimising the derived category D(X). In particular, the derived categoryD(X0)of a minimal model ofX should embed intoD(X). The rst basic

result into this direction is the description of the derived category of a blow-upπ:Xe →X in

a smooth centre due to Orlov [Orl92]. In particular, the pull-backπ∗:D(X)→D(Xe) is fully

faithful.

One concrete expected phenomenon is the DK Hypothesis of Bondal, Orlov, and Kawa-mata; see, for example [Kaw16, Sect. 2]. It concerns a birational correspondence

Z

q

p

X //X0

of smooth varieties. Then we should have:

• A fully faithful embedding Db(X),→Db(X0), if q∗KX ≤p∗KX0.

• A fully faithful embedding Db(X0),→Db(X), if q∗KX ≥p∗KX0.

• An equivalence Db(X0)∼=Db(X), in the op caseq∗KX =p∗KX0. This is proven in many instances; see [BO95], [Bri02], [Kaw02], [Nam03].

Given a equivalenceΦ : D(X)−∼=→D(X0)between varieties related by a op one can consider the op-op autoequivalence ΨΦ∈Aut(D(X)) where Ψ :D(X0)−∼=→ D(X) is the autoequiva-lence with the same FM kernel asΦbut in the reverse direction; compareSection 0.1.3. Often, the centre of the birational morphismq:Z→X gives rise to spherical orP-functors and the

autoequivalenceΨΦcan be described in terms of the associated twists. This is known as the op-op=twist principle and proven in many instances; see [Tod07], [BB15], [DW16], [DW15], [ADM15].

0.1.6 McKay correspondence

Canonical surface singularities are exactly the quotient singularities of the form A2/G for a

nite subgroupG⊂SL(2,C). These kind of singularities, also known as Kleinian singularities,

are in one-to-one correspondence with the ADE Dynkin diagrams. This correspondence is realised by considering the intersection graph of the exceptional (−2)-curves of the minimal resolutionY →A2/G.

McKay [McK80] observed that there is a bijection between the non-trivial irreducible representations of the group G and the irreducible components of the exceptional divisor of

the resolution Y. This bijection relates the decompositions of the tensor products of the

representations into irreducibles to the intersection graph of the exceptional divisor.

Gonzales-Sprinberg and Verdier [GV83] gave a conceptual explanation of this correspon-dence by constructing an isomorphismK(Y) ∼= KG(A2) between the Grothendieck group of

the minimal resolutionY and the Grothendieck group ofG-equivariant vector bundles on A2.

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smooth varietyM, is there a resolutionµ:Y →M/Gof singularities which is minimal in an

appropriate sense and such that the geometry ofY reects the representation theory ofGand

the way the group acts onM?

It turned out that a very promising candidate for such a resolution is the G-Hilbert scheme HilbG(M) which is (an irreducible component of) the moduli space of G-clusters on M; see

[IN96], [IN00]. A G-cluster is a zero-dimensional G-invariant subscheme Z ⊂ M such that

the space of functions O(Z), as a G-representation, equals the regular G-representation C[G].

Every free G-orbit inM together with its reduced subscheme structure is aG-orbit but there

are also non-reduced G-clusters supported on non-free G-orbits. There is indeed a birational

morphismτ: HilbG(M)→M/G, theG-HilbertChow morphism. Let[Z]∈HilbG(M)be the point representing a G-clusterZ⊂M. Thenτ([Z])∈M/Gis the point of the quotient lying

under the G-orbit on whichZ is supported. It follows easily from an inspection of

Gonzales-Sprinberg's and Verdier's proof that the minimal resolution Y →_A2_/G_for _G_⊂_SL(2_, C) can

be identied with HilbG(A2).

Theorem 0.1.4 ([BKR01]). LetGbe a nite group acting on a smooth quasi-projective variety M and letτ: HilbG(M)→M/Gbe the G-HilbertChow morphism. Assume that:

(i) The canonical bundle ωM is locally G-trivial: For every x ∈ M, the isotropy subgroup

Gx ⊂G acts trivially on the breωM(x).

(ii) The bres of τ are not too big: dim HilbG(M)×_M/GHilbG(M)

≤dim(M) + 1.

Then τ: HilbG(M) → M/G is a crepant resolution, which means that HilbG(M) is smooth andτ∗ωM/G ∼=ωHilbG_(M), and

Φ :=p∗q∗:D(HilbG(M))→DG(M) is an equivalence where

HilbG(M)←− Zq −→p M

are the projections from the universal family ofG-clusters Z ⊂HilbG(M)×M.

The equivariant derived category DG(M) on the right-hand side of the derived McKay

equivalenceΦ :D(HilbG(M))∼=DG(M)is dened asDG(M) :=Db(CohG(M))whereCohG(M) is the abelian category ofG-equivariant coherent sheaves. This category can be canonically

identied with the category of coherent sheaves on the quotient stack (or, depending on the preferred notion, quotient orbifold):

CohG(M)∼=Coh([M/G]) , DG(M)∼=D([X/G]).

This shows that we can interpret the derived McKay equivalence as an instance of a stacky/orbifold version of the DK hypothesis by considering the following op diagram of orbifolds

[Z/G]

q

}

p

#

e

Y //

%

!

[X/G]

π

{

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0.1.7 Categorication P4

The derived categoryD(X) controls other invariants of the smooth projective varietyX such

as the Grothendieck group K(X) or the cohomology H∗(X,C). The reason is that, by

The-orem 0.1.1, every autoequivalence is a FourierMukai transform. Hence, one can use its FM kernel in order to induce correspondences on the level of the Grothendieck groups and cohomol-ogy which are isomorphisms. There are also very interesting, though more subtle, connections of D(X) to the image of X in the Grothendieck ring of varieties and its motive; see for

ex-ample [Tab15], [KS16]. In particular, the motives with rational coecients of FourierMukai partners only dier by Tate twists; see [Orl05].

Hence, whenever other invariants such as the Grothendieck group or the cohomology of a variety carry an interesting additional structure, it is a very natural problem to lift this structure in an appropriate way to the level of the derived category. This problem is known as categorication.

One principal idea is that, whenever you see a canonical direct sum decomposition of the cohomology or K-theory, you look for a semi-orthogonal decomposition on the derived category that induces the decomposition. A basic example is the semi-orthogonal decomposition

D(Xe) =hD(X),D(Z), . . . ,D(Z)

| {z }

(c−1)times

i

where Xe is the blow-up of X in the smooth centre Z ⊂X of codimensionc; see [Orl92].

The second principal idea is that, whenever you see the action of a group or an algebra on the cohomology or K-theory, you look for a collection of FourierMukai transforms which descends to the action of a set of generators of the group or algebra (naive categorication). It would be nice if the relations between the generators can already be seen as isomorphisms between compositions of the FM-transforms (weak categorication) and even nicer if these isomorphisms themselves can be organised in an interesting way (strong categorication).

An example is given by anAn-chain of(−2)-curves on a surfaceX. Then reection on the

hyperplanes orthogonal to the classes of these curves dene an action ofS_non the cohomology

H∗(X,C). This lifts to a categorical action of the braid group Bn+1 on D(X) given by the

spherical twists along the structure sheaves of the curves; see [ST01].

0.1.8 Derived categories of Hilbert schemes of points on surfaces

One might have noticed that a large part of the results on derived categories summarised above concern varieties of low dimension. In higher dimensions not much is known besides for abelian varieties and varieties with (anti-)ample canonical bundle. One approach, followed by various authors including myself, is to study Hilbert schemes of points on surfaces, where one can use the broad knowledge concerning derived categories on surfaces. The hope is that, with results for this class of examples as an inspiration, one might be able to make more general statements regarding derived categories of higher-dimensional varieties.

Denition and basic properties

Given a smooth quasi-projective variety X and a number n ∈ _N, the Hilbert scheme (also known as Douady space) of n points on X is dened as the ne moduli space of

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denes the HilbertChow morphism µ:X[n] →X(n) :=Xn/S_n to the symmetric product of the variety. This makesX[n] a resolution of the quotient singularities of X(n). In particular,

the Hilbert schemes of points on surfaces are always smooth of dimensiondimX[n]= 2n; see

[Fog68]. ForX a smooth variety of higher dimension,X[n] is smooth only forn≤3.

As already mentioned inSection 0.1.3, ifXis a K3 surface, the associated Hilbert scheme is

one of the rare series of examples of a compact hyperkähler variety (also known as a irreducible holomorphic symplectic variety). A second series of examples are the generalised Kummer varieties KnAassociated to an abelian variety A. They are the codimension 2subvarieties of

A[n+1]whose points correspond to lengthn+ 1subschemes ofAwhose weighted support adds

up to zero.

The cohomology ofX[n]: Nakajima operators and the Heisenberg action

Göttsche [Göt90] gave a formula for the Betti numbers of the Hilbert schemes, i.e. the di-mensions of the graded pieces of H∗(X[n]_,

Q), in terms of the Betti numbers of the surface

X. It turned out that they coincide with the dimensions of the graded pieces of a certain

irreducible representation, called the Fock space, of the Heisenberg Lie algebrah_V associated to the cohomologyV :=H∗(X,Q) of the surfaceX.

The Lie algebra hV associated to a (graded) vector space V together with a (graded)

symmetric bilinear form h , i is, as a vector space, given by a copy of V for every non-zero

integer together with one extra base vectorc:

h_V :=Q·c⊕ M

n∈_Z\{0}

V .

Forβ ∈V, we denote its copy in then-th direct summandV ofh_V byaβ(n). The Lie bracket

ofh_V is dened by settingc to be central, which means [c, h] = 0for every h∈h_V, and

[aα(n), aβ(m)] :=δm,−nnhα, βi ·c= (

0 if m6=−n

nhα, βi ·c if m=−n. (3)

In the case of the cohomolgyV =H∗(X,Q), the bilinear form one considers is the intersection

pairinghα, βi=R_Xα∩β.

Inspired by Göttsche's formula, Nakajima [Nak97] and Grojnowski [Gro96] gave geometric constructions of an action of h_V on the cohomology space _HX := ⊕_`≥0H∗(X[`],Q). For

Nakajima's construction, the incidence schemes

Z`,n ={(x, ξ, ξ0)|ξ⊂ξ0, ξ and ξ0 only dier inx} ⊂X×X[`]×X[n+`] (4)

play a crucial role.

The Hilbert scheme of points as G-Hilbert schemes

Given a smooth quasi-projective surfaceX, there is a birational morphismX[n]99KHilbSn₍_Xn₎

where Sn acts by permutation of the factors on Xn. It identies the reduced length n

sub-schemes ofX with the freeS_n-orbits on Xn by mapping{x1, . . . , xn} 7→Sn·(x1, . . . , xn). In

his seminal work [Hai01], Haiman extended this to an isomorphism X[n]∼= HilbSn₍_Xn). His

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consequence is that this makes the Hilbert scheme of points accessible to the derived McKay correspondence. Indeed, one can check that the assumptions of Theorem 0.1.4 are satised. Hence, we get an equivalence

Φ :D(X[n])−→∼= DSn(X n₎_.

This equivalence allows to make various interesting constructions for the equivariant derived categoryDSn(X

n₎ _{and translate them into results concerning the derived category}_D₍_X[n]₎_.

Results on D(X[n])

The rst basic step into the direction of the study of problems P1 and P2 for the derived category of Hilbert schemes of points is the following theorem of Ploog.

Theorem 0.1.5 ([Plo07]). Let X, Y be smooth projective surfaces. Then

D(X)∼=D(Y) =⇒ D(X[n])∼=D(Y[n]).

Furthermore, for every smooth projective surfaceX, there is an injective group homomorphism

Aut(D(X)),→Aut(D(X[n])).

The proof crucially uses the derived McKay correspondence D(X[n]) ∼= DSn(X

n₎_{: By}

Orlov's Theorem 0.1.1 every autoequivalence D(X) −∼=→ D(Y) is a FourierMukai transform

FMP for some objectP ∈D(X×Y). One can show that its box power Pn∈DSn(X

n_×_Yn₎

induces an equivalenceDSn(X

n₎₋∼=_→_D

Sn(Y n₎_.

Similarly using the box product, one can producePn-objects onD(X[n]) out of spherical

objects on the surfaceX; see [PS14].

Note that there is no general construction analogous to Theorem 0.1.5 known for gener-alised Kummer varieties. In particular, forA and B abelian surfaces and n≥2 it is an open problem whether or not

D(A)∼=D(B) =?⇒ D(KnA)∼=D(KnB).

However, there is an armative answer forn= 1; see [Hos+03], [Ste07].

In regard of the categorication problem P4 and the results on the cohomology of the Hilbert schemes described in Section 0.1.8, it is desirable to construct a categorical action of the Heisenberg algebra on the derived categories of Hilbert schemes of points on surfaces. This was achieved by Cautis and Licata [CL12] in the case that the surface X is a minimal

resolution of a Kleinian singularity A2/G. This Heisenberg action also induces further

non-standard autoequivalences ofD(X[n]); see [CLS14]. The construction of [CLS14] uses the fact that the structure sheaves of the exceptional curves of the resolutionX→A2/Gare spherical

objects and can be seen as a generalisation of the construction of [PS14].

The derived categories of the Hilbert schemeX[n]and of its underlying surfaceXare always

connected by two canonical functorsF, F00:D(X)→D(X[n]), namely the FM transforms along

the ideal sheafF = FMIΞ and along the structure sheafF

00 ₌_FM

OΞ of the universal family of length nsubschemes Ξ⊂X×X[n].

InSection 0.1.3, we already discussed the fact that, forXa K3 surface,F is aPn−1-functor

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The images of vector bundles under the FM transform F00: D(X) → D(X[n]) are again vector bundles called tautological bundles. Not surprisingly, they play an important role for the geometry of Hilbert schemes of points on surfaces; see [Leh99; LS01; LS03; Sta16]. The compositionΦF00:D(X)→DSn(X

n₎ _{with the derived McKay equivalence was computed by}

Scala [Sca09a].

0.2 Summary of results

This thesis consists of 9 research papers of mine which I wrote between my PhD graduation in August of 2012 and January of 2017. Of these 9 papers, 4 are published in journals with peer-review and a further two are accepted for publication. The remaining 3 are all submitted to journals and available on the arXiv as well as on my personal website https://sites. google.com/site/andkrugmath. Four of the 9 papers are written jointly with coauthors.

Papers constituting this thesis

[1] Andreas Krug and Pawel Sosna. Equivalences of equivariant derived categories. J. Lond. Math. Soc. (2) 92.1 (2015), pp. 1940.

[2] Andreas Krug, David Ploog, and Pawel Sosna. Derived categories of resolutions of cyclic quotient singularities (2017). arXiv:1701.01331.

[3] Andreas Krug and Ciaran Meachan. Spherical functors on the Kummer surface. Nagoya Math. J. 219 (2015), pp. 18.

[4] Andreas Krug. Varieties withP-units (2016). To appear in Trans. Amer. Math.

Soc. arXiv:1604.03537.

[5] Andreas Krug and Pawel Sosna. On the derived category of the Hilbert scheme of points on an Enriques surface. Selecta Math. (N.S.) 21.4 (2015), pp. 13391360. [6] Andreas Krug. On derived autoequivalences of Hilbert schemes and generalized

Kummer varieties. Int. Math. Res. Not. IMRN 20 (2015), pp. 1068010701. [7] Andreas Krug. P-functor versions of the Nakajima operators (2014). arXiv:1405.

1006.

[8] Andreas Krug. Symmetric quotient stacks and Heisenberg actions (2015). To appear in Math. Z. arXiv:1501.07253.

[9] Andreas Krug. Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles (2016). arXiv:1612.04348.

Most of the results can be regarded as contribution towards the goals P1, P2, P3, and P4. A central theme is the McKay correspondence and the two series of varieties which are studied the most intensively are Hilbert schemes of points on surfaces and generalised Kummer varieties.

0.2.1 Equivariant derived functors and a necessary condition for derived McKay equivalences [1]

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every objectP ∈DG×H(X×Y) gives rise to an equivariant FM transformDG(X)→DH(Y).

These equivariant FM transforms are ubiquitous when dealing with the derived McKay corre-spondence and applications. In particular, the derived McKay equivalence itself is an equiv-ariant FM transform.

The rst part of the article [1] is a theoretical treatment of the question when a `usual' FourierMukai transform D(X) → D(Y) lifts to an equivariant one. We give a criterion for such a lift to exists and show that the lift inherits a large number of possible properties of the original FM transform: being an equivalence, fully faithful, spherical, or a P-functor. This

generalises results of the foundational paper [Plo07]. We are able to treat equivariant derived categories and equivariant FM transforms in the above sense (whereGis a nite subgroup of

Aut(X)) in the same way as equivariant derived categories with respect to a nite subgroup G⊂Pic(X) acting onD(X)by tensor products. This allows us to deduce criteria for lift and descend of FM transforms along Galois covers which generalise results of [BM98] and [LP15, Sect. 2].

The second part of the article [1] gives applications of the rst part in the context of the derived McKay correspondence. Probably the most interesting application is a criterion for a global quotient stack[X/G]not to have a smooth projective variety as a FM partner. Recall that a crucial assumption for the derived McKay correspondenceTheorem 0.1.4is thatωM is

locallyG-trivial. We prove the following partial converse

Theorem 0.2.1 ([1, Proposition 1.5.6 & Proposition 1.5.7]). Let M be a smooth projective

variety together with a nite subgroupG⊂Aut(M)such thatωM is not locallyG-trivial. Then

there is no smooth projective varietyY such thatD(Y)∼=DG(X) if one of the following holds: 1. As a non-equivariant bundle, ωM is trivial.

2. As a non-equivariant bundle, ωM is ample or anti-ample (and a mild technical

assump-tion holds).

Since the two cases in which the theorem holds are quite opposite, it makes sense to conjecture in general that[M/G] cannot have a smooth projective variety as an FM partner as soon asωM is not locallyG-trivial. As a corollary ofTheorem 0.2.1, we see that symmetric

quotient stacks of curves[Cn/S_n]never have varieties as FM partners. Further aspects of [1] which are worth being pointed out are:

• A very short proof, in the special case of smooth projective varieties, of Balmer's result [Bal05] that the derived category D(X) together with its tensor structure determines the scheme X; see [1,Subsection 1.3.5].

• There are two variants in the denition of spherical functors in the literature. Sometimes the equivalence FR∼=CFLis required to be of a certain form. We discuss under which

circumstances the a priori stronger condition follows as soon as there is any equivalence

FR∼=CFL; see [1,Remark 1.3.20].

0.2.2 McKay correspondence for cyclic quotients and categorical crepant resolutions [2]

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Indeed, we haveCm,n ∼=An/µm where the cyclic group µm acts on An by multiplication by

a primitive n-th root of unit. The blow-up Ye → C_m,n in the origin is a resolution of the

singularity and its derived category has been studied in [Abu16, Sect. 4], [IU15], and [Kaw16, Ex. 4].

We consider the following generalised set-up. Let µm act on a smooth variety X. We

assume that only1andµmoccur as the isotropy groups of the action and writeS :=Fix(µm)⊂

X for the xed point locus. Furthermore, we assume that µm acts diagonally with only one

eigenvalue on the normal bundle NS/X. Then the blow-upYe →X/µ_m inS is a resolution of

singularities. A particular case of this set-up, for m= 2, is the product X=T2 of a smooth

variety T. In this case, the resolution is given by the Hilbert scheme of two points: Ye ∼=T[2].

First, we prove that the blow-up resolution equals the µm-Hilbert scheme in our set-up: e

Y ∼= Hilbµm(X); see [2, Proposition 1.1].

Setting n := codim(S ,→ X) we obtain the following trichotomy which conrms the prediction of the DK hypothesis (see Subsection 0.1.5) in this case.

Theorem 0.2.2 ([2,Theorem 2.1.2 &Theorem 2.4.1]).

1. For m > n, there is a fully faithful embedding Φ : D(Ye) ,→ Dµm(X) and a

semi-orthogonal decomposition of Dµm(X) consisting ofΦ(D(Ye))and m−npieces equivalent

to D(S).

2. For n > m, there is a fully faithful embedding Ψ : Dµm(X) ,→ D(Ye) and a

semi-orthogonal decomposition of D(Ye) consisting ofΨ(D_µm(X))andn−mpieces equivalent

to D(S).

3. For n=m, we have D(Ye)∼=D_G(X).

We study the case m = n in further detail. The FM transforms D(S) → Dµm(X) and D(S) → D(Ye) which are fully faithful in the cases m > n and n > m, respectively, become

spherical functors for m = n. We are able to describe the tensor products induced by the

equivalence D(Ye)∼=D_G(X) on D(Ye) and D_G(X) in terms of the twists along these spherical

functors. Furthermore, we get a rst example of a stacky 'twist-twist=op' principle; see [2, Subsection 2.4.6]. In [2,Section 2.5], we introduce a general candidate for a categorical crepant resolution of a variety and test this candidate in our set-up of cyclic quotient singularities. In [2, Section 2.6], as an application of Theorem 0.2.2, we construct stability conditions on resolutions of Kummer threefolds.

Further aspects of [1] which are worth being pointed out are:

• We develop a theory of relative tilting objects in [2,Subsection 2.2.7]; compare [BB17b] for a similar approach.

0.2.3 Spherical functors on the Kummer surface [3]

Recall the discussion of Subsection 0.1.3: The FourierMukai transform along the ideal sheaf of the universal family induces a Pn−1-functor F: D(X) → D(X[n]) with cotwist [−2] for

n ≥ 2; see [Add16]. The same holds for the analogue FM transform FK:D(A) → D(KnA)

to the generalised Kummer surface for n≥2; see [Mea15]. Forn= 1 the universal family of

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The n = 1 case becomes much more interesting for the Kummer varieties where KA :=

K1A is the Kummer K3 surface. This is what we study in [3].

Theorem 0.2.3 ([3]). The FM transforms F, F00:D(A) →D(KA) along the ideal sheaf and the structure sheaf of the universal family are both split spherical functors with cotwist (−1)∗ ∈

Aut(D(A)).

This seems very surprising in light of Bridgeland's conjecture: For a K3 surface the group of derived autoequivalences should be generated by standard autoequivalences and twists along spherical objects (not functors); see Subsection 0.1.3. However, we are able to bring Theorem 0.2.3in accordance with Bridgeland's conjecture by also proving

Theorem 0.2.4 ([3]). Both spherical twistsTF, TF00∈Aut(D(KA))are compositions of twists along the structure sheaves of the 16 exceptional(−2)-curves (which are spherical objects) on the Kummer surface and tensor products by line bundles.

• It is conjectured that every spherical object in the derived category of a smooth projective variety has a non-empty orthogonal complement. In contrast, the images of our spherical functors have an empty orthogonal complement; see [3,Corollary 3.2.6].

0.2.4 Varieties with P-units [4]

As mentioned inSection 0.1.3, there are conjectured deep relations between P-functors and

hyperkähler varieties. The starting point of [4] is the observation that, for X a smooth

projective variety of dimension2n, we have

X is hyperkähler ⇐⇒ OX ∈D(X) is aPn-object.

This follows quite immediately from [HN11, Prop. A.1] but we give a dierent proof in [4, Subsection 4.3.3]. Furthermore, almost by denition, a smooth projective varietyXis a strict

CalabiYau variety if and only ifO_X ∈D(X) is a spherical object. One way to rephrase these two observations without referring to derived categories is: Let X be a variety with trivial

canonical bundle. Then

X is a strict CalabiYau of dimension k ⇐⇒ H∗(O_X)=∼C[x]/x2, degx=k ,

X is hyperkähler of dimension 2n ⇐⇒ H∗(O_X)=∼C[x]/xn+1, degx= 2.

In regard of this, it seems natural to consider, as a common generalisation of strict Calabi Yau and hyperkähler varieties, the class of varieties with trivial canonical bundle and the property that the algebra H∗(O_X) is generated by one element. IfX is a variety with ωX ∼=

OX and H∗(OX) ∼= C[x]/xn+1 where degx = k, we say that X has a Pn[k]-unit. Indeed,

this is equivalent to the condition that OX ∈ D(X) is a Pn[k]-object, which means that FMOX:D(pt) → D(X) is a P

n_{-functor with cotwist} _[₋_k_]_{. Using these notions, the classical} P-objects of Huybrechts and Thomas [HT06] become P[2]-objects and hyperkähler varieties

become varieties with P[2]-units. Furthermore, strict CalabiYau varieties of dimension k

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This raises the basic question for which values of n > 2 and k > 2 varieties with Pn[k]

-units exist. The rst simple observation is that k must be even since the cup product on

H∗(O_X) is graded commutative. We show that the question of existence of varieties with

Pn[k]-units for k ≥ 4 is closely related to the question of existence of higher dimensional

Enriques varieties. Such higher dimensional analogues of Enriques surfaces where studied by Boissière, Nieper-Wiÿkirchen, and Sarti [BNS11] and independently by Oguiso and Schröer [OS11]. They provide examples of Enriques varieties of dimension 4 and 6 as quotients of generalised Kummer varieties. We say that a variety is a strict Enriques variety if it is an Enriques variety in the sense of [BNS11] and in the sense of [OS11].

Theorem 0.2.5 ([4, Theorem 4.1.3]). Let n+ 1 =pν be a prime power. Then the following

are equivalent:

(i) There exists a variety with Pn[4]-unit,

(ii) There exists a variety with Pn[k]-unit for every even k,

(iii) There exists a strict Enriques variety of dimension2n.

For n+ 1 arbitrary, the implications (iii)=⇒(ii)=⇒(i) are still true and we also prove a statement which is only slightly weaker than (i)=⇒(iii); see [4,Theorem 4.5.8].

Corollary 0.2.6. For n = 2 and n = 3 there are examples of varieties with Pn[k]-units for

every even k∈N.

Our construction of varieties with P-units out of Enriques varieties can be seen as a

generalisation of a construction of higher dimensional CalabiYau varieties of Cynk and Hulek [CH07]; see [4, Remark 4.4.7]. Some further constructions are described in [4, Sub-section 4.6.1&4.6.2&4.6.3]. It would be very interesting to construct varieties withPn[k]-units

directly, maybe as moduli spaces of objects on CalabiYau categories and then employ The-orem 0.2.5 to obtain, possibly new, Enriques or even hyperkähler varieties; see [4, Subsec-tion 4.6.7] for some more details on this speculaSubsec-tion.

• We prove that the class of strict Enriques varieties is stable under derived equivalence; see [4,Subsection 4.6.5].

0.2.5 Semi-orthogonal decompositions on Hilbert schemes of points on sur-faces and induced autoequivalences [5]

We have already seen that, for X a K3 surface, the FourierMukai transform along the

uni-versal ideal sheaf F:D(X)→ D(X[n]) has very interesting properties. Thus, it makes sense to study this functor also for other surfaces.

Theorem 0.2.7 ([5,Theorem 5.1.2]). Let X be a smooth projective surface with pg=q= 0.

In other words, O_X ∈ D(X) is an exceptional object. Then, the FourierMukai transform

F =FMIΞ:D(X)→D(X

[n]₎ _{is fully faithful for every}_n_≥₂_.