Fourier-Mukai transforms in algebraic geometry

315 Pages·2006·1.391 MB·English

by  Daniel Huybrechts

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D.Huybrechts:Fourier-Mukaitransformsinalgebraicgeometry
Fourier–Mukai transforms in
algebraic geometry
D. HUYBRECHTS
Mathematisches Institut Universita¨t Bonn
CLARENDON PRESS • OXFORD
2006
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PREFACE
ThisbookisbasedonacoursegivenattheInstitutdeMath´ematiquesdeJussieu
in 2004 and once more in 2005. It was conceived as a first specialized course in
algebraicgeometry.Astudentwithabasicknowledgeinalgebraicgeometry,e.g.
a student having worked through the first three chapters of Hartshorne’s book
[45], should be able to follow the lectures without much trouble. Occasionally,
notions from other areas, e.g. singular cohomology, Hodge theory, abelian var-
ieties, K3 surfaces, were needed, which were then presented in a rather ad hoc
manner, tailor-made for the purposes of the course. With a few exceptions full
proofs are given. The exercises included in the text should help the reader to
gain a working knowledge of the subject.
What is this book about? Its principal character is the derived category of
coherent sheaves on a smooth projective variety. Derived categories of this type
have been known for many years. Although widely accepted as the right frame-
work for any kind of derived functors, e.g. cohomology groups, higher direct
images,etc.,theywereusuallyconsideredasratherformalobjectswithoutmuch
interestinginternalstructure.ContrarytothecohomologyandtheChowringof
a projective variety X, the derived category of coherent sheaves as an invariant
of X had not been investigated thoroughly. This has changed drastically over
the last ten years.
Theoriginofthetheoryastreatedherehowevergoesbacktocelebratedpapers
by Mukai, more than twenty years ago. He constructed geometrically motivated
equivalences between derived categories of non-isomorphic varieties. Also, over
many years the Moscow school had constantly worked on the description of
coherent sheaves on homogenous varieties, e.g. the projective space, Grassman-
nians, etc. On the other hand, Kontsevich’s homological mirror symmetry has
revivedtheinterestinthesequestionsoutsidethesmallcircleofexperts.Roughly,
Kontsevich proposed to view mirror symmetry as an equivalence of the derived
category of coherent sheaves of certain projective varieties with the Fukaya cat-
egory associated to the symplectic geometry of the mirror variety. Although we
deliberately do not enter into the details of this relation, it is this point of view
that motivates and in some sense explains many of the central results as well as
open problems in this area.
The derived category turns out to be a very reasonable invariant. Due to
results of Bondal and Orlov one knows that it determines the variety whenever
the canonical bundle is either ample or anti-ample. If this was true without
any assumptions on the positivity of the canonical bundle, the theory would
be without much interest. However, there is a region in the classification of
vi Preface
projectivevarietieswherethederivedcategoryturnsouttobelessrigidwithout
getting completely out of hand. The most prominent example was observed by
Mukaiintheveryfirstpaperonthesubject.HeshowedthatthePoincar´ebundle
inducesanequivalencebetweenthederivedcategoryofanabelianvarietyAand
(cid:1)
the derived category of its dual A (which in general is not isomorphic to A).
These results, to be discussed in detail in various chapters, naturally lead to
the question under which conditions two smooth projective varieties give rise to
equivalent derived categories. This is the central theme of this book.
One word on the choice of the material. Everything that did not have a dis-
tinctive geometric touch has been left out. In particular, questions related to
representation theory, e.g. of quivers, or to modules over (non-commutative)
rings, have not been touched upon. This choice is due to personal taste, lim-
itations by a one semester course and my own ignorance in some of these
areas.
Werefrainfromgivingalengthyintroductiontothecontentsofeverychapter.
A glance at the table of contents will give a first impression of which topics are
treated, and the remarks at the beginning of each chapter provide more details.
Thereaderfamiliarwiththegeneralyogaofderivedcategoriesandderivedfunc-
torsmaygodirectlytoChapter4or5andcomebacktosomeofthebackground
material collected in the first three chapters whenever needed.
Acknowledgements:IamintellectuallyindebtedtoA.Bondal,T. Bridgeland,
Y. Kawamata, S. Mukai, and D. Orlov. The overwhelming part of the theory as
presented here is due to them. The idea that this text could help to stimulate
newcomers to pursue research originated by them was the driving force during
the preparation of these notes.
I am particularly grateful to the Institut de Math´ematiques de Jussieu for
givingme(twice)theopportunitytoteachthecoursethisbookisbasedon.The
intellectual atmosphere at the institute has been very stimulating throughout
the whole project and I have fond memories of all the discussions I had with
my colleagues at the IJM during this time. In particular, I wish to thank J. Le
Potier and R. Rouquier.
Many people have made comments on earlier versions of the book. I wish
to thank U. Go¨rtz, M. Nieper-Wißkirchen, K. Oguiso, D. Ploog, P. Stellari,
R.Thomas,andD.Ben-Zvi.IammostgratefultoBahouaFuandJustinSawon
whocarefullyworkedthroughthebookandsentmedetailedlistsofcorrections,
questions, and remarks.
CONTENTS
1 Triangulated categories 1
1.1 Additive categories and functors 1
1.2 Triangulated categories and exact functors 11
1.3 Equivalences of triangulated categories 19
1.4 Exceptional sequences and orthogonal decompositions 24
2 Derived categories: a quick tour 27
2.1 Derived category of an abelian category 27
2.2 Derived functors 43
2.3 Spectral sequences 52
3 Derived categories of coherent sheaves 62
3.1 Basic structure 62
3.2 Spanning classes in the derived category 69
3.3 Derived functors in algebraic geometry 71
3.4 Grothendieck–Verdier duality 86
4 Derived category and canonical bundle – I 90
4.1 Ample (anti-)canonical bundle 91
4.2 Autoequivalences for ample (anti-)canonical bundle 99
4.3 Ample sequences in derived categories 101
5 Fourier–Mukai transforms 113
5.1 What it is and Orlov’s result 113
5.2 Passage to cohomology 123
6 Derived category and canonical bundle – II 136
6.1 Kodaira dimension under derived equivalence 136
6.2 Geometrical aspects of the Fourier–Mukai kernel 141
6.3 Nefness under derived equivalence 145
6.4 Derived equivalence versus birationality 147
6.5 Recap: Kodaira dimension, canonical ring, etc. 151
7 Equivalence criteria for Fourier–Mukai transforms 154
7.1 Fully faithful 154
7.2 Equivalences 159
7.3 Canonical quotients 162
8 Spherical and exceptional objects 166
8.1 Autoequivalences induced by spherical objects 166
8.2 Braid group actions 175
viii Contents
8.3 Beilinson spectral sequence 179
8.4 They go together 185
9 Abelian varieties 192
9.1 Basic definitions and facts 192
9.2 The Poincar´e bundle as a Fourier–Mukai kernel 201
9.3 Sl -action 204
2
9.4 Derived equivalences of abelian varieties 209
9.5 Autoequivalences of abelian varieties 223
10 K3 surfaces 228
10.1 Recap: K3 surfaces 228
10.2 Derived equivalence of K3 surfaces 232
10.3 Recap: Moduli spaces of sheaves 240
11 Flips and flops 246
11.1 Preparations: Closed embeddings and blow-ups 246
11.2 Derived categories under blow-up 254
11.3 The standard flip 258
11.4 The Mukai flop 263
12 Derived categories of surfaces 272
12.1 Recap: Enriques classification of algebraic surfaces 273
12.2 Minimal surfaces with kod=−∞,2 278
12.3 Surfaces with torsion canonical bundle 280
12.4 Properly elliptic surfaces 281
13 Where to go from here 286
13.1 McKay correspondence for derived categories 286
13.2 Homological mirror symmetry 289
13.3 D-branes and stability conditions 291
13.4 Twisted derived categories 295
References 299
Index 305
1
TRIANGULATED CATEGORIES
Thereaderfamiliarwiththebasicnotionsofabelianandderivedcategoriesmay
only need to browse through this section or skip it altogether. It will be much
more interesting to come back to the specific results discussed here when, in
the later chapters, they are actually applied to geometrically concrete problems.
However, the reader not feeling completely at ease with the formal language of
category theory should work through this chapter in order to be well prepared
for everything that follows.
Wehopethatseparatingresultsfromcategorytheoryfromtheotherchapters
rather than blending them in later when used, will help readers to understand
which part of the theory is really geometrical and which is more formal.
On the other hand, this chapter is not meant as a thorough introduction to
the subject. We only present those parts of the theory that are relevant in our
context.
We will not worry about any kind of set theoretical issues and will always
assume we remain in a given universe (or, as put in [39, p.58], ‘that all the
required hygiene regulations are obeyed’).
1.1 Additive categories and functors
We suppose that the reader is familiar with the notion of a category and of a
functor between two categories. For the reader’s convenience we briefly recall a
few central notions. If not otherwise stated all functors are covariant.
Definition 1.1 Let A and B be two categories. A functor F : A (cid:1)(cid:1)B is full
if for any two objects A,B ∈A the induced map
(cid:1)(cid:1)
F :Hom(A,B) Hom(F(A),F(B))
is surjective. The functor F is called faithful if this map is injective for all
A,B ∈A.
Amorphism F (cid:1)(cid:1)F(cid:2) betweentwofunctorsF,F(cid:2) :A (cid:1)(cid:1)B isgivenbymorph-
isms ϕ ∈ Hom(F(A),F(cid:2)(A)) for any object A ∈ A which are functorial in A,
A
i.e. F(cid:2)(f)◦ϕ =ϕ ◦F(f) for any f :A (cid:1)(cid:1)B.
A B
Definition 1.2 Two functors F,F(cid:2) : A (cid:1)(cid:1)B are isomorphic if there exists a
morphism of functors ϕ : F (cid:1)(cid:1)F(cid:2) such that for any object A ∈ A the induced
morphism ϕ :F(A) (cid:1)(cid:1)F(cid:2)(A) is an isomorphism (in B).
A
2 Triangulated categories
Equivalently, F and F(cid:2) are isomorphic if there exist functor morphisms ϕ :
F (cid:1)(cid:1)F(cid:2) and ψ :F(cid:2) (cid:1)(cid:1)F with ϕ◦ψ =id and ψ◦ϕ=id.
Definition 1.3 A functor F : A (cid:1)(cid:1)B is called an equivalence if there exists
a functor F−1 :B (cid:1)(cid:1)A such that F ◦F−1 is isomorphic to idB and F−1◦F is
isomorphic to idA. One calls F−1 an inverse or, sometimes, quasi-inverse of F.
Two categories A and B are called equivalent if there exists an equivalence
F :A (cid:1)(cid:1)B.
Clearly, any equivalence is fully faithful. A partial converse is provided by
Proposition 1.4 Let F : A (cid:1)(cid:1)B be a fully faithful functor. Then F is an
equivalence if and only if every object B ∈ B is isomorphic to an object of the
form F(A) for some A∈A.
Proof In order to define the inverse functor F−1, one chooses for any B ∈ B
an object A ∈A together with an isomorphism ϕ :F(A ) ∼ (cid:1)(cid:1)B. Then, let
B B B
(cid:1)(cid:1)
F−1 :B A
be the functor that associates to any object B ∈ B this distinguished object
A ∈AandforwhichF−1 :Hom(B ,B ) (cid:1)(cid:1)Hom(F−1(B ),F−1(B ))isgiven
B 1 2 1 2
by the composition of
Hom(B ,B ) ∼ (cid:1)(cid:1) Hom(F(A ),F(A )), f (cid:1) (cid:1)(cid:1) ϕ−1◦f ◦ϕ
1 2 B1 B2 B2 B1
and the inverse of the bijection
∼ (cid:1)(cid:1)
F :Hom(A ,A ) Hom(F(A ),F(A )).
B1 B2 B1 B2
The isomorphisms F ◦F−1 (cid:4) idB and F−1 ◦F (cid:4) idA are the ones that are
naturally induced by the isomorphisms ϕ . (cid:1)
B
The proposition immediately yields the
Corollary 1.5 Any fully faithful functor F : A (cid:1)(cid:1)B defines an equivalence
between A and the full subcategory of B of all objects B ∈B isomorphic to F(A)
for some A∈A. (cid:1)
InthefollowingpropositionweletFun(A)bethecategoryofallcontravariant
functors, i.e. the objects are functors F : Aop (cid:1)(cid:1)Set and the morphisms are
functor morphisms. Consider the natural functor
A (cid:1)(cid:1) Fun(A), A (cid:1) (cid:1)(cid:1) Hom( ,A)
Proposition 1.6 (Yoneda lemma) This functor A (cid:1)(cid:1)Fun(A) defines an
equivalence of A with the full subcategory of representable functors F, i.e. func-
(cid:1) (cid:1)(cid:1)
tors isomorphic to some Hom( ,A). In particular, A Hom( ,A) is fully
faithful.