noncommutative tori (writing in progress) · contents 1 introduction 7 2 operator algebras 9 2.1...

Noncommutative Tori(writing in progress)

Clarisson Rizzie Canlubo

February 10, 2015

Contents

1 Introduction 7

2 Operator Algebras 92.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Definitions and Properties . . . . . . . . . . . . . . . . . . . . 92.3 Transformation Groups C*-Algebras . . . . . . . . . . . . . . 92.4 Twisted Group C*-Algebras . . . . . . . . . . . . . . . . . . . 9

3 Deformation Quantization 113.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Symplectic and Poisson Manifolds . . . . . . . . . . . . . . . . 113.3 Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Topology of the Noncommutative Tori 134.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 Motivic Homotopy Theory . . . . . . . . . . . . . . . . . . . . 134.4 Covering Spaces and Fundamental Group . . . . . . . . . . . . 13

5 Kahler Geometry 155.1 Complex Geometry . . . . . . . . . . . . . . . . . . . . . . . . 15

5.1.1 Preliminaries of Complex Geometry . . . . . . . . . . . 155.1.2 Holomorphic Bundles . . . . . . . . . . . . . . . . . . . 15

5.2 Symplectic Geometry . . . . . . . . . . . . . . . . . . . . . . . 185.3 Kahler Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 18

6 Representation Theory 196.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3

6.2 Kirillov’s Orbit Method . . . . . . . . . . . . . . . . . . . . . 196.2.1 Symplectic Structures on Coadjoint Orbits . . . . . . . 196.2.2 Integral Coadjoint Orbits . . . . . . . . . . . . . . . . . 236.2.3 The Moment Map . . . . . . . . . . . . . . . . . . . . . 246.2.4 Polarizations . . . . . . . . . . . . . . . . . . . . . . . 24

6.3 Heisenberg Lie Algebra . . . . . . . . . . . . . . . . . . . . . . 246.4 Heisenberg Group . . . . . . . . . . . . . . . . . . . . . . . . . 246.5 The Canonical Commutation Relation . . . . . . . . . . . . . 246.6 Stone-von Neumann Theorem . . . . . . . . . . . . . . . . . . 246.7 Representations of the Heisenberg Group . . . . . . . . . . . . 246.8 Coadjoint Orbits and Representations . . . . . . . . . . . . . . 246.9 Heisenberg Modules . . . . . . . . . . . . . . . . . . . . . . . . 24

7 Algebraic Geometry 257.1 The Classical Case . . . . . . . . . . . . . . . . . . . . . . . . 25

7.1.1 Generalities on Sheaves . . . . . . . . . . . . . . . . . . 257.1.2 Coherent Sheaves . . . . . . . . . . . . . . . . . . . . . 287.1.3 Locally Free Sheaves . . . . . . . . . . . . . . . . . . . 30

7.2 The Category of Coherent Sheaves . . . . . . . . . . . . . . . 317.3 The Category of Quasicoherent Sheaves . . . . . . . . . . . . . 31

7.3.1 Ind Objects . . . . . . . . . . . . . . . . . . . . . . . . 317.4 Quasicoherent Sheaves . . . . . . . . . . . . . . . . . . . . . . 317.5 Noncommutative Divisors . . . . . . . . . . . . . . . . . . . . 317.6 Noncommutative Stacks . . . . . . . . . . . . . . . . . . . . . 31

8 K-theory 338.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

8.1.1 Swan-Serre Theorem . . . . . . . . . . . . . . . . . . . 338.2 Algebraic K-theory . . . . . . . . . . . . . . . . . . . . . . . . 338.3 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.4 Moduli space of Connections . . . . . . . . . . . . . . . . . . . 33

9 Cyclic Homology 35

10 Spin Geometry 3710.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.2 Dirac Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.3 Spectral Triples . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5

10.4 Spin Structures on the Noncommutative Tori . . . . . . . . . . 37

11 Mirror Symmetry 3911.1 Algebraic Cycles . . . . . . . . . . . . . . . . . . . . . . . . . 3911.2 Fukaya Categories . . . . . . . . . . . . . . . . . . . . . . . . . 3911.3 Quantum Cohomology . . . . . . . . . . . . . . . . . . . . . . 39

12 Quantum Symmetry 4112.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.2 Cocycle Deformation . . . . . . . . . . . . . . . . . . . . . . . 4112.3 Drinfel’d Twists . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.4 Quantized Heisenberg Algebra . . . . . . . . . . . . . . . . . . 41

13 Arithmetic on Noncommutative Tori 4313.1 Complex Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

13.1.1 Construction and Properties . . . . . . . . . . . . . . . 4313.1.2 Line Bundles over Complex Manifolds . . . . . . . . . 4413.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.1.4 First Chern Class, Dual Torus and the Poincare Bundle 5113.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.1.6 Type of a Line Bundle . . . . . . . . . . . . . . . . . . 55

13.2 Theta and Heisenberg Groups . . . . . . . . . . . . . . . . . . 5713.2.1 Theta Groups . . . . . . . . . . . . . . . . . . . . . . . 5713.2.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 6113.2.3 The Commutator Map . . . . . . . . . . . . . . . . . . 6113.2.4 Heisenberg Groups . . . . . . . . . . . . . . . . . . . . 6213.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 6413.2.6 Theta Functions and Schrodinger Representation . . . 65

13.3 Theta Functions on Noncommutative Tori . . . . . . . . . . . 6613.3.1 Theta Vectors . . . . . . . . . . . . . . . . . . . . . . . 6613.3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 70

13.4 Dual Noncommutative Torus . . . . . . . . . . . . . . . . . . . 7013.5 Vanishing Theorems . . . . . . . . . . . . . . . . . . . . . . . 70

14 Discrete Noncommutative Tori 71

A Bibliography 73

Noncommutative ToriClarisson Rizzie Canlubo, University of Copenhagen, 2014

6


Chapter 1

Introduction

The noncommutative torus is probably the most famous and worked outexample of a noncommutative space. In this notes, we will look into dif-ferent aspects of this object ranging from its geometric side to its analyticand arithmetic aspects. This notes assumes familiarity with graduate levelmathematics but because of a great span of topics involve, we will try to givea reasonable exposition of the mathematics involved.

CHAPTER 1. INTRODUCTION 8


Chapter 2

Operator Algebras

2.1 Preliminaries

2.2 Definitions and Properties

2.3 Transformation Groups C*-Algebras

2.4 Twisted Group C*-Algebras

CHAPTER 2. OPERATOR ALGEBRAS 10


Chapter 3

Deformation Quantization

3.1 Preliminaries

3.2 Symplectic and Poisson Manifolds

3.3 Index Theory

CHAPTER 3. DEFORMATION QUANTIZATION 12


Chapter 4

Topology of theNoncommutative Tori

4.1 Preliminaries

4.2 Homotopy

4.3 Motivic Homotopy Theory

4.4 Covering Spaces and Fundamental Group

CHAPTER 4. TOPOLOGY OF THE NONCOMMUTATIVE TORI 14


Chapter 5

Kahler Geometry

5.1 Complex Geometry

5.1.1 Preliminaries of Complex Geometry

In this section, we will give the basic materials in complex geometry that wewill be needing to go to the noncommutative case. One of the best book toconsult about complex geometry is the one by Huybrechts [20].

HHHHHHHHHHHHHHHHHHHHHHHHHHthis section not yet doneHHHHHHHHHHHHHHHHHHHHHHHHHH

5.1.2 Holomorphic Bundles

Consider the universal C∗-algebra realization of a noncommutative d-torusT dθ associated with the irrational d× d skew-symmetric matrix θ = (θij), i.e.elements of the form∑

(n1,...,nd)∈Zdα(n1, ..., nd)U

n11 · · ·U

ndd , α ∈ S(Zd).

The generators Ui, i = 1, ..., d are unitary operators on a Hilbert space sat-isfying the commutation relation UjUk = e2πiθjkUkUj.

By the Swan-Serre theorem, we can view projective modules over T 2θ as

the analogues of the usual vector bundles over the (commutative) 2-torus.How about holomorphic vector bundles? What is the suitable notion ofholomorphicity in projective modules? We will answer this in what follows.

CHAPTER 5. KAHLER GEOMETRY 16

Consider L = Rd as an abelian Lie algebra generated by δ1, ..., δd. Notethat L is acting on T dθ by δjUk = 2πiδjkUj where δjk denotes the Kroneckerdelta function. We say that a projective module P is holomorphic if there is

a connection ∇ defined on P , i.e. a linear map P ⊗ L ∇−→ P satisfying

∇X (f · e) = f · ∇Xe+ (δXf) · ewhere f ∈ T dθ , e ∈ P and δX corresponds to the differentiation along thedirection X ∈ L. The curvature of a connection ∇ is defined as

K∇(X, Y ) = [∇X ,∇Y ]−∇[X,Y ]

where X, Y ∈ L. Since L is an abelian Lie algebra, the curvature of anyconnection on a holomorphic projective module over T dθ can be computed as

K∇(X, Y ) = [∇X ,∇Y ] .

Remark. Let us justify that the above definition for a holomorphic projectivemodule make is the one suitable for our purpose. To do this, let us show thatover the (commutative) complex torus a vector bundle is holomorphic if andonly if its module of sections has a connection.

HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH

Let us look at a particular example. Consider an irreducible representa-tion of Heisenberg commutation relation

[∇j,∇k] = 2πiωjk (24)

for 1 6 j, k 6 d, as operators on a Hilbert space where ω = (ωjk) is ad × d skew-symmetric matrix. By the Stone-von Neumann theorem, thiscommutation relation has a unique irreducible representation up to unitaryequivalence. By putting ω is the form

λ1

λ2

. . .

λm−λ1

−λ2

. . .

−λm

(25)



we may realize the commutation relation (24) as

[∇j,∇j+m] = − [∇j+m,∇j] = 2πiλj for 0 6 j 6 m

[∇j,∇k] = 0 otherwise.

where d = 2m (assuming ω is a nondegenerate skew-symmetric matrix, dmust be even). The above commutation relation can be realized as operatorson S(Rm), the Schwarz space on Rm. Note that S(Rm) is not a Hilbertspace but is dense in L2(Rm). Let us denote by x1, ..., xm the usual euclideancoordinates in Rm. Then the operators ∇i, i = 1, ..., d can be realized as

∇j =

2πi xj· 1 6 j 6 m∂j−m m < j 6 2m

where xj· and ∂j stand for the operators that multiply by xj and differentiate

with respect to xj, respectively. Define Ui = e−∑jθij∇j

where θ = ω−1. Sincewe have written ω in the form (25), up to unitary equivalence θ must be ofthe form

−λ−11

−λ−12

. . .

−λ−1m

λ−11

λ−12

. . .

λ−1m

Thus, we have

Ui =

e−(i

√2/λi)∇i 1 6 i 6 m,

e(i√

2/λi)∇i m < i 6 2m.

If we denote by A the Lie algebra spanned by the operators ∇i we immedi-ately see that [A,A] 6 Z(A). By the Baker-Campbell-Hausdorff formula, wesee that for any X, Y ∈ A we have



eXeY = eX+Y+ 12

[X,Y ].

Thus, we immediately see that Uj and Uk commute if and only if |j−k| 6= m.

In the case k = j + 1, we have[− i√

2λj∇j,

i√

2λj∇k

]=(−2i2

λ2j

)2πiλj. Thus,

UjUk = e− i√2

λj∇j+ i

√2

λj∇k+ 1

2

((−2i2

λ2j

)2πiλj

)= e2πi(1/λj)UkUj

In any case, we have UjUk = e2πiθijUkUj. Hence, the operators Ui2mi=1

provide us a concrete model for T dθ . In addition, we also have the relation

[∇j, Uk] =

5.2 Symplectic Geometry

5.3 Kahler Geometry


Chapter 6

Representation Theory

6.1 Preliminaries

The presentation of the noncommutative torus as a universal C∗-algebra isa direct generalization of the classical case, namely the algebra of functionson the classical 2-torus is generated by two commuting unitaries. The pre-sentation of the noncommutative torus has a classical name, the canonicalcommutation relation or CCR for short. This relation is closely related tothe Heisenberg group H and its Lie algebra h. As we shall soon see, H andh have nice presentations. Though essentially different, we shall simply referto these presentations as the canonical commutation relations. We will makeuse of Kirillov’s theory of coadjoint orbits in what follows. They provide ageometric and cohomological foundation in the representation theory of Hand k.

6.2 Kirillov’s Orbit Method

6.2.1 Symplectic Structures on Coadjoint Orbits

In this section, we will describe a method to generate representations ofnilpotent groups. This method is known as the orbit method, popularizedby Kirillov and a good discussion can be found on his book [25]. We will usethis method to generate representations of the Heisenberg group.

Before we embark on this specific task, let us discuss the method in agreat generality. Let G be a Lie group and let g be its Lie algebra. We

CHAPTER 6. REPRESENTATION THEORY 20

identify g with the tangent space of G at the identity e and also with thespace of left-invariant vector fields on G. For g ∈ G, the inner automorphism

Ag : h 7→ ghg−1 of G is smooth. The differential dAg

∣∣∣e

: TeG −→ TeG of this

map at the identity e ∈ G is an automorphism of g = TeG. Let us denote by

Ad(g) = dAg

∣∣∣e. This construction defines a representation of G on g, known

as the adjoint representation, denoted by Ad and is given by

Ad : G −→ Aut(g)

g 7→ Ad g

The dual of this representation is known as the coadjoint representation,given explicitly by

Ad∗ : G −→ Aut(g∗)

g 7→ Ad∗(g) = (Ad(g−1))∗

Let us write K(g) for Ad∗(g) as is customary in the literature. This repre-sentation induces an action of G on g∗, known as the coadjoint action. Letus study the orbits of this action, which we shall simply refer to as coadjointorbits. In what follows, we will see that each coadjoint orbit has a naturalsymplecture structure on it. An immediate consequence is that coadjoint or-bits are of even dimension. For a brief introduction to symplectic geometry,one can refer back to section 5.2. A brief but well-written work by Wong[50] may serve as a good introduction but of course the standard refernce isKirillov [26].

We can view g and g∗ as Lie groups with the usual vector space structureson them and with this, the adjoint and the coadjoint maps are smooth. Letus denote by ad the differential of Ad and by k the differential of K. Themaps

ad : g −→ Aut(g) and k : g −→ Aut(g∗)

are also called the adjoint and the coadjoint maps. Explicitly, for any X ∈ gthe automorphism ad(X) is given by

ad(X)Y = [X, Y ]

for any Y ∈ g. Let us describe k explicitly. For any X ∈ g we have a mapk(X) : g∗ −→ g∗. Thus, for α ∈ g∗, k(X)(α) ∈ g∗. Let Y ∈ g. Then



k(X)(α)(Y ) = α(−ad(X)Y ) = α([Y,X]).

Given α ∈ g∗, let us denote by Oα its coadjoint orbit, i.e.

Oα = K(g)α : g ∈ G .

For simplicity, let us fixed α and its coadjoint orbit Oα all throughout thissection. Hence, it makes sense to just denote by Oα by O. By the orbit-stabilizer theorem, we have

O ∼= G/Gα

where Gα stands for the stabilizer of α in G. The group G acts smoothly ong∗ and hence O may be equipped with a submanifold structure. Using thecanonical projection ϕ : G −→ G/Gα, we may consider G as a fibre bundleover O ∼ G/Gα whose fiber at α is precisely Gα. Let gα denote the Liealgebra of the Lie subgroup Gα 6 G. Explicitly, gα = X ∈ g : k(X)α = 0.Using the differential version of the orbit-stabilizer theorem, we have an exactsequence

0 −−−→ gα −−−→ gdϕ−−−→ TαO −−−→ 0

where TαO denotes the tangent space at α. Thus, we have TαO ∼= g/gα.By the orbit-stabilizer theorem applied to the differential k of the coadjointrepresentation we get

TαO = k(X)α : X ∈ g .

Let us denote by X the image of X ∈ g under the canonical projection

gdϕ−→ g/gα. We are now ready to define the symplectic structure on the coad-

joint orbit O. Consider the skew-symmetric bilinear form Bα on g definedby

Bα(X, Y ) = α([X, Y ]).

Let us compute for the kernel of this bilinear form. Note that

X ∈ ker Bα ⇔ Bα(X, Y ) = 0,∀Y ∈ g⇔ α [X, Y ] = 0,∀Y ∈ g



⇔ α(ad(X)Y ) = 0, ∀Y ∈ g⇔ −(k(X)α)(Y ) = 0,∀Y ∈ g.

This shows that k(X)α = 0 and hence, ker Bα = gα. Next, we claim that if αand β belong to the same coadjoint orbit then they define the same bilinearform up to a twist by an automorphism. To make this precise, supposeα, β ∈ O. This means that for some g ∈ G, β = K(g)α and so, we have

Bα(X, Y ) = α([X, Y ])

= K(g−1)K(g)α([X, Y ])

= (K(g)α)(Ad(g) [X, Y ])

= (K(g)α([Ad(g)X,Ad(g)Y ]))

= BK(g)α(Ad(g)X,Ad(g)Y )

= Bβ Ad(g)⊗2(X, Y )

Theorem 1. The bilinear form ωα(X, Y ) := Bα(X, Y ) is a G-invariantsymplectic form on O.

Proof. First of all, let us show that ωα is well-defined. Suppose X and Y areleft-invariant vector fields on G such that X = Y . Then X − Y ∈ ker Bα

and so Bα(X,Z) = Bα(Y, Z) for any Z ∈ g. Since Bα is essentially definedusing the Lie bracket in a linear fashion, it is bilinear and skew-symmetric.This implies that ωα is bilinear and skew-symmetric as well.

Let us show that ωα is G-invariant. Let g ∈ G. Note that for any

X, Y ∈ g, we have X = Ad(g)X and Y = Ad(g)Y . Thus, we have

ωα(X, Y ) = ωα(Ad(g)X, Ad(g)Y )

= Bα(Ad(g)X,Ad(g)Y )

= Bβ(X, Y ) where β = K(g−1)α

= ωβ(X, Y )

Thus, ωα defines a G-invariant 2-form on O which is obviously nondegen-erate. What is left to show is that it is closed. There is a direct and muchsimpler proof of this in [26] but we will present one that is more geometric(found in the same book).



Consider the fiber bundle Gϕ−→ O. Let σ be the pull-back of the nonde-

generate differential 2-form ωα, i.e. σ = ϕ∗ωα.Let Lg denote the left translation of G by g ∈ G. Let Θ be the g-valued

differential 1-form on G defined as follows: ∀g ∈ G, Θ(g) ∈ Ω1(G, g) suchthat for any left-invariant vector field X ∈ g we have Θ(g)(X) = dLg−1(X).Note that Θ is left-invariant and hence the equation Θ(e)(X) = X completedetermines Θ. Consider the left-invariant real-valued 1-form θ on G definedby θ = −α Θ. We claim that σ = dθ.

Let X, Y be left-invariant vector fields on G. Note that X, Y can beviewed as elements of g. To make things clear, we will denote X, Y byXo, Y o if we want to refer to them as elements of g. By definition of theexterior derivative, we have

dθ(X, Y ) = Xθ(Y )− Y θ(X)− θ [X, Y ] .

Left-invariance of X, Y implies that θ(X) and θ(Y ) are constant functions.Thus, the first two terms of the above equation vanish. Now, left-invarianceof [X, Y ] and of Θ implies that

−θ [X, Y ] = α Θ [X, Y ] = α [Xo, Y o] = ϕ∗ωα [Xo, Y o] = σ [X, Y ] .

Thus, dθ = σ. This proves the claim.Note that ϕ is a submersion and hence dϕ is surjective. Thus, the dual

map ϕ∗ is injective. Now, ϕ∗dωα = d (ϕ∗ωα) = dσ = d2θ = 0. Hence,dωα = 0, i.e. the nondegenerate differential 2-form ωα is closed.

6.2.2 Integral Coadjoint Orbits

Integrality condition has several nice consequences for coadjoint orbits. Butbefore we present them, let us recall some relevant notion in the theory ofsingular homology for smooth manifolds.

By a singular k-cycle on a smooth manifold M , we mean a formal R-linearcombination

C =∑i

ciϕi(Mi)

where Mi are smooth k-dimensional manifolds and ϕi : Mi −→M are smoothmaps. We say that a coadjoint orbit O is integral if the associated symplecticfrom ω has the following property:



for any singular 2-cycle C in O,∫C

ω ∈ Z.

The following theorem presents the geometric and representation theoreticimplications of integrality for coadjoint orbits.

Theorem 2. Let G be a simply connected Lie group. Then TFAE:

(a) The coadjoint orbit O is integral.

(b) There is a G-equivariant complex line bundle L over O with a G-equivariant Hermitian connection ∇ whose curvature is κ(∇) = 2πiσ.

(c) For every α ∈ O, there is a 1-dimensional representation χ of theconnected component Go

α of Gα such that χ(expX) = e2πiα(X).

Proof.

6.2.3 The Moment Map

6.2.4 Polarizations

6.3 Heisenberg Lie Algebra

6.4 Heisenberg Group

6.5 The Canonical Commutation Relation

6.6 Stone-von Neumann Theorem

6.7 Representations of the Heisenberg Group

6.8 Coadjoint Orbits and Representations

6.9 Heisenberg Modules


Chapter 7

Algebraic Geometry

7.1 The Classical Case

Similar to the previous chapters, before we discuss the main meat of thischapter let us describe the classical case. In this section, we will reviewsome basic notions involve in laying the foundation of the modern algebraicgeometric aspects of noncommutative geometry in general. Let’s start ofwith the theory of sheaves. A great exposition can be found on Tennison[46].

7.1.1 Generalities on Sheaves

Let X be a topological space and let C be a category. A presheaf on X withvalues in C is a C-valued functor on X satisfying the following:

(a) for every U ⊆o X, F (U) ∈ Ob C,

(b) for every inclusion V ⊆ U , we have a morphism resUV : F (U) −→ F (V )in C (called restriction morphisms),

(c) for every U ⊆o X, resUU is the identity morphism,

(d) given open sets W ⊆ V ⊆ U , we have resVW resUV = resUW .

In most of our interesting examples, the category C is concrete like the cat-egory of sets, abelian groups, rings, etc. In case the category is concrete,elements of F (U) are called sections of F over U and elements of F (X) arecalled global sections.

CHAPTER 7. ALGEBRAIC GEOMETRY 26

Remark. Note that X can be viewed as a category whose objects are its opensets and the morphism between such objects are taken to be inclusions. Withthis interpretation, we can view a presheaf as a contravariant functor fromX to C.

Let I be a directed set (i.e. a set with a defined reflexive and transitiverelation 6) such that for every pair α, β ∈ I there is an element γ ∈ I suchthat α, β 6 γ. A directed system on a category C indexed by I is a collectionUαα∈I of objects in C satisfying

(a) for every pair of index α 6 β, there is a morphism ρβα : Uα −→ Uβ,

(b) for every α ∈ I, ραα is the identity morphism on Uα,

(c) for every α 6 β 6 γ, ργα = ρβα ργβ.

Note that a topological space X is a directed set, ordered by inclusion. Inthis case, we see that a presheaf F with values on C is just a directed systemon C indexed by the open sets of X.

Let Uαα∈I be a directed system on C. A target in the given directed sys-

tem is an object V in C together with a collection of morphismsUα

σα→ Vα∈I

for which the following family of diagrams commute.

Uβ

σβ

Uα

σα

''

ρβα

77

V

The direct limit of a directed system is a target U , ηα : Uα −→ Uα∈I sat-

isfying the following universal property: given any target V ,Uα

σα→ Vα∈I

there is a unique morphism U −→ V such that the following family of dia-grams commute.



U

Uα

σα

''

ηα

77

V

We will write U = lim−→Uα

Uα. The direct limit of a directed system, if it

exists, is unique up to isomorphisms.Let F be a presheaf on a topological space X with values in C. Let x ∈ X.

Then the collection F (U) : x ∈ U ⊆o X forms a directed system. The stalkof F at x, denoted by Fx is

Fx = lim−→U3x

F (U).

We are now going to define what a sheaf is. For this purpose, we willrestrict our attention to a concrete category C for reasons that will be ap-parent in the succeeding definitions. The presheaf F on X with values on aconcrete category C is a sheaf if it satisfies the following conditions.

(Separation) Let U ⊆o X covered by open sets Uαα∈I . If s, t ∈ F (U)such that resUUα(s) = resUUα(t) for every α ∈ I then s = t.

(Gluing) Let U ⊆o X covered by open sets Uαα∈I . Suppose there is acollection of sections sαα∈I such that sα ∈ F (Uα) and resUUα∩Uβ(sα) =

resUUα∩Uβ(sβ) for any α, β ∈ I then there is an s ∈ F (U) such that

resUα (s) = sα for all α ∈ I.

Sheaves form a category and to fully appreciate this, we should define themorphisms between sheaves. Let F and G be sheaves on X. A morphismϕ : F −→ G is a natural transformation between the functors F and G, i.e.

a morphism F (U)ϕ(U)−→ G(U) associated to every U ⊆o X commuting with

the restriction maps, i.e. a family of morphisms ϕ(U) : U ⊆o X such that



F (V )ϕ(V )

//

resUV

G(V )

resUV

F (U)ϕ(U)

// G(U)

commutes for any U, V ⊆o X. Note that the above definition also make sensefor presheaves.

Let U ⊆o X. Consider a sheaf F over X. Since open sets in U areopen sets in X as well, we can construct a sheaf over U using F called therestriction of F on U , denoted by F

∣∣U

, by simply gathering those sectionsand restriction morphisms on F that concerns open subsets of U .

7.1.2 Coherent Sheaves

Let X be a topological space together with a sheaf of rings OX . The pair(X,OX) is called a ringed space. We call OX the structure sheaf of X. Aringed space is called a locally ringed space if all the stalks of the structuresheaf are local rings. Examples of locally ringed spaces are the familiarsheaves of continuous C-valued functions on X, denoted by C(X); or if Xis a manifold then C∞(X) is a structure sheaf for a locally ringed spacestructure on X.

Consider a locally ringed space (X,OX). An OX-module M over X is asheaf of modules over X satisfying the following conditions

(a) for every U ⊆o X, M(U) is an OX(U)-module,

(b) given open sets V ⊂ U , the diagram commutes

OX(U)×M(U) //

resUV ×RESUV

M(U)

resUV ×RESUV

OX(V )×M(V ) // M(V )



The stalk Mx of M over x ∈ X will be a module over (OX)x. Thecollection of allOX-modules over a given locally ringed space forms an abeliancategory. The morphisms are what you might expect them to be. Specifically,

a morphism Fψ−→ G between two OX-modules F and G is a sheaf morphism

that is compatible with the OX-module structure in the following sense: forany U ⊆o X, the diagram

OX(U)× F (U) mod //

id×ψ

F (U)

ψ

OX(U)×G(U)mod

// G(U)

commutes. The morphism ψ descends to module morphisms on the stalks ofF and G. Let us denote by HomOX (F,G) the collection of all OX-modulemorphisms between F and G. Note that HomOX (OX , F ) is a ring and assuch, it is isomorphic to F (X). To see this, note that a morphism from OXto F is completely determined by the image of 1 ∈ OX(X) in F (X). Ina similar spirit, a map OSX −→ F is completely determined by choosing afamily of global sections sii∈S of F . We say that F is generated by theglobal sections sii∈S if the map OSX −→ F determined by these sections isa sheaf surjection, i.e. surjective on all stalks.

A quasicoherent sheaf F over a ringed space (X,OX) is an OX-modulesuch that for every x ∈ X, there is a neighborhood x ∈ U and indexing setsI and J such that

OJX

∣∣∣∣∣U

ζ−−−−→ OIX

∣∣∣∣∣U

ω−−−−→ F

∣∣∣∣∣U

−−−−→ 0 (23)

is an exact sequence of sheaves. Note that this definition make sense sinceOU -modules form an abelian category. The indexing sets I and J depend onx ∈ X. They may be infinite. If for any x ∈ X, the indexing set I is finite,we say that F is of finite type. If both I and J can be chosen to be finitesets then F is said to be finitely presented.

An OX-module F is said to be coherent if the following are satisfied:

(a) F is of finite type, i.e. for every x ∈ X, there is a neighborhood x ∈ U , apositive integer n and a surjective morphism of sheaves OnX

∣∣U−→ F

∣∣U

,



(b) for any U ⊆o X, any morphism of sheaves OmX∣∣U−→ F

∣∣U

has kernel offinite type.

HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH insertpropositions here HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH-HHHHHH

7.1.3 Locally Free Sheaves

An OX-module F is said to be free if it is isomorphic to OIX for some indexingset I. In this case, we define the rank of F to be the cardinality of I. Ifin OX-module F every point x ∈ X has a neighborhood U such that therestriction of F on U is free, then F is said to be locally free. It is not hardto see that if X is connected, the rank of every such restrictions are thesame and hence, it make sense to define this constant to be the rank. Ourinterest in locally free sheaves lies in the next theorem. But before that, letus describe an important sheaf.

Given a complex manifold X, consider the sheaf OX whose sections overU ⊆o X are the holomorphic functions on U . The restriction morphismsfrom U to V ⊆o U is the one that restricts a holomorphic function on U toV . Note that if X is a compact then OX(X) = C, i.e. the only holomorphicfunctions on a compact manifold are the constant functions. However, OX isnontrivial since it has nonconstant holomorphic functions on a proper openset. If we mention a locally free sheaf over a complex manifold withoutany reference to a particular structure sheaf, it will be understood that thestructure sheaf taken is the sheaf of holomorphic functions.

Theorem 3. Finite rank locally free sheaves on a complex manifold X is ina one-to-one correspondence with vector bundles over X.

Proof. Let Eπ−→ X be a vector bundle, say of rank n. Let us construct a

sheaf F as follows. For any U ⊆o X, take F (U) as the space of all holomor-phic sections of π over U , i.e. maps s : U −→ E such that π s(x) = x. Letus show that F is locally free of rank n. Let x ∈ X. Using a local trivial-ization for E, x has a neightborhood U ⊆o X such that π−1(U) ∼= U × Cn.Thus, F (U) can be identified with holomorphic maps s : U −→ Cn. Hence,

F (U) ∼= OnX∣∣∣U

. This proves that F is locally free of rank n.

Conversely, consider a locally free sheaf F on X. Without loss of general-ity, assume X is connected. Then F is of constant rank, say n. Consider an



open covering Uii∈I of X such that F is locally free on each Ui, i.e there

are isomorphisms F (Ui)ϕi−→ OnX(Ui). Let Uij = Ui∩Uj. Then ϕij := ϕ−1

j ϕiis an automorphism of F (Uij). Note that F (Uij) ∼= OnX(Uij) and hence, ϕijcan be thought of a matrix of holomorphic functions on Uij. Let E

′be the

disjoint union of all Ui × Cn. Let us identify points (x, v) and (x,w) on E′

if there is a ϕij such that v = ϕijw. After identifying, we get a complex

vector bundle Eπ−→ X whose local trivializations are given by ϕii∈I . The

projection π is simply the projection on the first local coordinate.Note that the above constructions are inverses of each other in the sense:

Let ∼ denote the identification on E′. If we take the local sections of

E′/ ∼ π−→ X we will get the locally free sheaf F we started with.

Remark. The method of constructing functions on overlap of open coverswill play a very prominent role in constructing global objects out of local ones.This is a simple over view of sheaf cohomology.

7.2 The Category of Coherent Sheaves

7.3 The Category of Quasicoherent Sheaves

In [37], Polishchuk studied the quasicoherent sheaves on the noncommuta-tive tori. He defined quasicoherent sheaves as ind-objects in the category ofholomorphic projective modules over the noncommutative torus. In the clas-sical set up, quasicoherent sheaves are ind-objects in the category of coherentsheaves over a locally ringed-space X.

7.3.1 Ind Objects

7.4 Quasicoherent Sheaves

7.5 Noncommutative Divisors

7.6 Noncommutative Stacks


Chapter 8

K-theory

8.1 Preliminaries

8.1.1 Swan-Serre Theorem

8.2 Algebraic K-theory

8.3 Rigidity

8.4 Moduli space of Connections

CHAPTER 8. K-THEORY 34


Chapter 9

Cyclic Homology

CHAPTER 9. CYCLIC HOMOLOGY 36


Chapter 10

Spin Geometry

10.1 Preliminaries

10.2 Dirac Operators

10.3 Spectral Triples

10.4 Spin Structures on the Noncommuta-

tive Tori

CHAPTER 10. SPIN GEOMETRY 38


Chapter 11

Mirror Symmetry

11.1 Algebraic Cycles

11.2 Fukaya Categories

11.3 Quantum Cohomology

CHAPTER 11. MIRROR SYMMETRY 40


Chapter 12

Quantum Symmetry

12.1 Preliminaries

12.2 Cocycle Deformation

12.3 Drinfel’d Twists

12.4 Quantized Heisenberg Algebra

see Rosenberg [40]

CHAPTER 12. QUANTUM SYMMETRY 42


Chapter 13

Arithmetic onNoncommutative Tori

13.1 Complex Tori

13.1.1 Construction and Properties

Before discussing the case of the noncommutative tori, we will recall sev-eral facts about the commutative case. This chapter benefited much fromBirkenhake-Lange [4]. Let V denote the an n-dimensional complex vectorspace and let Λ be a lattice of maximal rank. Note that Λ acts freely andproperly discontinuously on V by translations. The quotient X = V/Λ iscalled a complex torus which inherits a complex Lie group structure from V .Connectedness of V implies that of X. Maximality of the rank of Λ impliesthat we can choose the generators g1, ..., g2n of Λ so that X is faithful imageof a subset of the bounded convex hull of 0, g1, ..., g2n and so, X is compact.In fact, we have the following strong result:

Proposition 1. Any connected compact complex Lie group is a complextorus.

Note that it is known in Lie theory that any compact, connected, abelianLie group has a structure of a torus. In our present case, the complex struc-ture ensures commutativity. This uses a maximum modulus principle typeof argument. It is worth mentioning that if we view V as an abelian Liealgebra then the map π is nothing but the exponential map.

CHAPTER 13. ARITHMETIC ON NONCOMMUTATIVE TORI 44

The quotient map Vπ−→ X is clearly a covering map and since V is simply

connected, it is the universal covering space of X. Furthermore, the group ofall deck transformations of the covering π is Λ, the fundamental group π1(X)of X is Λ. The group Λ being a lattice means that Λ ∼= Z2n. Note that loopsin a path-connected topological space can be viewed as 1−cells and for thisreason, there is a corresponding homomorphism from the fundamental groupπ1(X) ofX to the first simplicial homology groupH1(X,Z). In fact, H1(X,Z)is isomorphic to the abelianization of π1(X). In our case, π1(X) is alreadyabelian for a complex torus X, thus H1(X,Z) ∼= π1(X). A homomorphismX −→ Y of complex tori is a holomorphic group homomorphism.

13.1.2 Line Bundles over Complex Manifolds

Let us consider a more general situation–that of a complex manifold X.Let X

π→ X be its universal cover. Via the covering map π, X becomes acomplex manifold. Let us determine which line bundles L whose pull backπ∗L is trivial. Note that in the case of a complex tori this is a lot easier, anyline bundle will have trivial pull back along the universal covering map. Tosee this, consider the exponential sequence

0 −→ Z −→ OVexp(2πi·)−→ O∗V −→ 1.

where V is any vector space and OV is the sheaf of holomorphic functionson V . From the induced long exact sequence we have the following exactsequence:

H1(V,OV ) −→ H1(V,O∗V ) −→ H2(V,Z).

By the ∂-Poincare lemma (Griffiths-Harris [15]), we have H1(V,OV ) = 0and a since V is contractible we have H2(V,Z) = 0. Thus, H1(V,O∗V ) = 0.This applies to our present situation since the universal covering space of acomplex torus is a vector space. All we have to show is that the set of allline bundles over any complex M is H1(M,O∗M).

Before we proceed, let us have a little digression about the set of allline bundles over a complex manifold M . This set is forms a group wheremultiplication is tensor product and inversion is taking duals. The identityis C viewed as the trivial line bundle. This is called the Picard group of M ,denoted by Pic(M). Note that this group can be defined in general for anyringed space over any field where the concept of invertible sheaves (or line



bundles) make sense. To finish the case of a complex torus, we have thefollowing.

Proposition 2. Pic(M) ∼= H1(M,O∗M).

Proof. Let Lp→M be a line bundle. Consider an open cover Uα of M and

the following trivialization of L with respect to this open cover.

φα : LUα −→ Uα × C

Here, LUα = p−1 (Uα). Consider the corresponding transition functions

gαβ =(φα φ−1

β

) ∣∣∣Uα.

By definition of a complex manifold, these functions are holomorphic. They

are also nonvanishing since they can be realized as Uα∩Uβgαβ−→ GL1(C) ∼= C∗.

Hence, gαβ defines a Cech 1-cochain on M . More importantly, they satisfythe following conditions

gαβ · gβα = 1gαβ · gβγ · gγα = 1

(1)

which means that gαβ defines a 1-cocycle. Conversely, given a collectionof functions gαβ ∈ O∗ (Uα ∩ Uβ) satisfying conditions (1), we can construct aline bundle L with transition functions gαβ as follows. Consider the disjointunion of all Uα×C. We glue Uα×C and Uβ ×C along Uα ∩Uβ via the map

z × Cgαβ−→ z × C defined by gαβ (z, ζ) = (z, gαβ(z)ζ).

Given a line bundle L with the above data,φ′α = fα · φα : fα ∈ O∗ (Uα)

also defines a trivialization of L. The corresponding transition functions g

′

αβ

is given by

g′

αβ =

(fαfβ

)gαβ ⇐⇒ g

′

αβ · g−1αβ = fα · f−1

β .

Note that any trivialization of L over Uα arise this way. Thus, two col-lections of transition functions gαβ and

g′

αβ

define the same line bundle

if there are functions fα ∈ O∗ (Uα) satisfying the above relation. Thus,g′

αβ

and gαβ define the same line bundle if their difference gαβ

′·g−1αβ

is a coboundary. This proves that Pic(M) is in bijection with H1(M,O∗M).



To see that this gives a group homomorphism, note that if the correspond-ing transition functions to L and L

′are gαβ and

g′

αβ

, respectively then

the associated transition functions to L ⊗ L′ and L−1 = L∗ aregαβ · g

′

αβ

and

g−1αβ

, respectively.

Remark. In a more modern language, we say that the obstruction for a linebundle L over M to be trivial is the first Cech cohomology group H1(M,O∗M).

Let us consider the general complex manifold X with universal cover X.By definition, H0(O∗

X) is the multiplicative group of nonvanishing holomor-

phic functions on X. The fundamental group π1(X) acts by deck trans-formations on X and this induces a π1(X)-module structure on H0(O∗

X).

A 1-cocycle of π1(X) with values in H0(O∗X

) is a holomorphic function f :

π1(X)× X −→ C∗ satisfying the following cocycle condition

f(λµ, x) = f(λ, µx)f(µ, x), ∀ λ, µ ∈ π1(X), x ∈ XThe set of all 1-cocycle form an abelian group under multiplication of func-tions. Let us denote this group by Z1

(π1(X), H0

(O∗X

)). Authors that are

more inclined in arithmetic tend to call elements of Z1(π1(X), H0

(O∗X

))as

factors of automorphy. Let h ∈ H0(O∗X

). Consider f(λ, x) = h(λx)h(x)−1.

It is a matter of simple computation that functions of the latter form con-stitute a subgroup of Z1

(π1(X), H0

(O∗X

)). This subgroup is denoted by

B1(π1(X), H0

(O∗X

))and its elements are called coboundaries. We define

the group H1(π1(X), H0

(O∗X

))as the following quotient.

H1(π1(X), H0

(O∗X

))=

Z1(π1(X), H0

(O∗X

))B1(π1(X), H0

(O∗X

)) .Proposition 3. There is a homomorphism Z1

(π1(X), H0

(O∗X

))−→ H1 (X,O∗X).

Proof. Let f ∈ Z1(π1(X), H0

(O∗X

)). The fundamental group π1(X) acts

holomorphically on the trivial bundle X × C −→ X by

λ · (x, z) = (λx, f(λ, x)z) ∀ λ ∈ π1(X), x ∈ X, z ∈ C.

In fact, the cocycle condition is the one that guarantees that the above rela-tion indeed, defines an action. Furthermore, this action is free and properly



discontinuous. Thus, the orbit space L =(X × C

)/π1(X) has a complex

manifold structure. The first projection map X × C −→ X induces a mapL −→ X making L into a holomorphic line bundle over X.

To see that the above construction defines a homomorphism, considerthe trivial bundle in another form, namely X × (C⊗ C) −→ X. Given two1-cocycles h and g, π1(X) acts on the above bundle by

λ·(x, z ⊗ y) = (λx, h(λ, x)z ⊗ g(λ, x)y) ∀ λ ∈ π1(X), x ∈ X, z, y ∈ C.

Note that the orbit space of the above action is the same as the orbit spaceof the earlier construction with the 1-cocycle f replaced by hg.

The universal covering map Xπ−→ X induces a map H1 (X,O∗X)

π∗−→H1(X,O∗

X

). The kernel of this map answers the question: which line bun-

dles over X has a trivial pull-back along the universal covering map? Thefollowing proposition identifies these line bundles with the first cohomologygroup of the fundamental group of X.

Proposition 4. There is a canonical isomorphism

H1(π1(X), H0

(O∗X

)) ϕ1−→ ker(

H1 (X,O∗X)π∗−→ H1

(X,O∗

X

)).

Proof. The above construction of line bundles over X from the trivial bundleover X and the previous proposition show that we have a homomorphism ofgroups

Z1(π1(X), H0

(O∗X

)) ψ−→ ker(

H1 (X,O∗X)π∗−→ H1

(X,O∗

X

)).

We claim that this map factorises via H1(π1(X), H0

(O∗X

)). Once proven,

we will let ϕ1 be the cofactor of the projection Z1(π1(X), H0

(O∗X

))−→

H1(π1(X), H0

(O∗X

))in ψ. To prove the claim, we need to show that the

line bundles over X arising from elements of B1(π1(X), H0

(O∗X

))are all

trivial. However, we have a convenient way of handling line bundles in termsof transition functions.

Since Xπ−→ X is a covering map, we can find an open covering Uα of

X such that for every α, π−1(Uα) is the disjoint union of open sets in X eachof which are biholomorphic to Uα. For each α, choose Wα among this disjoint



collection. Let πα be the restriction of π on Wα. Since π1(X) acts freely onX, for every i, j there exists unique γij ∈ π1(X) such that π−1

j (x) = γijπ−1i (x)

for every x ∈ Ui ∩ Uj. Uniqueness of such elements imply that γijγjk = γik.Suppose f ∈ B1

(π1(X), H0

(O∗X

)). Then there is an h ∈ H0

(O∗X

)such that

f(λ, x) = h(λx)h(x)−1. The homomorphism ψ maps f to the cocycle gijdefined by gij(x) = h(γijπ

−1i (x))h(π−1

i (x))−1 for every i, j and x ∈ Ui ∩ Uj.We can express gij = h(π−1

j )h(π−1i )−1 which then implies that gij is a Cech

1-coboundary, i.e. gij = 0 in H1 (X,O∗X). Hence, ψ(f) = 0. This provesthe claim.

All that is left to show is that ϕ is an isomorphism. To do this, let us

construct the inverse. Let L ∈ ker(

H1 (X,O∗X)π∗−→ H1

(X,O∗

X

)), i.e. π∗L

is a trivial line bundle over X. Let α : π∗L −→ X×C be a trivialization. Theaction of π1(X) on X induces automorphisms of π∗L. For every λ ∈ π1(X),via α we get an automorphism Aλ of X ×C. This automorphism must be ofthe form Aλ(x, z) = (λx, Cz) where C is a complex number holomorphicallydepending on λ and x, i.e. C = f(λ, x) for some f : π1(X) × X −→ C isholomorphic. As before, the equation Aλµ = AλAµ guarantees that f is a1-cocycle in Z1

(π1(X), H0

(O∗X

)). Suppose β : π∗L −→ X × C be another

trivialization. Then βα−1(x, z) = (x, h(x)z) where h ∈ H0(X,O∗X

). Let Bλ

be the automorphism of X × C associated to λ ∈ π1(X). Then

Bλ(x, z) = (βα−1)Bλ(βα−1)(x, z) = (λx, h(λx)f(λ, x)h−1(x)z)

This shows that the class of f in H1(π1(X), H0

(O∗X

))does not depend on

the trivialization of π∗L. This gives us a map

ker(

H1 (X,O∗X)π∗−→ H1

(X,O∗

X

))−→ H1

(π1(X), H0

(O∗X

)).

Seeing that this map is the inverse of ϕ completes the proof.

Remark. By the previous proposition, if we take X to be a complex torusthen

H1(π1(X), H0

(O∗X

)) ∼= H1 (X,O∗X) .

The previous proposition is a special case of the following result. Thisasserts the existence of maps relating the group cohomology of the funda-mental group π1(X) with values in the π1(X)-module of global sections of



the pull-back of any sheaf along the universal cover and the Cech cohomol-ogy of that sheaf. This may sound horrible but once stated precisely, onecan see the beauty behind this result. We will follow the discussion foundin Mumford [32] with a slight modernization in the language. First, let usconsider a discrete group G acting freely and properly discontinously on anice space Y (when we speak of a nice space, we mean a space in whichall the peripheral constructions are possible). Let X = Y/G. Note that Yis a covering space of X with deck transformation G. The covering map isprecisely the canonical projection p. Let F denote a sheaf of abelian groupsover X. Then we have the following result.

Proposition 5. For n > 0, there are canonical homomorphisms

Hn(G, H0 (Y, p∗F)

) φn−→ Hn (X,F)

Proof. By virtue of p being a covering map and G acting properly discontin-uously on Y , we can choose an open covering Vi of X satisfying

(1) ∀ Vi, ∃ Ui ⊆o Y such that p−1(Vi) =⋃g∈G

gUi and p∣∣Ui

: Ui'−→ Vi.

(2) ∀ i, j, there is at most one g ∈ G such that Ui ∩ gUj 6= ∅. If suchelement exists, denote this by gij.

Let us define a map in the level of group and Cech cochains.

ψn : Cn(G, H0 (Y, p∗F)

)−→ Cn (Vi ,F)

(ψnf)i0,i1,...,in = res∣∣YU Ti0

(f(gi0,i1 , gi1,i2 , ..., gin−1,in

))where U = Ui0 ∩ Ui1 ∩ ... ∩ Uin and Ti0 is the following composition

H0 (Y, p∗F)res−→ H0 (Ui0 , p

∗F)(p∗)−1

−→ H0 (Vi0 ,F)

If we denote by ∂ and ∂ the Cech and the group coboundary operators thenby a simple computation we see that ∂ψn = ψn+1∂. Thus, the maps ψndescend to maps between cohomology groups

Hn(G, H0 (Y, p∗F)

) φn−−−−→ Hn (X,F) .



Remark. 1. In the previous proposition, if we take Y to be the universalcovering of X then G = π1(X) and in this case, we get Proposition 4.

2. If the sheaf F is acyclic i.e. all the cohomology groups vanish, then themaps φn are all isomorphism. For a detailed proof of this, see Mumford[32].

Let X be a complex manifold and L a line bundle over X. As be-fore, let us denote by X the universal covering of X. Then the action of

π1(X) on X defines a natural action of π1(X) on H0(X, π∗L

)by (γf) (x) =

f(γx). Note that the functions fixed by this action are the functions con-stant along fibers of X

π−→ X. Thus, we have a natural isomorphism

H0(X, π∗L

)π1(X) ∼= H0 (X,L). If π∗L is trivial (which is the case with com-

plex tori) and α : π∗L −→ X×C is a trivialization then we have the isomor-

phism H0(X, π∗L

)∼= H0

(X, X × C

). Now, let f ∈ Z1

(π1(X), H0

(O∗X

))be the 1-cocycle associated to L with respect to the trivialization α. Then

elements of H0(X, X × C

)π1(X)

are holomorphic functions θ : X −→ Csatisfying

θ (γx) = f(γ, x)θ (x) (2)

for any γ ∈ π1(X) and x ∈ X. Hence, sections of L over X can be thoughtof as holomorphic functions on X satisfying equation (2). Such functions arecalled theta functions and they are one of the main object of discussion ofthe upcoming sections. Note that the identification of H0 (X,L) as functionssatisfying (2) obviously depend on a choice of a trivialization. However, aswe saw in the proof of Proposition 4 a different choice of a trivialization willgive us a 1-cocycle which is in the same cohomology class as f . Let us endthis part with the following remark on the complex analysis on complex tori.

Remark. It is well known in complex geometry that holomorphic functionson a compact connected complex manifold must be constant (a generalizationof Louville’s Theorem). One of the important principles of noncommutativegeometry (and algebraic geometry as well) is that a nice space is determinedby its algebra of functions. In the case of compact connected complex mani-folds, this is certainly a dead end since all of them have the same algebra offunctions. The reason behind using general line bundles over complex man-ifolds is that their sections provide us the space to work with when doing



complex analysis. The existence of non-constant holomorphic sections of linebundles is due to the fact that line bundles can be twisted.

13.1.3 Exercises

1. Show that for a complex vector space V we have H1(V,OV ) = H2(V,Z) =0.

2. Let Alt2 (Λ,Z) denote the space of alternating Z-valued 2-form on Λ.Show that the map α : Z2 (Λ,Z) −→ Alt2 (Λ,Z) defined by αF (λ, µ) =F (λ, µ)− F (µ, λ) induces an isomorphism H2(Λ,Z) ∼= Alt2 (Λ,Z).

3. In Proposition 5, show that φ1 (f)ij = f(λij, π

−1i

)and φ2 (f)ijk =

f(λij, λjk, π

−1i

). Show that in the case of a complex torus X = V/Λ,

φ1 and φ2 are isomorphisms.

13.1.4 First Chern Class, Dual Torus and the PoincareBundle

Let X = V/Λ be a complex torus. Consider the exponential sequence

0 −−−→ Z −−−→ OXexp(2πi·)−−−−−→ O∗X −−−→ 1.

Then there is a long exact sequence in cohomology

−−−→ H1 (X,Z) −−−→ H1 (OX) −−−→ H1 (O∗X)c1−−−→ H2 (X,Z) −−−→ .

Given a line bundle L ∈ H1 (O∗X), we call c1(L) ∈ H2 (X,Z) its first Chernclass. By exercise 13.1.3(2), elements of H2 (X,Z) can be thought of asalternating Z-valued 2-form on Λ. Furthermore, via a canonical isomorphismwe can express the first Chern class of L as the following alternating form

EL (λ, µ) = g (µ, v + λ) + g (λ, v)− g (λ, v + µ)− g (µ, v) (3)

for all λ, µ ∈ Λ and v ∈ V where f = e2πig is the 1-cocycle associated with Land g : Λ× V −→ C is chosen to be holomorphic. Details of this claim canbe constructed from the exercises at the end of this section.

Let us define what a dual complex torus is. Let X = V/Λ be a complex

torus. Let V denote the space of all C-antilinear maps V −→ C. Let



Λ =φ ∈ V | Im φ(λ) ∈ Z, ∀λ ∈ Λ

.

Then Λ is a lattice in V . We define the dual complex torus X of X asX = V /Λ. Note that X is a complex torus of the same dimension as X.Using a couple of identifications, one can show that is an involution.

Proposition 6. The homomorphism V −→ Hom(Λ,C1), φ 7→ exp(2πi Im φ(·))induces an isomorphism X

∼−→ Pic0(X).

We leave the proof of the above proposition to the reader. Some hints canbe found in the exercises. Proposition 6 says among other things that theline bundles in Pic0(X) are parametrized by X which is also a complex tori.In fact, these line bundles can be viewed as foliations in the direction of X ofa larger line bundle over X × X called the Poincare bundle. More precisely,a Poincare bundle P is a holomorphic line bundle over X × X satisfying thefollowing

1. P|X×L ' L for every L ∈ X,

2. P|0×X is trivial.

Poincare bundles exist and for a proof, a nice treatment is available inBirkenhake-Lange [4]. It is unique up to isomorphisms. Moreover, it isuniversal in the sense that if there is a complex manifold T and a line bundleQ over X × T satisfying

1. Q|X×t ∈ Pic0(X) for every t ∈ T ,

2. Q|0×T is trivial,

then there is a unique holomorphic map ψ : T −→ X such that Q '(id× ψ)∗P . The natural choice for the map ψ is the map t 7→ Q|X×t.Using this map, we asked the reader to prove the claim about universality.

Let X = V/Λ be a complex torus. Let NS(X) denote the image ofH1 (O∗X)

c1−→ H2 (X,Z). This is called the Neron-Severi group of X. Ele-ments ofNS(X) can be realized as Hermitian forms on V such that ImH(Λ,Λ) ⊆Z (see exercise 13.1.5 (4)). Let C1 = z ∈ C| ‖z‖ = 1. A semicharacter χfor H is a map χ : Λ× Λ −→ C1 which satisfies



χ(λ+ µ) = χ(λ)χ(µ)exp(πi Im H(λ, µ))

for all λ, µ ∈ Λ. Note that the usual C1-characters of Λ are the semicharactersof 0 ∈ NS(X). Let us denote by P(Λ) the set of all pairs (H,χ) whereH ∈ NS(X) and χ is a semicharacter of H. Note that this is a group withrespect to the following operation: (H1, χ1)(H2, χ2) = (H1+H2, χ1χ2). Given(H,χ) ∈ P , define

a(H,χ)(λ, v) = χ(λ)exp(π H(v, λ) +

π

2H(λ, λ)

). (4)

It is left to the reader to show that a(H,χ) is in Z1(Λ, H0 (O∗V )

). Let L (H,χ)

be the line bundle determined by a(H,χ). This familiar construction defines ahomomorphism P(Λ) −→ Pic(X). In fact we have the following proposition

Proposition 7. The following diagram commutes

P(Λ)

p

""

// Pic(X)

c1

NS(X)

where p is the projection onto the first factor.

Remark. It is not at all obvious that the map p is surjective. To prove this,we have to show that every H ∈ NS(X) has a semicharacter. This is left forthe reader to settle and some hints are provided at the end of this section.

The main result of this section is the following theorem which says, amongother things, that any line bundle L over X is of the form L(H,χ) for some(H,χ) ∈ P(Λ). We will not give the proof of this theorem but one mayconsult Birkenhake-Lange [4].

Theorem [Appell-Humbert]Let X = V/Λ be a complex torus and let Pic0(X) = ker c1. Then we have



the following isomorphism of exact sequences

1 // Hom (Λ,C)

'

// P (Λ)

'

// NS(X) // 0

1 // Pic0(X) // Pic(X) // NS(X) // 0

13.1.5 Exercises

1. Show that there is a homomorphism δ : H1(Λ, H0 (O∗V )

)−→ H2 (Λ,Z)

such that δf (λ, µ) = g (µ, v + λ) − g (λ+ µ, v) + g (λ, v) where f =exp (2πig). Hint: Use the exact sequence

0 −−−→ Z = H0 (V,Z) −−−→ H0 (OV )exp(2πi·)−−−−−→ H0 (O∗V ) −−−→ 1.

2. With φ1 and φ2 as in exercise 13.1.3(3), δ as above and c1 the firstChern class, show that the following commutes

H1(Λ, H0 (O∗V )

)φ1

δ //H2 (Λ,Z)

φ2

H1 (O∗X)c1 // H2 (X,Z)

3. Using the above exercises, prove that the first Chern class of a linebundle can be identified with the alternating form defined in equation(3).

4. Show that the Neron-Severi groupNS(X) can be identified with the ad-ditive group of Hermitian formsH : V×V −→ C such that ImH(Λ,Λ) ⊆Z.

5. Show that a(H,χ) as defined in (4) satisfies the cocycle condition.

6. Show that to every H ∈ NS(X) has a semicharacter. Hint: Choosea line bundle whose first Chern class is H then choose an associated2-cocycle for this line bundle.



7. Let us define 〈, 〉 : V ×V −→ R by 〈φ, v〉 = Im φ(v). Show that this isnondegenerate. using this, prove Proposition 6.

8. A homomorphism Xf−→ Y between complex tori induces a homomor-

phism Yf−→ X between the dual complex tori. Show that is an

involution and (contravariantly) functorial, i.e.

(a)X = X,

f = f ,

(b) if X1f−→ X2 and X2

g−→ X3 are homomorphisms of complex torithen

gf = f g.

(c) and idX = idX .

9. Show that preserves exactness of sequences (although in reverse order)i.e. is an exact functor.

13.1.6 Type of a Line Bundle

Let X = V/Λ be a complex tori of dimension g and L a line bundle over Xwith first Chern class H. The imaginary part Im H of the Hermitian formH is an alternating integer-valued 2-form on Λ. By [30], we can find a basisfor Λ such that E is represented by the matrix D

−D

(12)

where D = diag (d1, d2, ..., dg) with di positive integers such that di|di+1. Theintegers d1, d2, ..., dg are uniquely determined by E and Λ and hence, by L.These integers (or D itself) will be refered to as the type of the line bundleL.

We say the a line bundle is nondegenerate ifH is nondegenerate. Likewise,we will say that a line bundle is positive-definite if its associated Hermitianform H is positive-definite. Another term widely used by algebraic geometersfor a positive-definite line bundle is ample. A polarization of X is the firstChern class c1(L) of a nondegenerate line bundle L over X. The type ofpolarization will refer to the type of L. A polarization of type (1, 1, ..., 1)



is called principal. By an abelian variety X, we mean a polarized complextorus and the pair (X,L) will be referred to as a polarized abelian variety.A homomorphism of polarized abelian varieties f : (X,L) −→ (Y,M) is aholomorphic Lie group homomorphism f : X −→ Y such that f ∗c1(M) =c1(L).

Remark. A complex subtorus of a complex torus X = V/Λ is a quotientW/Λ

′such that Λ

′6 Λ is a sublattice of even rank contained in the vector

subspace W of V . A complex subtorus of an abelian variety is an abelianvariety.

Let us end this section by defining several notions of equivalence for linebundles. The first one is the usual equivalence of line bundles but to distin-guish it from the next two, we will call it topological equivalence althoughthis must not be mistaken form homeomorphism. Two line bundles L andM over X are said to be topologically equivalent if there is a homeomorphism

Lφ−→M for which the following diagram commutes

L

φ//M

Xid //X

.

Note that consequently, the map φ sends a fibre over a point to itself. Wefurther require that such restriction is C-linear. We denote this equivalencebetween L and M as L 'M .

Before we define the next equivalences, let us define several relevant struc-tures. By an analytic space, we mean a locally ringed space (X,OX) suchthat every point has a neighborhood isomorphic to an analytic variety aslocally ringed spaces.

The second equivalence is called analytic equivalence. We say that the linebundles L and M are analytically equivalent if there is a connected analyticspace T , a line bundle Σ on X × T and points l,m ∈ T such that

Σ∣∣∣∣X×l

' L andΣ∣∣∣∣X×m

'M. (8)

Proposition 8. The following are equivalent.



(a) L and M are analytically equivalent,

(b) L⊗M−1 ∈ Pic0(X),

(c) c1(L) = c1(M).

Proof. (b ⇒ a) Suppose L ⊗ M ∈ Pic0(X). Consider the line bundle

Q = P⊗p∗M over X×X where P is the Poincare bundle and X×X p−→ Xis the projection onto the first factor. Note that Q|X×L⊗M−1 ' L andQ|X×0 'M . This shows that L and M are analytically equivalent.

(a⇒ c) Let L and M be analytically equivalent line bundles. Let Σ be theline bundle over X × T giving the equivalence. Define the map

T −→ H (X,Z) , t 7→ c1(Σ|X×t).

Note that this map is continuous. Since T is connected and ˇX,Z is discrete,the map above is constant. Thus, c1(L) = c1(M).

(c⇒ b) This is immediate from the fact that ker c1 = Pic0(X).

The third notion of equivalence is algebraic equivalence. The definitionis almost the same except that we require the space T to be algebraic in-stead of being analytic. We will not dwell on this equivalence deeper butit is important to note that algebraic equivalence is stronger than analyticequivalence.

13.2 Theta and Heisenberg Groups

13.2.1 Theta Groups

Let X be an abelian variety and L a line bundle over X. Let x ∈ X. Anautomorphism of L over x is a biholomorphic map φ : L −→ L such that:



(1) the following diagram commutes

L

φ// L

Xtx //X

where tx is the translation of X by x,

(2) the map φy : Ly −→ Lx+y is C-linear for every y ∈ X.

We call the set of all automorphisms of L over points of X as the thetagroup of L, denoted by θ(L). The theta group of L is indeed a group underthe following operation

(ϕ1, x1) (ϕ2, x2) = (ϕ1ϕ2, x1 + x2)

for any (ϕ1, x1) , (ϕ2, x2) ∈ θ(L). Let us denote by K(L) the subgroup ofX consisting of points x such that t∗xL ' L. Then we have the followingproposition.

Proposition 9. The following sequence is an exact sequence

1 −−−−→ C∗ i−−−−→ θ(L)p−−−−→ K(L) −−−−→ 0

where i(ζ) = (ζ·, 0) and p(φ, x) = x.

Proof. Let (ϕ, x) ∈ θ(L). By definition, t∗xL is the fiber product of the

L −→ X and Xtx−→ X. Hence, there is a unique isomorphism ϕ of line

bundles

L

ϕ

##

ϕ

t∗xL

// L

Xtx // X



Thus, t∗xL ' L which implies that x ∈ K(L). Thus, the map p is well-defined.Let x ∈ K(L). Then there is an isomorphism ϕ : L −→ t∗xL over X. Themap

Lϕ−→ t∗xL = X ×X L

proj−→ L

is an automorphism of L over x. This shows that p is surjective. Let ψ :L −→ L be an automorphism of L over X. By definition, the followingdiagram commutes

L

ψ// L

Xid //X

and that ψx : Lx −→ Lx is C-linear. Note that Lx ∼= C and hence, ψx is justmultiplication by a nonzero scalar ζx. This gives us a holomorphic functionX −→ C given by x 7→ ζx. Since X is a compact complex manifold, this mapmust be constant. Since t∗0 = id, this shows that the kernel of p is preciselythe image of i. This concludes the proof.

Remark.

(1) Any automorphism of L over points of X commutes with (z·, 0) whichmeans that θ(L) is a central extension of K(L) by C∗.

(2) If we let θ(L) be the set of all bundle isomorphism L −→ t∗xL over Xthen according to the proof of Proposition 9, there is a bijection betweenθ(L) and θ(L). The bijection induces a group structure on the latter by

ϕ1 · ϕ2 = (t∗x2ϕ1)ϕ2

where Lϕi−→ t∗xi ∈ θ(L) for i = 1, 2.

Let Vπ−→ X be the canonical projection. Note that π∗L is the trivial line

bundle over V . Let Λ(L) = π−1(K(L)). Assume L = L(H,χ). For α ∈ C∗and w ∈ Λ(L), define the map ψα,w : V × C −→ V × C by



ψα,w(v, t) = (v + w, α exp(πH(v, w))t),

which is an automorphism of V ×C over w ∈ Λ(L) 6 V (see exercise 1.2.2(1)).Let Ψ(L) denote the set of all these automorphisms. This is a group withthe following group multiplication

ψα,w · ψβ,u = ψαβexp(πH(u,w)),w+u.

Then we have the following exact sequence

1 −−−−→ C∗ j−−−−→ Ψ(L)q−−−−→ Λ(L) −−−−→ 0 (5)

with j(ζ) = ψζ,0 and q(ψα,w) = w. Similarly, Ψ(L) is a central extension ofΛ(L) by C∗. Consider the associated 2-cocycle aL to L = L(H,χ) as definedin (4). The map

sL : Λ −→ Ψ(L), sL(λ) = ψaL(λ,0),λ (6)

is a injective section of q over Λ. Using exercise 1.2.2(4), we can show thatsL(Λ) is inside the center of Ψ(L). This implies that from the exact sequence(5) we have the following exact sequence

1 −−−−→ C∗ −−−−→ Ψ(L)/sL(Λ) −−−−→ Λ(L)/Λ −−−−→ 0 (7)

Theorem 4. The following is an isomorphism of exact sequences

1 // C∗ // Ψ(L)/sL(Λ)

'

// Λ(L)/Λ // 0

1 // C∗ // θ(L) // K(L) // 0

We encourage the reader to chase diagrams to prove the isomorphism inTheorem 4. Note that the sequence (5) is just the pull-back of the sequencein Proposition 9 via the map Λ −→ K(L). Note that Ψ(L) depend only onthe polarization H of X while θ(L) depend on a particular line bundle in thealgebraic equivalence induced by the polarization H.



13.2.2 Exercises

1. For α ∈ C∗ and w ∈ Λ(L), show that the map ψα,w : V ×C −→ V ×Cis an automorphism of V × C over w ∈ Λ(L) 6 V .

2. Show that the sequence in (5) is an exact sequence and that it definesa central extension for Λ(L).

3. Show that the map sL as defined in (6) is an injective group homomor-phism.

4. Let L = L(H,χ). Show that Λ(L) = v ∈ V | Im H(v,Λ) ⊆ Z.

5. Show that line bundles L and M are analytically equivalent if and onlyif there is a point x ∈ X such that L ' t∗xM .

6. Define a map σ : Ψ(L) −→ θ(L) such that the kernel-reduced mapσ′: Ψ(L)/sL(Λ) −→ θ(L) is the map in the diagram in Theorem 4.

13.2.3 The Commutator Map

From the previous section, we saw that θ(L) and Ψ(L) are central exten-sions of K(L) and Λ(L), respectively by C∗. In other words, we may writeθ(L)/C∗ = K(L) and Ψ(L)/C∗ = Λ(L). Given x, y ∈ K(L), let us chooseliftings x1, y1 ∈ θ(L). Then x1y1x

−11 y−1

1 is in C∗. To see this, x1, y1 commutesin θ(L) since K(L) is abelian. This means that x1y1x

−11 y−1

1 represents the 0element in θ(L)/C∗ which is C∗. This shows that x1y1x

−11 y−1

1 ∈ C∗. Now, letus show that the value of x1y1x

−11 y−1

1 is independent of the chosen liftings.But this is immediate since any other liftings of x and y are just non-zeromultiple of x1 and y1 and hence, will give us the same commutator. Let usdefine eL(x, y) to be this well-defined value, i.e.

eL : K(L)×K(L) −→ C∗, eL(x, y) = x1y1x−11 y−1

1

We can also define a similar map Λ(L)× Λ(L) −→ C∗, which we denoteby the same symbol eL by eL(x, y) = eL(π(x), π(y)) for any x, y ∈ Λ(L).

Proposition 10. Let H represent the first Chern class of L. Then for anyu, v ∈ Λ we have

eL(u, v) = e−2πi Im H(u,v).



Proof.

eL(u, v) = ψ(α,u)ψ(β,v)ψ( 1αexp(π H(u,u)),−u)ψ( 1

βexp(π H(v,v)),−v) = ψ(exp(−2π ImH(u,v)),0)

Remark.

(1) From the previous proposition, we have

(a) eL(x+ y, z) = eL(x, z)eL(y, z)

(b) eL(x, y) = eL(y, x)−1

(c) eL(x, x) = 1

i.e. eL is a multiplicative alternating form.

(2) By the previous proposition together with the Appell-Humbert theorem,we have

(a) eL⊗M = eL · eM on K(L) ∩K(M),

(b) if L and M are algebraically equivalent then eL = eM .

13.2.4 Heisenberg Groups

Let H ∈ NS(X) be a polarization of type D = diag(d1, d2, ..., dn). LetH(D) = C∗ × K(D) where K(D) = Zn/DZn ⊕ Zn/DZn. Let us define agroup structure onH(D) as follows. Let z1, ..., z2n be the standard generatorsfor K(D). Define

eD(zi, zj) =

exp(−2πi

di) j = n+ i

exp(2πidi

) i = n+ j

1 otherwise

(9)

Now, for any (α, x, u) , (β, y, w) ∈ H(D), define

(α, x, u) · (β, y, w) =(αβ eD(x,w), x+ y, u+ w

)(10)

We ask the reader to prove that this defines a group structure on H(D).Moreover, we have the following result.



Proposition 11. The following sequence is exact.

1 −−−−→ C∗ i−−−−→ H(D)p−−−−→ K(L) −−−−→ 0 (11)

where i(α) = (α, 0, 0) and p(α, x, y) = (x, y).

Let PicH(X) denote the class of line bundles with first Chern characterH, i.e. analytically equivalent line bundles of polarization H. We will seethat the theta groups θ(L) of such line bundles are all isomorphic to theHeisenberg group H(D) in a strong way. Before showing this, recall fromsection 13.1.6 that we can find a basis for Λ such that the alternating formE representing the first Chern character is of the form (12). Let this basis be

λ1, ..., λn, µ1, ..., µn. Then 1d1λ1, ...,

1dnλn,

1d1µ1, ...,

1dnµn constitutes a generat-

ing set for K(L). The following then defines an isomorphism between K(L)and K(D).

K(L)b−→ K(D), b(

1

diλi) = zi, b(

1

diµi) = zn+i (13)

Moreover, the map (13) turns out to be a symplectic isomorphism in thesense that b∗eD = eL. Hints can be found in the following set of exercises toprove this claim. Now, we have a commutative diagram

1 // C∗ // θ(L) // K(L)

b

// 0

1 // C∗ // H(D) // K(D) // 0

(14)

Now, if we can find a group homomorphism θ(L)σ−→ H(D) completing the

commutative diagram (14), by the short five lemma we see that σ must bean isomorphism. We will work this is out in the sequel.

Let L = L(H,χ) and let E be the corresponding alternating 2-form.Consider a decomposition Λ = Λ1 ⊕ Λ2 into isotropic subgroups relativeto E. We may take for example Λ1 = 〈λ1, ..., λn〉 and Λ2 = 〈µ1, ..., µn〉.Decompose V accordingly, V = V1 ⊕ V2. Define

χ0(v) = eπi E(v1,v2) (15)



where v = v1 + v2 according to the decomposition above. Then we have

χ0(v + w) = χ0(v)χ1(w)eπi E(v,w)e−2πi E(v2,w1)

which implies that χ0|Λ is a semicharacter for H. Let L0 = L(H,χ0). It canbe shown that there exists a point c ∈ V such that any L ∈ PicH(X) is thepullback of L0 along the translation tc. See exercise 13.2.5(4). Now, defineac : Λ(L) −→ C∗ as

ac(w) = χ0(w)e2πi Im H(c,w)+π2H(w,w).

Define a map Ψ(L)Bc−→ H(D) as follows

Bcψα,w =(α ac(w)−1, b(w1), b(w2)

). (16)

This map is a group homomorphism that factors through θ(L). This pro-duces the desired map σ. Thus, we have the following isomorphism of exactsequences

1 // C∗ // θ(L)

σ

// K(L)

b

// 0

1 // C∗ // H(D) // K(D) // 0

(17)

13.2.5 Exercises

1. Show that the map defined in (9) is a 2-cocycle in K(D) with valuesin C1. Use this to show that the operation defined on (10) defines agroup structure on H(D).

2. Show that for any (α, x1, x2) , (β, y1, y2) ∈ H(D) we have

(α, x1, x2) (β, y1, y2) (α, x1, x2)−1 (β, y1, y2)−1 =(eD (x1 + x2, y1 + y2) , 0, 0

).

Use this to show that b∗eD = eL.



3. Using 13.2.2(5), show that for any L ∈ PicH(X) we can find a pointc ∈ V unique up to translation by points of Λ such that L = t∗cL0 whereL0 is the line bundle defined by (15).

4. Show that the map (16) defines a group homomorphism that factorsthrouh θ(L).

13.2.6 Theta Functions and Schrodinger Representa-tion

Let X = V/Λ be an abelian variety and L = L(H,χ) a line bundle overX. Let s ∈ H0(L) be a section and let (φ, x) ∈ θ(L). From the followingcommutative diagram,

Lφ

// L

X

s

OO

tx //X

φst−x

OO

we see that φst−x ∈ H0(L). This defines an action of θ(L) on H0(L)

θ(L)ρ−→ GL(H0(L)).

Note that C∗ acts on H0(L) by scalar multiplication. This means that therein an induced projective representation

K(L)p−→ PGL(H0(L)).

such that the following commutes

1 // C∗ // θ(L)

ρ

//K(L)

p

// 0

1 // C∗ // GL(H0(L)) // PGL(H0(L)) // 1

.



The canonical map σ : Ψ(L) −→ θ(L) induces a representation of Ψ(L) onH0(L) by composing with ρ, call this ρ

′.

Theorem 5. The representations ρ and ρ′

are irreducible.

Let (X,H) be a polarized abelian variety of typr D = diag(d1, ..., dn).Let C(Zn/DZn) be the group-algebra of the finite group G = Zn/DZn. Thedelta functions δg, g ∈ G defines a basis for C(Zn/DZn). The Heisenberggroup H(D) acts on C(Zn/DZn) by

(α, x, y) γ(g) = α eD(g, y)γ(g + x)

for any (α, x, y) ∈ H(D) and γ ∈ C(Zn/DZn). This action induces a repre-sentation

ρs : H(D) −→ GL(C(Zn/DZn)),

called the Schrodinger representation.

Theorem 6. There is an isomorphism H0(L)β−→ C(Zn/DZn) such that the

following diagram commutes

θ(L)× H0(L)

(σ,β)

// H0(L)

β

H(D)× C(Zn/DZn) // C(Zn/DZn)

where σ is the isomorphism in (17).

The previous theorem says that the representation of the theta group ontheta functions is equivalent to the Schrodinger representation.

13.3 Theta Functions on Noncommutative Tori

13.3.1 Theta Vectors

Let θ be a skew-symmetric n × n-matrix. Consider the noncommutatived-torus T dθ as the space of operators of the form



f =∑n∈Zd

fnUn

where fn ∈ S(Zd) and the operators Un constitutes a full set of lineargenerators satisfying UnUm = eπinθmUn+m. We can simplify this descriptionby considering multiplicative unitary generators U1, ..., Ud satisfying

UiUj = e2πiθijUjUi. (18)

Note that specifying a particular set of unitary operators U1, ..., Ud satisfyingequation (18) over some Hilbert space H gives us a T dθ -module. Let usconsider an irreducible representation of the canonical commutation relation

[∇i,∇j] = 2πiωij (19)

on some Hilbert space where additionally, we require the ∇i’s to be skew-Hermitian operators. Here, (ωij) is a nondegenerate skew-symmetric matrix.For simplicity, we assume θ to be nondegenerate amd that θ−1 = ω. Sinceθ is skew-symmetric, d = 2m. Take for example, the case when ω is thefollowing skew-symmetric matrix I

−I

.

With this, we may realize the canonical commutation relation as

[∇i,∇i+m] = − [∇i+m,∇i] = 2πi for 0 6 i 6 m

[∇i,∇j] = 0 otherwise,

which has a familiar representation by operators on S(Rm), the Schwarz spaceon Rm and whose variables are denoted by x1, ..., xm. The representation isas follows: consider the operators ∇i defined as

∇i =

2πi xi· 1 6 i 6 m∂i−m m < i 6 2m

where xi· and ∂i refers to the operators that multiply by xi and differen-tiates with respect to xi, respectively. Integrating these operators, we get



the following operators Ui = e−∑jθij∇j

. It is easy to check that equations(18) are satisfied. One can also check that we have the following relations[∇i, Uj] = 2πiδijUj which implies that

∇α (f · e) = f · ∇αe+ δαf · e

for any f =∑n∈Zd

fnUn ∈ T dθ and e ∈ S(Rm). Here, δα denotes the differential

of the translation of Rd along α. Thus, ∇1, ...,∇d determines a connection∇ by

∇

(∑i

aixi, e

)= ai

∑i

∇ie.

The derivations δ1, ..., δd span an abelian Lie algebra L and this acts on T dθby δαUβ = 2πiδαβUα where the latter δ in the previous equation refers to theKroneker delta. By the defining commutation relation for∇i, we immediatelysee that the connection ∇ has constant curvature [∇,∇] = 2πi.

Consider the complexification L = L⊕ iL of the Lie algebra L. Note thatL is a complex abelian Lie algebra. A complex structure on L is a decompo-sition of L into conjugate subspaces L1,0 and L0,1, i.e. L1,0 = L0,1. We mayview L0,1 as the imaginary subspace. This subspace is of real dimension d.Let δ1, ..., δd be a basis for L0,1. The basic vectors δ1, ..., δd can be written interms of δ1, ..., δd via a d×d-complex matrix (hij). Now, a complex structureon a general T dθ -module E is a collection of commuting linearly independentT dθ -derivations ∇1, ..., ∇d on E , i.e.

∇α (f · e) = f · ∇αe+ δαf · e

and[∇i, ∇j

]= 0 for all 0 6 i, j 6 d.

Likewise, these derivations define a connection ∇ on E which is flat. Such aconnection is called a ∂-connection. An element e ∈ E is said to be holomor-phic if ∇Xe = 0 for any X ∈ L0,1.

Remark. We have constructed a T dθ -module structure on S(Rm) equippedwith a connection ∇ with constant curvature. Our definition of a com-plex structure on a general T dθ -module E requires the existence of a flat ∂-connection on E. One may ask, how can we construct a flat connection on



S(Rm) given the information at hand. From the connection ∇ = (∇i) andthe derivations δi = hijδj, we can construct

∇i =∑j

hij∇j, (20)

a ∂-connection. Now, the question is: when is ∇ flat? We will address thisin what follows.

We can consider L = L ⊕ iL ' Cd to be a symplectic manifold withsymplectic structure induced by ω. A Lagrangian submanifold of a symplecticmanifold is one in which the symplectic form vanishes. If the subspace L0,1

is a Lagrangian submanifold of L then the connection defined in (20) is flat.We ask the reader to prove this using several hints provided in the exercisesbelow. In the case of the T dθ -module S(Rm), we can construct a ∂-connection∇ associated to any Lagrangian submanifold Ω in the same fashion. Wewill denote by SΩ the module S(Rm) equipped with the complex structureinduced by the Lagrangian submanifold Ω.

The symplectic form ω determines a sesquilinear pairing (v, u) =∑i

viωijuj.

A subspace Ω of L is said to be positive if Im (v, v) > 0 for all 0 6= v ∈ Ω.We are now ready to prove the following:

Theorem [Schwarz, 2001]A holomorphic vector exists on SΩ if and only if Ω is a positive Lagrangiansubmanifold.

Proof. To make things simple, let us consider the standard symplectic struc-ture on L determined by the basis vectors x1, ..., xm, p1, ..., pm. Let Ω be aLagrangian submanifold of L. Since Ω is of dimension m, it must project sur-jectively onto the linear span ofm of the vectors x1, ..., xm, p1, ..., pm. Withoutloss of generality, assume these are p1, ..., pm. In this case, Ω is given by thefollowing system of equation

pi + Ωijxj = 0, i = 1, ...,m (21)

where Ωij is a complex matrix. A function τ (x1, ..., xm) is holomorphic if itsatisfies the following equation(

∂

∂xi+ 2πiΩijxj

)τ = (∇i + Ωij∇j) τ = 0



Solving these differential equations we see that

τ (x1, ..., xm) = e−2πixiΩijxj (22)

From exercise 13.3.2(2), we see that if the Lagrangian submanifold Ω is pos-itive then the imaginary part of (Ωij) is positive which then would implythat equation (22) defines a Schwarz function. Thus, τ ∈ SΩ Using the sameexercise, we see that if Ω is not positive then (22) does not define a functionon S(Rm), i.e. no holomorphic τ belongs to SΩ.

Remark. Note that from the proof of the above theorem, we see that if such aholomorphic vector exists, it must be unique up to a constant multiple. Thisfollows from the fact the the solution of the differential equation we solveabove has a unique solution up to a constant multiple. For the remainder ofthis section, we will denote this holomorphic vector τ by ϑΩ,θ. This is whatSchwarz’ regards as the analogue of theta functions on the noncommutativetorus.

13.3.2 Exercises

1. Show that if Ω = 〈δ1, ..., δd〉 is a Lagrangian submanifold of L then[∇i,∇j] = 0. Hint: derive a useful relation involving the matrix (hij)and the skew-symmetric matrix (ωij).

2. Show that the Lagrangian submanifold defined by equation (21) is pos-itive if and only the imaginary part of (Ωij) is positive. Hint: computeIm (pi, pi) for all i.

13.4 Dual Noncommutative Torus

13.5 Vanishing Theorems


Chapter 14

Discrete Noncommutative Tori

Appendix A

Bibliography

[1] A. Atanasov, Algebraic Stacks, 2011.http://www.math.harvard.edu/ nasko/documents/stacks.pdf.

[2] D. Auroux, A Beginner’s Introduction to Fukaya Categories, 2013.http://arxiv.org/pdf/1301.7056v1.pdf.

[3] C. Birkenhake and H. Lange, Complex tori, vol. 177 of Progress inMathematics, Birkhauser Boston, Inc., Boston, MA, 1999.

[4] , Complex abelian varieties, vol. 302 of Grundlehren der Mathema-tischen Wissenschaften [Fundamental Principles of Mathematical Sci-ences], Springer-Verlag, Berlin, second ed., 2004.

[5] B. Blackadar, Operator algebras, vol. 122 of Encyclopaedia of Math-ematical Sciences, Springer-Verlag, Berlin, 2006. Theory of C∗-algebrasand von Neumann algebras, Operator Algebras and Non-commutativeGeometry, III.

[6] P. Bressler and Y. Soibelman, Mirror Symmetry and DeformationQuantization, 2008. http://arxiv.org/abs/hep-th/0202128.

[7] A. Connes, C*-algebres et geometrie differentielle, C. R. Acad. Sci.Paris Ser. A-B, 290 (1980), pp. 599–604.

[8] A. Connes and M. Rieffel, Yang-mills for non-commutative two-tori, Proceedings of the Conf. on Operator Algebras and MathematicalPhysics, University of Iowa, 62 (1987), pp. 237–266.

73

[9] D. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry,American Mathematical Society, US, 1999.

[10] F. D’Andrea, G. Fiore, and D. Franco, Modulesover the Noncommutative Torus and Elliptic Curves, 2013.http://arxiv.org/abs/1307.6802.

[11] B. I. Dundas, M. Levine, P. A. Østvær, O. Rondigs, andV. Voevodsky, Motivic Homotopy Theory. Lectures at a SummerSchool in Nordfjordeid, Norway, Springer, Berlin, 2007.

[12] S. Echterhopf, R. Nest, and H. Oyono-oyono, Principal Non-commutative Torus Bundles, 2007. http://arxiv.org/abs/0810.0111.

[13] K. Fukaya, Y. G. Oh, K. Ono, and G. Tian, Symplectic Geometryand Mirror Symmetry, World Scientific Publishing Co., Singapore, 2001.

[14] D. Goswami, Quantum Group of Isometries in Classical and Noncom-mutative Geometry, 2007. http://arxiv.org/abs/0704.0041.

[15] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and AppliedMathematics.

[16] R. Hartshorne, Algebraic Geometry, Springer, US, 1977.

[17] P. J. Hilton and U. Stammbach, A Course on Homological Algebra,2nd ed., Springer-Verlag, New York, 1996.

[18] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas,C. Vafa, R. Vakil, and E. Zaslow, Mirror Symmetry, AmericanMathematical Society, Cambridge, MA, 2003.

[19] R. Howe, On the role of the heisenberg group in harmonic analysis,Bulletin of the American Mathematical Society, 3 (1980), pp. 821–843.

[20] D. Huybrechts, Complex Geometry, Springer, Germany, 2005.

[21] H. Kajiura, Noncommutative tori and mirror symmetry, Research In-stitute for Mathematical Sciences, 1587 (2008), pp. 27–72.

[22] A. Kapustin, K.-G. Schlesinger, and M. Kreuzer, HomologicalMirror Symmetry, Springer, Germany, 2009.

[23] M. Khalkhali, Basic Noncommutative Geometry, European Mathe-matical Society Publishing House, Zurich, 2010.

[24] E. Kim and H. Kim, A topological mirror symmetry on noncommu-tative complex two-tori, Journal of Korean Mathematical Society, 43(2006), pp. 951–965.

[25] A. Kirillov, Elements of the Theory of Representations, Springer, Ger-many, 1972.

[26] A. A. Kirillov, Lectures on the orbit method, vol. 64 of GraduateStudies in Mathematics, American Mathematical Society, Providence,RI, 2004.

[27] M. Kontsevich, Homological algebra of mirror symmetry, Proceedingsdu Congres International de Zurich, 1 (1995), pp. 101–139.

[28] M. Kontsevich, Deformation Quantization of Poisson Manifolds I,1997. http://arxiv.org/abs/q-alg/9709040.

[29] J.-L. Loday, Cyclic Homology, Springer-Verlag, Berlin, 1991.

[30] S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann,Skew-symmetric matrix polynomials and their Smith forms, 2012.http://eprints.ma.man.ac.uk/1849/.

[31] M.Rieffel, The cancellation theorem for projective modules over ir-rational rotation c*-algebras, Proc. London Math. Soc. (3), 47 (1983),pp. 205–302.

[32] D. Mumford, Abelian Varieties, Oxford University Press, London,1970.

[33] D. Mumford, Tata lectures on theta. III, Modern Birkhauser Clas-sics, Birkhauser Boston, Inc., Boston, MA, 2007. With collaboration ofMadhav Nori and Peter Norman, Reprint of the 1991 original.

[34] M. Pimsner and D. Voiculescu, Exact sequences for k-groups andext-groups of certain cross-product c*-algebras, Journal of Operator The-ory, 4 (1980), pp. 93–118.

[35] A. Polishchuk, Classification of Holomorphic Vector Bundles on Non-commutative Two-Tori, 2003. http://arxiv.org/abs/math/0308136.

[36] A. Polishchuk, Noncommutative two-tori with real multiplication asnoncommutative projective varieties, Journal of Geometry and Physics,50 (2003), pp. 162–187.

[37] A. Polishchuk, Quasicoherent Sheaves on Complex NoncommutativeTwo-Tori, 2006. http://arxiv.org/abs/math/0506571.

[38] A. Polishchuk and A. Schwarz, Categories of Holo-morphic Vector Bundles on Noncommutative Two-Tori, 2002.http://arxiv.org/abs/math/0211262.

[39] M. Rieffel, Projective modules over higher-dimensional noncommuta-tive tori, Canadian Journal of Mathematics, 40 (1987), pp. (2)257–338.

[40] A. Rosenberg, Noncommutative Algebraic Geometry and Representa-tions of Quantized Algebras, Kluwer Academic Publishers, The Nether-lands, 1995.

[41] J. Rosenberg, Rigidity of K-theory under Deformation Quantization,1996. http://arxiv.org/abs/q-alg/9607021.

[42] A. Schwarz, Theta functions on noncommutative tori, Letters inMathematical Physics, 58 (2001), pp. 81–90.

[43] S. Shnider and S. Sternberg, Quantum Groups, International PressInc., Hong Kong, 1993.

[44] Y. Soibelman, Quantum Tori, Mirror Symmetry and DeformationTheory, 2000. http://arxiv.org/pdf/0011162v3.pdf.

[45] , Mirror symmetry and noncommutative geometry of a-infinity-categories, Journal of Mathematical Physics, 45(10) (2004), pp. 3742–3757.

[46] B. Tennison, Sheaf Theory, Cambridge University Press, US, 1975.

[47] R. Thomas, The Geometry of Mirror Symmetry, 2005.http://arxiv.org/pdf/0512412v1.pdf.

[48] J. J. Venselaar, Classification of Spin Structures on the Noncommu-tative n-Torus, 2011. http://arxiv.org/pdf/1003.5156v4.pdf.

[49] P. Woit, Topics in Represntation Theory: The Moment Map and theOrbit Method. http://www.math.columbia.edu/ woit/notes23.pdf.

[50] K. Wong, Introduction to Symplectic Geometry, 2007.http://www.math.cornell.edu/ kywong/report3.pdf.

noncommutative tori (writing in progress) · contents 1 introduction 7 2 operator algebras 9 2.1...

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