Complexified Symplectomorphisms and Geometric Quantization Master

Complexiﬁed Symplectomorphisms and Geometric Quantization

Miguel Barbosa Pereira

Thesis to obtain the Master of Science Degree in Master in Mathematics and Applications

Supervisor(s): Prof. João Luís Pimentel Nunes

Examination Committee Chairperson: Prof. Miguel Tribolet de Abreu Supervisor: Prof. João Pimentel Nunes Member of the Committee: Prof. José Manuel Vergueiro Monteiro Cidade Mourão Member of the Committee: Prof. Sílvia Nogueira da Rocha Ravasco dos Anjos

November 2017 ii This thesis is dedicated to my parents and to my brother.

iii iv Acknowledgments

I would like to thank my advisor, Prof. Joao˜ Nunes, for all the time he took helping me, from my ﬁrst semester as a mathematics student in Riemannian geometry, up to now in my thesis. He was always available for help and was always extremely patient. I also want to express my gratitude to all the teachers of the courses I took during these last two years, for all the knowledge they gave me in class, and for helping me when I had questions about the material outside of the class. This thesis started as a project for the Centre for Mathematical Analysis, Geometry, and Dynamical Systems, that awarded me a scholarship one year and a half ago. So I would like to thank CAMGSD for this opportunity to learn and do math in a way that is closer to the professional level. I also want to thank the members of the Examination Committee, Prof. Miguel Abreu, Prof. Jose´ Mourao,˜ and Prof. S´ılvia Anjos for their time reading and evaluating my thesis, and for their helpful corrections. During the time I was in Lisbon, I had the opportunity to meet new people and spend time with many excellent friends, who enriched my life. They would be too many to list, but I want to thank my friends from aerospace engineering, especially Tiago Silva, Manuel Barjona and Telmo Pires who encouraged me to become a better student, my neighbors from B1 for making it the best ﬂoor in RDP, as I do want to thank my friends from my class Z in Braga for the past 9 years. A special mention goes to Andre´ Pereira, for helping me in many occasions. Coming to study to a different city was not an easy experience for me, and I couldn’t have done it without the continuous support of my parents. When I decided to change from engineering to maths, they gave me complete freedom to do so. They have always been there for me. A big thank you, Mom and Dad. I want to thank my brother Pedro for all the memes and for putting up with my explanations of the math I was learning. Pedro, every aspiring physicist should know what a cotangent bundle is. Thank you to all the other members of my family, uncles, and cousins for caring for me. A special thank you to my grandmother Maria, for all the meals she cooked for me, but also for being a caring, strong and inspiring person. Finally, if you are reading this, I want to thank you, and I hope that the text is clear, organized and I hope that it is helpful to your studies.

v vi Resumo

Nesta tese estudamos geometria complexa, simplectica´ e Kahler.¨ O cap´ıtulo1 e´ introdutorio´ e apresentamos os preliminares teoricos´ relevantes para o resto da tese. No cap´ıtulo2 apresentamos um resumo de [14]. Partindo de uma variedade K ahler,¨ atuamos na sua estrutura complexa via pullback pelo fluxo de um campo vetorial Hamiltoniano. Provamos que as metricas´ Kahler¨ resultantes formam uma curva numa variedade Riemanniana (de dimensao˜ infinita) cujos pontos sao˜ potenciais Kahler,¨ e que esta curva e´ na realidade uma geodesica´ relativamente a` metrica´ (que e´ a metrica´ de Mabuchi). No cap´ıtulo3 estudamos estruturas K ahler¨ no fibrado cotangente de um grupo de Lie G. Explicamos as estruturas Kahler¨ conhecidas de Hall e Kirwin e de Kirwin, Mourao˜ e Nunes, e relacionamos as duas. Em [12] os autores provam que e´ poss´ıvel definir estruturas Kahler¨ em T ∗G usando funçoes˜ complexificadoras Ad-invariantes. Provamos que a condiçao˜ de Ad-invarianciaˆ pode ser substitu´ıda por um conjunto de equaçoes˜ mais geral. No cap´ıtulo5, damos um exemplo de quantizaç ao˜ geometrica´ de T ∗G. Introduzimos um fibrado de ∗ linha L em T G e uma polarizaçao˜ Pτ definida por fazer um pullback da polarizaçao˜ vertical pelo fluxo complexificado de um campo vetorial Hamiltoniano. A partir destas estruturas, definimos um espaço de

Hilbert cujos elementos sao˜ as secc¸oes˜ integraveis´ e Pτ -polarizadas de L.

Palavras-chave: geometria Kahler,¨ quantizaçao˜ geometrica,´ metrica´ de Mabuchi, grupo de Lie, fibrado cotangente, difeomorfismos complexificados

vii viii Abstract

In this thesis, we study complex, symplectic and Kahler¨ geometry. Chapter1 is introductory and presents the relevant theoretical background for the rest of the thesis. In chapter2 we present an overview of [14]. Starting with a K ahler¨ manifold, we act on its complex structure by pulling back along the complexified flow of an Hamiltonian vector field. We prove that the resulting Kahler¨ metrics form a curve in an (infinite dimensional) Riemannian manifold whose points are Kahler¨ potentials and that this curve is actually a geodesic with respect to the metric (which is the Mabuchi metric). In chapter3 we study K ahler¨ structures on the cotangent bundle of a Lie group G. We explain the known Kahler¨ structures of Hall and Kirwin and of Kirwin, Mourao,˜ and Nunes and relate the two. In [12], the authors prove that one can define Kahler¨ structures on T ∗G using Ad-invariant complexifier functions. We prove that the condition of Ad-invariance can be replaced by a more general set of equations. In chapter5, we give an example of geometric quantization of T ∗G. We introduce a line bundle L on ∗ T G and a polarization Pτ defined by pulling back the vertical polarization along the complexified flow of an Hamiltonian vector field. From these structures, we define a Hilbert space whose elements are integrable Pτ -polarized sections of L.

Keywords: Kahler¨ geometry, geometric quantization, Mabuchi metric, Lie group, cotangent bundle, complexiﬁed diffeomorphisms

ix x Contents

Acknowledgments...... v Resumo...... vii Abstract...... ix List of Tables...... xiii Nomenclature...... xv Glossary...... 1

1 Theoretical Background1 1.1 Symplectic Manifolds...... 1 1.2 Moser’s Theorem...... 3 1.3 Complex Geometry...... 3 1.3.1 Almost complex manifolds. Complex manifolds...... 3 1.3.2 Splittings and forms on complex manifolds...... 5 1.3.3 When is an almost complex manifold a complex manifold?...... 6 1.3.4 Compatible structures. Kahler¨ manifolds...... 7 1.3.5 Cohomology results on Kahler¨ manifolds...... 8 1.4 Lie groups and Lie algebras...... 11 1.4.1 Representations...... 11 1.4.2 Adjoint representations...... 12 1.5 Structure functions and structure constants...... 13

2 Geodesics on the Space of Kahler¨ Metrics of a Manifold 15 2.1 Lie Series and complexified flows...... 15 2.2 Action of a complexified analytic flow on a complex structure...... 18 2.3 Restriction to Hamiltonian flows...... 19 2.4 Action on Kahler¨ structures...... 19 2.5 Geodesics on the Space of Kahler¨ metrics...... 24 2.6 Symplectic picture and complex picture...... 25 2.7 The path of Kahler¨ metrics (generated by a complex flow of an Hamiltonian vector field) is a geodesic...... 26

xi 3 Kahler¨ structures on the Cotangent Bundle of a Lie Group 31 3.1 T ∗G is diffeomorphic to G × g ...... 31 3.2 Kahler¨ structures on T ∗G ...... 32 3.2.1 Standard Kahler¨ structure on T ∗G ...... 32 3.2.2 The Kahler¨ structure of Hall and Kirwin...... 33 3.2.3 The Kahler¨ structure of Kirwin, Mourao˜ and Nunes...... 34 3.2.4 Problems to study...... 35 3.3 Useful coordinates based on the diffeomorphism T ∗G =∼ G × g ...... 35 3.4 Relation between theorems 3.2.2 and 3.2.4...... 41 3.5 The theorem of Hall and Kirwin in the case of a Lie group...... 43 3.5.1 Attempt at proving conjecture 3.5.1...... 43

3.5.2 ωβ as a pullback of ωST by a diffeomorphism...... 45 3.6 The theorem of Kirwin, Mourao˜ and Nunes in the case of non Ad-invariant h ...... 47 τ 3.6.1 ϕh is a diffeomorphism. The complex structure J ...... 47 ∗ 3.6.2 A basis for T1,0(T G) ...... 48 3.6.3 Equations for T ∗G being Kahler¨ in terms of h ...... 49

4 A Short Overview of Geometric Quantization 53 4.1 Introduction and motivation...... 53 4.2 General set up...... 54

5 An Example of Quantization of the Cotangent Bundle of a Lie Group 55 5.1 General setup...... 55 5.2 A prequantum line bundle for T ∗G ...... 56 5.3 A polarization for T ∗G ...... 57 5.4 T ∗G is foliated by Kahler¨ manifolds...... 63

5.5 Computation of the P¯τ -polarized sections...... 67

5.6 The inner product of Hτ . An unitary isomorphism of Hilbert spaces...... 69

Bibliography 75

A Moser’s Theorem 77 A.1 Lie derivative of time dependent vector ﬁeld in terms of an isotopy...... 77 A.2 Proof of Moser’s Theorem...... 78

xii List of Tables

4.1 Comparison between classical and quantum mechanics...... 53

xiii xiv Nomenclature

H Hilbert space

N Nijenhuis tensor

P Polarization g Lie algebra

Greek symbols

α Differential form

∆ Laplacian

κ Kahler¨ potential

∇ Connection or gradient (depending on context)

ω Symplectic form

t φX Flow of the vector ﬁeld X

τ Complex time

θ Symplectic potential

Roman symbols

Ad Adjoint representation of Lie group ad Adjoint representation of Lie algebra f Function on a manifold

G Lie group g or h·, ·i Riemannian metric

J Complex structure

M Manifold

R Curvature of a connection

xv s Section of bundle

U Open set

X Vector ﬁeld

Subscripts

LX Lie derivative with respect to the vector ﬁeld X

Superscripts

α# associated vector to the covector α

X[ associated covector to the vector X

xvi Chapter 1

Theoretical Background

1.1 Symplectic Manifolds

Deﬁnition 1.1.1. Let M be a manifold. A symplectic form on M is a 2-form ω ∈ Ω2(M) that is closed and nondegenerate. A symplectic manifold is a pair (M, ω), where M is a manifold, and ω ∈ Ω2(M) is a symplectic form.

Deﬁnition 1.1.2. A symplectomorphism of two symplectic manifolds (M, ωM ) and (N, ωN ) is a diffeo- ∗ morphism ϕ : M → N such that ϕ ωN = ωM .

Theorem 1.1.3 (Darboux). Let (M, ω) be a symplectic manifold. For all p ∈ M, there exists a coordinate neighborhood of p, (U, x1, ...xn, y1, ..., yn), such that on U

n X ω = dxi ∧ dyi. (1.1) i=1

In particular, the dimension of M is even.

Proof. See [6], page 233.

Remark 1.1.4 (Comparison between Riemannian and Symplectic Geometries).

• Riemannian Geometry: there exist local coordinates centered at p such that g is the Euclidean inner product at p.

• Symplectic Geometry: there exist local coordinates centered at p such that ω is the canonical symplectic form in a neighborhood of p.

• The analogous to Darboux’s Theorem in Riemannian Geometry (i.e. g being the Euclidean inner product in a neighborhood of p) is false. If it were true, the curvature would always be zero.

• Darboux’s Theorem implies that there is no analogue of curvature in symplectic geometry. Since all symplectic manifolds of the same dimension are locally symplectomorphic, there cannot exist any local invariant.

1 Proposition 1.1.5. Let (M, ω) be a symplectic manifold of dimension 2n. Then M is orientable, with volume form ωn = ω ∧ ... ∧ ω. | {z } n times

Proof. We will prove that ∀p ∈ M, (ω ∧ ... ∧ ω)p 6= 0. Let p ∈ M. By Darboux’s Theorem, there exist coordinates around p such that n X ω = dxi ∧ dyi. i=1 Then, ω ∧ ... ∧ ω = dx1 ∧ dy1 ∧ ... ∧ dxn ∧ dyn 6= 0 at p.

Example 1.1.6. Let M be an n-dimensional manifold. We will give the cotangent bundle of M, T ∗M, 1 n 1 n the structure of a symplectic manifold. If (U, x , ..., x ) is a coordinate chart on M, then (dx )p, ..., (dx )p ∗ ∗ Pn i form a basis of Tp M. Then, any element α of Tp M can be given as α = i=1 pi(dx )p. This defines a ∗ ∗ 1 n 1 n local coordinate chart on T M, (T U, x , ..., x , p1, ..., pn), where the (x , ..., x ) are the coordinates of 1 n p = π(α) in M, and (p1, ..., pn) are the coefficients of α in the basis dx , ..., dx . Define θ ∈ Ω1(T ∗M) pointwise by:

∗ θα : Tα(T M) −→ R (1.2) v 7−→ α(dπ · v).

Pn i ∂ Pn ∂ θ is the canonical symplectic potential. If v = a i + bi , then i=1 ∂x i=1 ∂pi

  n n  X ∂ X ∂ θ (v) = α(dπ · v) = α dπ aj + b α   ∂xj j ∂p  j=1 j=1 j

 n  n  n  X ∂ X X ∂ = α aj = p (dxi) aj  ∂xj  i p  ∂xj  j=1 i=1 j=1

| {z∗ } ∈ Tp M n n n n ! X X X ∂ X ∂ = p ai = p (dxi) aj + b . i i α ∂xj j ∂p i=1 i=1 j=1 j=1 j

| {z ∗ } | {z } ∈ Tα(T M) v

Pn i Therefore, in local coordinates θ can be written as θ = i=1 pidx . Deﬁne the canonical symplectic form ω ∈ Ω2(T ∗M) by ω = −dθ. The local expression for ω is then

n ! n X i X i ω = −d pidx = dx ∧ dpi. i=1 i=1

ω is closed and nondegenerate. Therefore (T ∗M, ω) is a symplectic manifold.

Definition 1.1.7. Let h ∈ C∞(M). The Hamiltonian vector field defined by h is the unique vector field

Xh satisfying ω(Xh, ·) = dh.

∞ Deﬁnition 1.1.8. Let f, g ∈ C (M). Their Poisson bracket, denoted {f, g}, is {f, g} = Xf · g.

2 ∞ Remark 1.1.9. C (M) equipped with the Poisson bracket is a Lie algebra. Also, the map f 7→ Xf is a Lie algebra homomorphism.

1.2 Moser’s Theorem

Let M be a manifold.

Deﬁnition 1.2.1. An isotopy is a map ρ: M × R −→ M such that:

•∀ t ∈ R ρt : M −→ M deﬁned by ρt(p) = ρ(p, t) is a diffeomorphism.

• ρ0 = idM

Deﬁnition 1.2.2. A time dependent vector ﬁeld on M is an assignment R 3 t 7→ Xt ∈ X(M) that is smooth in t.

Remark 1.2.3. We can think of an isotopy as being the ﬂow of a time dependent vector ﬁeld:

• An isotopy deﬁnes a time dependent vector ﬁeld, by the formula

dρ t = X ◦ ρ . (1.3) dt t t

d −1 In other words, Xt is given at a point p by (Xt)p = ds ρs ρt (p) . s=t

• Conversely, if Xt is a time dependent vector ﬁeld, and if M is compact (or Xt has compact support),

then Xt can be integrated to obtain an isotopy.

The next theorem, by Moser, provides an answer to the question: If M is a manifold and ω0, ω1 are symplectic forms, does there exist a symplectomorphism ϕ :(M, ω0) −→ (M, ω1)?

2 Theorem 1.2.4 (Moser). Let M be a compact manifold, ω0, ω1 ∈ Ω (M) be symplectic forms, such that

2 (i) [ω0] = [ω1] ∈ H (M, R);

(ii) ∀t ∈ [0, 1] ωt := (1 − t)ω0 + tω1 is symplectic.

∗ Then there exists an isotopy ρt : M −→ M such that ρt ωt = ω0 ∀t ∈ [0, 1]

Proof. See appendixA.

1.3 Complex Geometry

1.3.1 Almost complex manifolds. Complex manifolds

Deﬁnition 1.3.1. An almost complex structure on a manifold M is a vector bundle isomorphism 2 J : TM −→ TM such that J = −idTM . An almost complex manifold is a pair (M,J) where M is a manifold and J is an almost complex structure on M.

3 Deﬁnition 1.3.2. A complex manifold of complex dimension n is given by the data:

• M a topological space (which is Hausdorff and 2nd countable),

n •A = {ϕα : Uα ⊂ M −→ Vα ⊂ C }α∈I a complex atlas, satisfying:

• M = ∪α∈I Uα ;

•∀ α ∈ I, ϕα is an homeomorphism;

•∀ α, β ∈ I such that Uα ∩ Uβ 6= ∅,

−1 n n ϕβ ◦ ϕα : ϕα(Uα ∩ Uβ) ⊂ C −→ ϕβ(Uα ∩ Uβ) ⊂ C

−1 is biholomorphic (this means that both ϕβ ◦ ϕα and its inverse are holomorphic);

n −1 •A is maximal: if ψ : U ⊂ M −→ V ⊂ C is an homeomorphism such that ∀α ∈ I ψ ◦ ϕα is biholomorphic, then ψ ∈ A.

Theorem 1.3.3. Any complex manifold admits an almost complex structure.

Proof. Let M be a complex manifold. We start by giving a local deﬁnition of J. Let (U, z1, ..., zn) be a complex coordinate chart on M. If zj = xj + iyj, then for all p ∈ U,

∂ ∂ T M = span , : j = 1, ..., n . p R ∂xj ∂yj

Deﬁne

Jp : TpM −→ TpM ∂ ∂ 7−→ ∂xj ∂yj ∂ ∂ 7−→ − . ∂yj ∂xj

2 Then clearly Jp = −id. Now, we prove that J is a globally well deﬁned object. Let (U, z1, ..., zn), (V, w1, ..., wn) be coordinate j j j j neighborhoods such that U ∩ V 6= ∅. We will show that J|U = J|V on U ∩ V . If z = x + iy and w = uj + ivj, since the transition functions are biholomorphic, they satisfy the Cauchy-Riemann equations:

∂uk ∂vk  =  j j ∂x ∂y (1.4) ∂uk ∂vk  = − .  ∂yj ∂xj

4 We will now show that J|U = J|V .

n ! ∂ X ∂uk ∂ ∂vk ∂ J| = J| + V ∂xj V ∂xj ∂uk ∂xj ∂vk k=1 n X ∂uk ∂ ∂vk ∂ = − ∂xj ∂vk ∂xj ∂uk k=1 n X ∂vk ∂ ∂uk ∂ ∂ = + = ∂yj ∂vk ∂yj ∂uk ∂yj k=1

∂ ∂ where we used the Cauchy-Riemann equations. Analogously, J|V ∂yj = − ∂xj . Therefore J|U = J|V , and J is well deﬁned globally.

1.3.2 Splittings and forms on complex manifolds

Definition 1.3.4. Let (M,J) be an almost complex manifold. Consider the complexified tangent bundle of M, TM ⊗ C. In TM ⊗ C, J has ±i as eigenvalues. Define the following spaces:

T1,0M := (+i) − eigenspace of J = holomorphic tangent vectors, (1.5a)

T0,1M := (−i) − eigenspace of J = anti-holomorphic tangent vectors, (1.5b)

1,0 ∗ T M := (T1,0M) = complex-linear cotangent vectors, (1.5c)

0,1 ∗ T M := (T0,1M) = complex-antilinear cotangent vectors. (1.5d)

Also, deﬁne k M l,m Ω (M; C) := Ω (M) = k-forms on M with complex values, (1.6) l+m=k where l ! m ! ^ ^ Ωl,m(M) := sections of the vector bundle T 1,0M ∧ T 0,1M . (1.7)

Consider the projections πl,m :Ωk(M; C) −→ Ωl,m(M). Deﬁne ∂,∂¯ by:

∂ := πl+1,m ◦ d:Ωl,m(M) −→ Ωl+1,m(M), (1.8a)

∂¯ := πl,m+1 ◦ d:Ωl,m(M) −→ Ωl,m+1(M). (1.8b)

If M is a complex manifold with almost complex structure J, then this splitting can be written explicitly in terms of local coordinates. Recall that:

∂ ∂ T M = span , : j = 1, ..., n , p R ∂xj ∂yj ∂ ∂ T M ⊗ = span , : j = 1, ..., n , p C C ∂xj ∂yj

∂ ∂ ∂ ∂ J = and J = − . ∂xj ∂yj ∂yj ∂xj

5 Using these equations, the spaces in deﬁnition 1.3.4 become:

( ) 1 ∂ i ∂ T M := (+i) − eigenspace of J = span − , (1.9a) 1,0 C 2 ∂xj 2 ∂yj | {z } j=1,...,n := ∂ ∂zj ( ) 1 ∂ i ∂ T M := (−i) − eigenspace of J = span + , (1.9b) 0,1 C 2 ∂xj 2 ∂yj | {z } j=1,...,n := ∂ ∂z¯j 1,0 ∗ j j T M := (T1,0M) = span {dx + idy }j=1,...,n, (1.9c) C | {z } =dzj 0,1 ∗ j j T M := (T0,1M) = span {dx − idy }j=1,...,n. (1.9d) C | {z } =dz¯j

And if (U, z1, ..., zn) is a complex coordinate neighborhood,

  l,m  X J K ∞  Ω (U) = bJ,K dz ∧ dz¯ : bJ,K ∈ C (U; C) . (1.10) |J|=l,|K|=m 

1.3.3 When is an almost complex manifold a complex manifold?

We saw that any complex manifold is an almost complex manifold. Question: Is the converse true? More precisely, given (M,J) an almost complex manifold, does there exist a complex structure A such that A deﬁnes J? Answer: Provided by a theorem of Newlander-Nirenberg.

Deﬁnition 1.3.5. Let (M,J) be an almost complex manifold. Its Nijenhuis tensor N is a tensor of type (2, 1) given by N (X,Y ) := [JX,JY ] − J[X,JY ] − J[JX,Y ] − [X,Y ].

Deﬁnition 1.3.6. Let (M,J) be an almost complex manifold. J is said to be integrable if M admits a complex structure A such that the almost complex structure deﬁned by A is J (in the sense of theorem 1.3.3).

Theorem 1.3.7. (Newlander-Nirenberg) Let (M,J) be an almost complex manifold with Nijenhuis tensor N . Then, the following are equivalent:

(i) J is integrable;

(ii) N = 0;

(iii) d = ∂ + ∂¯;

(iv) ∂¯2 = 0;

2,0 (v) π ◦ d|Ω0,1 = 0.

Proof. See theorem 5.7.4. in [11], and theorem 7.4. and proposition 8.2. in [13].

6 1.3.4 Compatible structures. Kahler¨ manifolds

Proposition 1.3.8. Let M be a manifold. Let ω ∈ Ω2(M) be a symplectic form on M, g be a Riemannian metric on M, and J be an almost complex structure on M. Then, the following are equivalent:

(i) g(·, ·) = ω(·,J·);

(ii) ω(·, ·) = g(J·, ·);

(iii) J(·) =g ˜−1(˜ω(·)).

(Where g˜, ω˜ are the linear isomorphisms given by g˜(u) = g(u, ·) and ω˜ = ω(u, ·)).

Proof. The result follows from computation, and from using the properties of each of the three structures 2 (in particular using that J = −idTM ).

Deﬁnition 1.3.9. If the conditions of proposition 1.3.8 are true, (ω, J, g) is called a compatible triple. If (M, ω) is a symplectic manifold, and J is an almost complex structure on M, then J is compatible (with ω) if g(·, ·) := ω(·,J·) is a Riemannian metric.

AKahler¨ manifold is at the same time a symplectic manifold, a complex manifold and a Riemannian manifold, in a way that the three structures are compatible:

Deﬁnition 1.3.10. A Kahler¨ manifold is a Manifold M equipped with a symplectic form ω, a complex structure J, and a Riemannian metric g such that (ω, J, g) is a compatible triple. If (M, ω, J, g) is a Kahler¨ manifold, we say that ω is a Kahler¨ form.

Theorem 1.3.11 (A. C. da Silva, [4]). Given a complex manifold (M,J), a form ω ∈ Ω2(M) is Kahler¨ if and only if ∂ω = 0, ∂ω¯ = 0, and if it is given locally by

n i X ω = h dzj ∧ dz¯k, (1.11) 2 jk j,k=1 where ∀p (hjk(p)) is a positive deﬁnite hermitian matrix.

Proof. See [4], pages 90-92.

Deﬁnition 1.3.12. Let (M, ω, J, g) be a Kahler¨ manifold, and U ⊂ M be open.

• κ ∈ C∞(M; R) is a (global) Kahler¨ potential if ω = i∂∂κ¯ ;

∞ ¯ • κ ∈ C (U; R) is a local Kahler¨ potential if ω|U = i∂∂κ.

∞ 2 2 Lemma 1.3.13. Let (M, ω, J, g) be a Kahler¨ manifold. Let h ∈ C (M). Then, kXhk = k∇hk .

Proof.

h∇h, ∇hi = dh(∇h) = ω(Xh, ∇h) = hJXh, ∇hi

= dh(JXh) = ω(Xh,JXh) = hXh,Xhi

7 1.3.5 Cohomology results on Kahler¨ manifolds

Deﬁnition 1.3.14. Let M be a compact Kahler¨ manifold. We deﬁne:

• The Hodge-∗ operator: if {ω1, ..., ω2n} is a positively oriented orthonormal coframe, then ∗(ω1 ∧ ... ∧ ωk) = ωk+1 ∧ ... ∧ ω2n.

• An inner product of forms:

k k h·, ·i:Ω (M) × Ω (M) −→ R (1.12) Z (α, β) 7−→ α ∧ ∗β. M

• Adjoints with respect to the inner product: the adjoint of d, d∗, is uniquely speciﬁed by the property hdα, βi = hα, d∗βi. Analogously, we deﬁne ∂∗ and ∂¯∗.

• The Laplacian, ∆: Ωk(M) −→ Ωk(M), is given by ∆ = dd∗ + d∗d.

Proposition 1.3.15. ∆ = dd∗ + d∗d = 2(∂¯∗∂¯ + ∂¯∂¯∗).

Proof. See [13], page 103.

Proposition 1.3.16. The following are equivalent:

(i) ∆ω = 0;

(ii) dω = 0 and d∗ω = 0;

(iii) ∂ω¯ = 0 and ∂¯∗ω = 0.

Proof. We prove only the ﬁrst equivalence. The proof of the second one is analogous. (⇐=): If dω = 0 and d∗ω = 0 then ∆ω = 0 by deﬁnition. (=⇒): Assume that ∆ω = 0. Then:

0 = h∆ω, ωi = hdd∗ω + d∗dω, ωi

= hdd∗ω, ωi + hd∗dω, ωi = hd∗ω, d∗ωi + hdω, dωi

= kd∗ωk2 + kdωk2.

Therefore, both d∗ω and dω are zero.

Deﬁnition 1.3.17. A form is harmonic if its Laplacian is zero. The set of harmonic forms of a given type is denoted Hl,m(M): Hl,m(M) := {α ∈ Ωl,m(M) : ∆α = 0}. (1.13)

Theorem 1.3.18 (Hodge-Dolbeaut decomposition). Let M be a compact Kahler¨ manifold. Then, Ωl,m(M) decomposes as a direct sum of the following subspaces:

Ωl,m(M) = Hl,m(M) ⊕ ∂¯Ωl,m−1(M) ⊕ ∂¯∗Ωl,m+1(M). (1.14)

8 This decomposition is orthogonal with respect to h·, ·i.

Proof. See [8], pages 84-100.

Lemma 1.3.19 (∂¯-Lemma). Let M be a complex manifold. Let ω ∈ Ω0,1(M) be such that ∂ω¯ = 0. Then, ω is locally ∂¯ exact. More preciselly, for all p ∈ M there exists a neighborhood V of p and a complex ∞ ¯ valued function on V , φ ∈ C (V ; C), such that ω|V = ∂φ.

Proof. See [8], pages 25-27.

Lemma 1.3.20 (Global i∂∂¯-Lemma). Let M be a compact Kahler¨ manifold. Let ω be an exact, real, type (1, 1) form. Then, there exists a φ ∈ C∞(M) such that ω = i∂∂φ¯ .

Proof. ω = dα, for some α ∈ Ω1(M). Decomposing α in its (1, 0), and (0, 1) parts, ω = ∂α1,0 + ∂α¯ 1,0 + ∂α0,1 + ∂α¯ 0,1. Since ω is of type (1, 1), ∂α1,0 and ∂α¯ 0,1 are zero. Applying the Hodge-Dolbeaut decomposition theorem to α0,1: α0,1 = γ + ∂η¯ + ∂¯∗ξ.

Where γ ∈ Ω1,0(M) is harmonic, η ∈ C∞(M; C) and ξ ∈ Ω2,0(M). We now prove that ∂¯∗ξ = 0.

0 = ∂α¯ 0,1 = ∂γ¯ + ∂¯∂η¯ +∂¯∂¯∗ξ, |{z} |{z} =0 =0

0 = h∂α¯ 0,1, ξi = h∂¯∂¯∗ξ, ξi = h∂¯∗ξ, ∂¯∗ξi =⇒ ∂¯∗ξ = 0.

So, α0,1 = γ + ∂η¯ , and therefore α = γ + ∂η¯ +γ ¯ + ∂η,¯

ω = dα = dγ +d∂η¯ + dγ¯ +d∂η¯ |{z} |{z} =0 =0 = ∂∂η¯ + ∂∂¯ η¯ = ∂∂η¯ − ∂∂¯η¯ = i∂∂¯(2 Im η).

Lemma 1.3.21 (Existence of local Kahler¨ potentials). Let M be a complex manifold. Let ω be a closed, real, type (1, 1) form. Then, ω is locally i∂∂¯-exact. More precisely, for all p ∈ M there exists a neighbor- ∞ ¯ hood V of p and a function on V φ ∈ C (V ) such that ω|V = i∂∂φ.

Proof. ω is closed, so it can be given locally as ω = dα. α = α1,0 + α0,1, where α1,0 = α0,1 because α is real. Since dα is of type (1, 1), both ∂α1,0 and ∂α¯ 0,1 are zero. The ∂¯-Lemma applied to α0,1 implies that locally, α0,1 = ∂ψ¯ (hence α1,0 = ∂ψ¯). Then, 2 Im ψ is a local Kahler¨ potential:

ω = dα = ∂α¯ 1,0 + ∂α0,1

= ∂∂¯ ψ¯ + ∂∂ψ¯ = −∂∂¯ψ¯ + ∂∂ψ¯

= ∂∂¯(ψ − ψ¯) = i∂∂¯(2 Im ψ).

9 Lemma 1.3.22. Let M be a compact connected Kahler¨ manifold, and φ ∈ C∞(M) such that ∂∂φ¯ = 0. Then φ is constant.

Proof. Recall the following deﬁnitions, present in [13], page 101:

L:Ωp,q(M) −→ Ωp+1,q+1(M) (1.15)

α 7−→ ω ∧ α,

Λ = L∗ :Ωp,q(M) −→ Ωp−1,q−1(M). (1.16)

Then, it is a fact, also stated in [13], page 103, that

i∂¯∗ = Λ∂ − ∂Λ. (1.17)

With this information, we can conclude that ∆φ = 0:

∆φ = 2(∂¯∗∂¯ + ∂¯∂¯∗)φ (proposition 1.3.15)

= 2∂¯∗∂φ¯ (φ is of type 0, 0)

= 2i(∂Λ∂¯ − Λ∂∂¯)φ (equation (1.17))

= −2iΛ∂∂φ¯ (φ is of type 0, 0)

= 0.

Using proposition 1.3.16, we conclude that dφ = 0, and since M is connected, φ is constant.

Lemma 1.3.23. Let M be a complex manifold, and φ ∈ C∞(M) such that ∂∂φ¯ = 0. Then φ is locally of the form φ = f + f¯, where f is holomorphic and f¯ is anti-holomorphic.

¯ ¯ ¯ Proof. Since d(∂φ) = ∂∂φ = 0, ∂φ is locally exact. Therefore, we can cover M by open sets {Ui} such ∞ ¯ that for each i there exists a gi ∈ C (Ui; C) such that ∂φ|Ui = dgi. Considering the type of the forms, ¯ we conclude that ∂gi = 0, so gi is anti-holomorphic. Let hi = φ|Ui − gi. Then

¯ ¯ ¯ ∂hi = ∂(φ|Ui − gi) = ∂φ|Ui − dgi = 0,

which implies that hi is holomorphic. Since φ|Ui is real,

¯ ¯ gi + hi = φ|Ui = φ|Ui =g ¯i + hi.

Therefore, ¯ gi − hi =g ¯i − hi.

The left hand side is a anti-holomorphic function. The right hand side is a holomorphic function. ¯ Then, both are holomorphic and anti-holomorphic, and therefore both are locally constant: gi − hi =

ci ¯ g¯i − hi = ci, where ci is a real locally constant function. Deﬁne fi = hi + 2 . Then, fi is holomorphic, fi

10 is anti-holomorphic, and

c c f + f¯ = h + i + h + i = h + h¯ + c = h + g = φ| . i i i 2 i 2 i i i i i Ui

1.4 Lie groups and Lie algebras

1.4.1 Representations

Deﬁnition 1.4.1. Let V be a ﬁnite dimensional vector space. Then

GL(V ) := {T : V −→ V | T is a linear isomorphism} (1.18) is a Lie group under composition, called the general linear group of V . Its Lie algebra is the general linear Lie algebra of V :

gl(V ) = {T : V −→ T | T is linear} = End(V ), (1.19) where the Lie bracket is the commutator as linear operators:

[T,S] = TS − ST.

Deﬁnition 1.4.2. A representation of a Lie group G on a vector space V is a Lie group homomorphism λ: G −→ GL(V ).A representation of a Lie algebra g on a vector space V is a Lie algebra homomorphism λ: g −→ gl(V ).

Deﬁnition 1.4.3. A unitary representation of a Lie group is a representation λ: G −→ GL(V ) such that the vector space V is a Hilbert space, and such that λ(g) is unitary ∀g ∈ G.

Deﬁnition 1.4.4. Let λ: G −→ GL(V ) be a representation of a Lie group. A subrepresentation of λ is a subspace W of V , such that ∀g ∈ G λ(g)(W ) ⊂ W . A representation λ of a Lie group on V is said to be irreducible if its only subrepresentations are {0} ⊂ V and V itself.

Deﬁnition 1.4.5. Let λ: G −→ GL(V ) and µ: G −→ GL(W ) be two unitary representations. λ and µ are equivalent unitary representations if there exists a Hilbert space isomorphism φ : V −→ W such that φ˜ ◦ λ = µ, where φ˜ is the following map:

φ˜: GL(V ) −→ GL(W ) (1.20)

T 7−→ φ ◦ T ◦ φ−1.

It is easy to see that this deﬁnes an equivalence relation over the set of unitary representations of G.

11 1.4.2 Adjoint representations

Deﬁnition 1.4.6. Let G be a Lie group. Deﬁne:

• the action of G on itself by conjugation:

Ψ: G −→ Aut(G) (1.21)   Ψg : G −→ G g 7−→   ; h 7−→ ghg−1

• the adjoint representation of G:

Ad: G −→ GL(g) (1.22)

g 7−→ (Adg := deΨg : g −→ g);

• the adjoint representation of g:

ad: g −→ End(g) (1.23)   adX : g −→ g X 7−→   . Y 7−→ [X,Y ]

The following proposition is a list of properties of the adjoint representations. In particular, it shows that Ad is a representation of G on g and that ad is a representation of g on g.

Proposition 1.4.7.

• Ψ is a Lie group homomorphism: Ψgh = Ψg ◦ Ψh;

−1 • Ψg is a Lie group isomorphism: Ψg(a)Ψg(b) = Ψg(ab) and (Ψg) = Ψg−1 ;

• Ad is a Lie group homomorphism: Adgh = AdgAdh;

−1 • Adg is a Lie algebra isomorphism: Adg[X,Y ] = [AdgX, Adg,Y ] and (Adg) = Adg−1 ;

• ad is a Lie algebra homomorphism: ad[X,Y ] = [adX , adY ];

• adX is a Lie algebra derivation: adX [Y,Z] = [adX Y,Z] + [Y, adX Z].

Proof. The proof of each equation follows from the deﬁnitions. The proofs of the statements about Ad use the properties of the derivative and that Ψg = LgRg−1 , which implies that Adg = dLgdRg−1 . The proof of the statements about ad make use of the Jacobi identity.

12 Remark 1.4.8. A Lie group homomorphism Φ: G −→ H induces an homomorphism on the Lie algebras

φ: g −→ h, given by φ = deΦ. Also, the following diagram commutes:

φ=d Φ g e h

exp exp

G Φ H

For a proof of these facts, see [9] page 60.

The next proposition shows that the homomorphism induced by Ad on the Lie algebras is ad.

Proposition 1.4.9. Let G be a Lie group with Lie algebra g. Then, ad = deAd. Therefore, the homomorphism induced by Ad on the Lie algebras is ad.

Proof. Let X,Y ∈ g. We will show that deAd(X)(Y ) = ad(X)(Y ) = [X,Y ].

d deAd(X)(Y ) = deψexp(tX)Y dt t=0 d d = ψexp(tX) exp(sY ) dt t=0 ds s=0 d d = exp(tX) exp(sY ) exp(−tX) dt t=0 ds s=0 d d d d = exp(tX) exp(sY ) − exp(sY ) exp(tX) = V. dt t=0 ds s=0 dt t=0 ds s=0

We claim that V = [X,Y ]. To see this, let f ∈ C∞(G) and let X,Y ∈ X(G) be the left invariant vector ﬁelds deﬁned by X and Y . Then

d d d d V · f = f exp(tX) exp(sY ) − f exp(sY ) exp(tX) dt t=0 ds s=0 dt t=0 ds s=0 d d d d = f exp(tX) exp(sY ) − f exp(sY ) exp(tX) dt t=0 ds s=0 ds s=0 dt t=0 d d = Y · f (exp(tX)) − X · f (exp(sY )) dt t=0 ds s=0 = X · (Y · f) (e) − Y · (X · f) (e)

= [X,Y ] · f.

1.5 Structure functions and structure constants

Definition 1.5.1. Let M be a differentiable manifold and {X1, ..., Xn} be a field of frames on an open k set of M. The structure functions of this frame, Cij, are defined by the equation:

n X k [Xi,Xj] = CijXk. (1.24) k=1

k k Note that since the Lie bracket is antisymmetric, so are the structure functions: Cij = −Cji. We now give some properties of these functions.

13 1 n Proposition 1.5.2. If {ω , ..., ω } is the ﬁeld of coframes dual to {X1, ..., Xn}, then

n 1 X dωi + Ci ωj ∧ ωk = 0. (1.25) 2 jk j,k=1

Proof.

i i i i dω (Xl,Xm) = Xl(ω (Xm)) − Xm(ω (Xl)) − ω ([Xl,Xm])

 n  i i X j = −ω ([Xl,Xm]) = −ω  ClmXj j=1 n 1 X = −Ci = − Ci ωj ∧ ωk(X ,X ). lm 2 jk l m j,k=1

Proposition 1.5.3. In the case where M = G is a Lie group and the Xi are left invariant, the structure functions are constant.

k k Proof. We prove that Cij(g) = Cij(e) for all g ∈ G.

n X k Cij(g)(Xk)g = [Xi,Xj]g = [dLgXi, dLgXj]g k=1 n X k = dLg[Xi,Xj]e = dLg Cij(e)(Xk)e k=1 n X k = Cij(e)(Xk)g, k=1

k k which implies that Cij(g) = Cij(e) since the (Xk)g form a basis of TgG.

14 Chapter 2

Geodesics on the Space of Kahler¨ Metrics of a Manifold

In this section we present a brief overview of [14]. In the first sections we explain how it is possible to change the complex structure of a compact Kahler¨ manifold (acting with the ”complex flow” of an Hamiltonian vector field), in such a way that the resulting manifold is still Kahler.¨ In the last sections we explain how this change of Kahler¨ metric can be seen as a change in symplectic form, and how it determines a curve in a space of Kahler¨ metrics. Then we give a proof of the result that this curve is in fact a geodesic with respect to a certain metric on the space of Kahler¨ metrics.

2.1 Lie Series and complexiﬁed ﬂows

Let (M,J0) be a compact, complex manifold.

Definition 2.1.1. Let S be a real analytic tensor field on M, and X be a real analytic vector field on M.

Denote by LX the Lie derivative with respect to the vector ﬁeld X. Let τ ∈ C. We deﬁne the exponential of τLX applied to S as the formal Lie Series:

∞ X τ k eτLX S := Lk (S). (2.1) k! X k=0

The deﬁnition given above is a formal one. We now prove that the exponential converges to a tensor ﬁeld on M.

Theorem 2.1.2. For all S real analytic tensor ﬁeld and X real analytic vector ﬁeld, there exists a T such that if t is real and |t| < T , then etLX S converges, and

tLX t ∗ e S = (φX ) S, (2.2)

t where φX is the ﬂow of X.

15 dk t ∗ k t ∗ Proof. We start by showing by induction that dtk ((φX ) S) = LX ((φX ) S). (k = 1) : d d d t ∗ t+s ∗ s ∗ t ∗ t ∗ (φX ) S = (φX ) S = (φX ) (φX ) S = LX ((φX ) S) (2.3) dt ds s=0 ds s=0

t ∗ t t s t+s (k =⇒ k + 1) : Note that (φX ) X = X, because the curve φX is tangent to X and φX ◦ φX = φX . t ∗ Therefore, LX and (φX ) commute:

t ∗ t ∗ t ∗ (φ ) LX S = L t ∗ (φ ) S = LX (φ ) S. (2.4) X (φX ) X X X

dk+1 d dk d ((φt )∗S) = ((φt )∗S) = Lk ((φt )∗S) dtk+1 X dt dtk X dt X X d = (φt )∗Lk S = L (φt )∗Lk S = (φt )∗Lk+1S, dt X X X X X X X

dk t ∗ k t ∗ Where we used the induction hypothesis. This proves that dtk ((φX ) S) = LX ((φX ) S). In particular,

k d t ∗ k (φX ) S = LX S. (2.5) dtk t=0

t Since X is real analytic, then φX is real analytic in t and as a map on M. Since S is also real analytic, t ∗ t ∗ (φX ) S is real analytic in t and as a tensor ﬁeld on M. Let p ∈ M. We can expand (φX ) S around t = 0, p. More precisely, for all p in M there exists Vp an open neighborhood of p and a Tp ∈ R such that

∀t : |t| < Tp ∞ X tk (φt )∗S = S (2.6) X k! k k=0

1 n in Vp, where the Sk are some tensors in Vp whose coefﬁcients are power series in x , ..., x centered in (0, ..., 0), that are convergent in Vp. Let {Vp1 , ..., VpN } be a covering of M by open sets as we just described, and take T = min{Tp1 , ..., TpN }. Let t be such that |t| < T . In each Vpj , it is true that

k k ∞ k k d t ∗ d X t LX S = (φX ) S = Sk = Sk (2.7) dtk t=0 dtk t=0 k! k=0

P∞ tk k t ∗ which implies that k=0 k! LX (S) converges to (φX ) S in each Vpj . We conclude that if |t| < T then P∞ tk k t ∗ k=0 k! LX (S) converges to (φX ) S in all of M, because the Vpj form an open covering of M.

Theorem 2.1.3. For all S real analytic tensor ﬁeld and X real analytic vector ﬁeld, there exists a T such that if |τ| < T , then eτLX S converges.

P∞ τ k k Proof. Let T be as in theorem 2.1.2. We start by noticing that k=0 k! LX (S) converges if and only P∞ τ k k 1 m 1 m if k=0 k! LX (S)(X1, ..., Xl, ω , ..., ω ) converges for all p ∈ M, X1, ..., Xl ∈ TpM and ω , ..., ω ∈ ∗ 1 m ∗ Tp M. Let p ∈ M, X1, ..., Xl ∈ TpM and ω , ..., ω ∈ Tp M. For ease of notation, denote ak = 1 k 1 m k! LX (S)(X1, ..., Xl, ω , ..., ω ). As a consequence of the last theorem,

∞ X k |τ| < T and τ ∈ R =⇒ akτ converges. (2.8) k=0

16 P∞ k From complex analysis, the power series k=0 akτ has a radius of convergence R, that satisﬁes:

∞ X k |τ| < R =⇒ akτ converges, (2.9) k=0 ∞ X k |τ| > R =⇒ akτ diverges. (2.10) k=0

For a proof of this, see the Abel power series theorem, for example in [1] page 38. Conditions (2.8) and (2.10) imply that R ≥ T . Condition (2.9) implies that

∞ X τ k |τ| < T =⇒ Lk (S)(X , ..., X , ω1, ..., ωm) converges, (2.11) k! X 1 l k=0 which completes the proof.

Remark 2.1.4. Applying eτLX to holomorphic tensors can be thought of as performing a pullback by a complexiﬁcation of the ﬂow of X.

We now prove some properties of the exponential.

Proposition 2.1.5. If all the series in each equation converge, then

• if S, R are tensor ﬁelds, eτLX (S ⊗ R) = eτLX S ⊗ eτLX R; (2.12)

• if S is a tensor ﬁeld of type (l, m),

τLX 1 m τLX τLX τLX τLX 1 τLX m e (S(X1, ..., Xl, ω , ..., ω )) = e S(e X1, ..., e Xl, e ω , ..., e ω ); (2.13)

• if Y,Z are vector ﬁelds, eτLX [Y,Z] = eτLX Y, eτLX Z . (2.14)

Proof. We start by proving that the equations are true if τ = t is real. In this case, each exponential is in fact a pullback. The equations we have to prove take the form:

t ∗ t ∗ t ∗ (φX ) (S ⊗ R) = (φX ) S ⊗ (φX ) R, (2.15)

t ∗ 1 m t ∗ t ∗ t ∗ t ∗ 1 t ∗ m (φX ) (S(X1, ..., Xl, ω , ..., ω )) = (φX ) S((φX ) X1, ..., (φX ) Xl, (φX ) ω , ..., (φX ) ω ), (2.16) t ∗ t ∗ t ∗ (φX ) [Y,Z] = (φX ) Y, (φX ) Z . (2.17)

Which are known properties of the pullback. Therefore each equation is true if τ is real. Performing complex analytic continuation on each side of each equation, we conclude that the equations are true for complex τ.

Remark 2.1.6. Equations (2.12), (2.13) and (2.14) can also be proven using the following method. Expand each exponential in a series. On the left side of the equation, use a formula for the Lie derivative

17 of a product (Leibniz rule). Then use Cauchy’s formula for the product of series to write the obtained sum as a product of series, which will be the right side of the equation.

2.2 Action of a complexiﬁed analytic ﬂow on a complex structure

Theorem 2.2.1 (Mourao˜ and Nunes, [14]). Let (M,J0) be a compact, complex manifold, and X be a real analytic vector ﬁeld on M. There exists a T > 0 such that for all τ ∈ B(0,T ), there exists an integrable almost complex structure Jτ satisfying:

1 n (i) (Deﬁnition of Jτ ) Let p ∈ M, and let (Uα, z0 , ..., z0 ) be a J0-holomorphic coordinate neighborhood

of p. Then, there exists an open neighborhood Vα,p of p such that:

• p ∈ Vα,p ⊂ Vα,p ⊂ Uα;

• Vα,p has compact closure;

j τX j • the series zτ := e · z0 are uniformly convergent on Vα,p;

1 n • (Vα,p, zτ , ..., zτ ) is a Jτ -holomorphic coordinate neighborhood of p.

(ii) There exists a unique biholomorphism ϕτ :(M,Jτ ) −→ (M,J0) that satisﬁes the following property: 1 n for all p ∈ M, there exists (Uα, z0 , ..., z0 ) a J0-holomorphic coordinate neighborhood of p and there 1 n exists (Vα,p, zτ , ..., zτ ) a Jτ -holomorphic coordinate neighborhood satisfying the conditions of (i), j j such that ϕτ (Vα,p) ⊂ Uα and zτ = z0 ◦ ϕτ .

Proof. See theorems 2.5 and 2.6. in [14].

Remark 2.2.2.

−1 • ϕτ being a biholomorphism means that Jτ = dϕτ ◦ J0 ◦ dϕτ . This condition can also be stated in

terms of the holomorphic functions: f is J0-holomorphic if and only if f ◦ ϕτ is Jτ -holomorphic.

j j j j τX¯ j • Performing complex conjugation on equation z0 ◦ ϕτ = zτ , we obtain that z¯0 ◦ ϕτ =z ¯τ = e · z¯0.

• The biholomorphisms ϕτ do not depend only on τ, but also on the initial complex structure J0. If J is

any complex structure on M, denote by ϕτ,J the unique biholomorphism ϕτ,J :(M,Jτ ) −→ (M,J) j τX j such that z ◦ ϕτ,J = e · z (in this notation, the biholomorphisms of the previous theorem

are written ϕτ = ϕτ,J0 ). If τ, σ ∈ C, then the following diagram is a commutative diagram of biholomorphisms: ϕτ,Jσ (M,Jτ+σ) (M,Jσ)

ϕτ+σ,J0 ϕσ,Jτ ϕσ,J0

(M,Jτ ) (M,J0) ϕτ,J0

We now note that:

t ∗ j tX j ◦ If t ∈ R, since (φX ) z = e · z , then ϕt is the ﬂow of X:

t ϕt = φX , (2.18)

18 and therefore it does not depend on J0. In this case, ϕt+s = ϕt ◦ ϕs.

◦ If τ, σ ∈ C \ R, then in general it is not true that ϕτ+σ = ϕτ ◦ ϕσ, which means that ϕ is not a

ﬂow. What we can say about about ϕτ+σ is that

ϕτ+σ = ϕτ+σ,J0 = ϕτ,J0 ◦ ϕσ,Jτ (2.19a)

= ϕσ+τ,J0 = ϕσ,J0 ◦ ϕτ,Jσ . (2.19b)

2.3 Restriction to Hamiltonian ﬂows

We now consider the case where (M, ω0,J0, g0) is a compact Kahler¨ manifold, with all three structures an real analytic. Let h ∈ C (M). Consider the Hamiltonian vector ﬁeld Xh deﬁned by h.

Proposition 2.3.1. Let f ∈ Can(M). Suppose that eτXh ·f is well deﬁned. Then, the Hamiltonian vector

ﬁeld of the function eτXh · f is given by:

τLXh XeτXh ·f = e · Xf . (2.20)

k Proof. We start by proving by induction on k that L Xf = X k . Xh Xh (f)

(k = 1) : LXh Xf = [Xh,Xf ] = X{h,f} = XXh(f)

k+1 k k k k k+1 (k =⇒ k + 1) : L Xf = LXh LX Xf = LXh XX (f) = [Xh,XX (f)] = X{h,X (f)} = X Xh h h h h Xh (f)

Now we prove the result.

∞ τ k ∞ τ k X X k τLX X τX = X k = X k = L X = e h · X . e h ·f P∞ τ Xk(f) X f Xh f f k=0 k! h k! h k! k=0 k=0

Note that in the case τ ∈ R, equation 2.20 is the known statement that symplectomorphisms preserve Hamiltonian vector ﬁelds.

1 n 1 n Proposition 2.3.2. Let (U, z , ..., z ) be a J0-complex coordinate chart on M and let (V, zτ , ..., zτ ) be a 1 n Jτ coordinate chart deﬁned by (U, z , ..., z ). Then on V we have that

τLX e h X j = X j , (2.21a) z zτ

τL¯ X e h X j = X j . (2.21b) z¯ z¯τ

Proof. The result follows from using proposition 2.3.1.

2.4 Action on Kahler¨ structures

In section 2.2 we gave a procedure that changes the complex structure of M from J0 to Jτ . Consider this procedure applied to (M, ω0,J0, g0). We will obtain a new structure (M, ω0,Jτ ).

19 Question: Is the resulting structure Kahler?¨ In other words, is it true that gτ := ω0(·,Jτ ·) is a Rieman- nian metric? Answer: At least for small values of |τ|, the answer is yes, provided by the following theorem.

Theorem 2.4.1 (Mourao˜ and Nunes, [14]). Let (M, ω0,J0, g0) be a compact Kahler¨ manifold, with ω0, J0, an g0 real analytic. Let h ∈ C (M). Then, there exists a T > 0 such that:

(i) For all τ ∈ B(0,T )(M, ω0,Jτ , gτ ) is a Kahler¨ manifold, where:

• Jτ is the complex structure deﬁned by applying theorem 2.2.1 to (M, ω0,J0, g0) with the vector

ﬁeld Xh;

• gτ := ω0(·,Jτ ·).

(ii)(K ahler¨ potential for (M, ω0,Jτ , gτ )) For all p ∈ M there exists:

1 n • (Uα, z0 , ..., z0 ) a J0-holomorphic coordinate neighborhood of p;

• κ0 : Uα −→ R a local Kahler¨ potential for (M, ω0,J0);

• Vα,p an open set such that:

◦ p ∈ Vα,p ⊂ Vα,p ⊂ Uα;

◦ Vα,p has compact closure;

◦ for all τ ∈ B(0,T ), ϕτ (Vα,p) ⊂ Uα;

◦ for all τ ∈ B(0,T ), κτ deﬁned by

i θ := (∂ − ∂¯ )κ (2.22a) 2 0 0 0 Z t sXh αt := e (θ(Xh))ds (2.22b) 0 ατ := unique complex analytic continuation of αt (2.22c) i ψ := − eτXh · κ + τh − α (2.22d) τ 2 0 τ

κτ := −2 Im ψτ (2.22e)

is well deﬁned on Vα,p (because the Lie series in (2.22b), (2.22c) and (2.22d) are uni-

formly convergent), and is a local Kahler¨ potential for (M, ω0,Jτ ).

Proof. Let T 0 be as in theorem 2.2.1. Along the proof, we will restrict T 0 as we need to. 0 (i): We need to prove that for all τ ∈ B(0,T ), ω0(·,Jτ ·) is symmetric and positive deﬁnite. Recall that symmetry is equivalent to ω0 being of type (1, 1). Since ω0 is of type (1,1) with respect to J0 and is preserved by Xh, it is of type (1, 1) with respect to Jτ :

!! ∂ ∂ τL ∂ ∂ ω , = e Xh ω , = 0. (2.23) 0 j ∂zk 0 j k ∂zτ τ ∂z0 ∂z0

0 0 ω0(·,J0·) is positive deﬁnite. By continuity, we can restrict T such that for all τ ∈ B(0,T ) ω0(·,Jτ ·) is positive deﬁnite.

20 1 n (ii): Let p ∈ M. There exists (Uα, z0 , ..., z0 ) a J0-holomorphic coordinate neighborhood of p that is the domain of a Kahler¨ potential κ0 : U −→ R. Let Vα,p be such that:

• p ∈ Vα,p,τ ⊂ Vα,p,τ ⊂ Uα;

• Vα,p,τ has compact closure.

By a theorem of Grobner and Knapp (theorem 3 of [7]), there exists Tα,p > 0 such that for all τ ∈

B(0,Tα,p) the series deﬁning κτ is uniformly convergent on Vα,p. Restrict Tα,p so that ϕτ (p) ∈ Uα, and shrink Vα,p so that ϕτ (Vα,p) ⊂ Uα. We have just proven that for all p ∈ M, there exists

1 n • (Uα, z0 , ..., z0 ) a J0-holomorphic coordinate neighborhood of p;

• κ0 : Uα −→ R a local Kahler¨ potential for (M, ω0,J0);

• Tα,p > 0, Vα,p an open set such that:

◦ p ∈ Vα,p ⊂ Vα,p ⊂ Uα;

◦ Vα,p has compact closure;

◦ for all τ ∈ B(0,Tα,p), ϕτ (Vα,p) ⊂ Uα;

◦ for all τ ∈ B(0,Tα,p), the series deﬁning κτ is uniformly convergent on Vα,p.

0 Let {Vαj ,pj }j=1,...,N be a ﬁnite open cover of M. Take T = min{Tα1,p1 , ..., TαN ,pN ,T }. This is the T of the statement of the theorem, since each p belongs to some Vαj ,pj . We now prove that κτ is a Kahler¨ potential for (M, ω0,Jτ ). The proof of this is a lengthy computation. We will divide it into parts.

Part 1: we prove that θ is real and that dθ = −ω0.

i θ¯ = − (∂¯ − ∂ )κ = θ 2 0 0 0

i i i i θ = ∂ κ − ∂¯ κ =⇒ θ(1,0) = ∂ κ and θ(0,1) = − ∂¯ κ 2 0 0 2 0 0 2 0 0 2 0 0 | {z } | {z } ∈Ω(1,0)(M) ∈Ω(0,1)(M) i i dθ = − (∂ + ∂¯ )(∂¯ − ∂ )κ = − (∂ ∂¯ − ∂¯ ∂ )κ = −i∂ ∂¯ κ = −ω 2 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0

P∞ τ k k−1 Part 2: we prove that ατ = k=1 k! Xh (θ(Xh)).

∞ Z t Z t X uk α = euXh (θ(X ))du = Xk(θ(X ))du t h k! h h 0 0 k=0 ∞ ∞ X 1 Z t X 1 tk+1 = ukdu Xk(θ(X )) = Xk(θ(X )) k! h h k! k + 1 h h k=0 0 k=0 ∞ X tk = Xk−1(θ(X )) k! h h k=1

Therefore, the complex analytic continuation of αt is

21 ∞ X τ k α = Xk−1(θ(X )). τ k! h h k=1

Part 3: we prove that ∀k ≥ 2 : Lk θ = (dι )kθ. We proceed by induction on k. Xh Xh (k = 2) :

L2 θ = (dι + ι d)2θ Xh Xh Xh

= dιX dιX θ + dιX ιX dθ +ιX dd ιX θ + ιX dιX dθ h h h h |{z} h |{z} h h h |{z} =−ω0 =0 =−ω0 2 = (dιXh ) θ − d(ω(Xh,Xh)) − ιXh ddh

2 = (dιXh ) θ

(k =⇒ k + 1) :

Lk+1θ = (dι + ι d)Lk θ = (dι + ι d)(dι )kθ Xh Xh Xh Xh Xh Xh Xh k+1 k k+1 = (dιXh ) θ + ιXh d(dιXh ) θ = (dιXh ) θ

Part 4: we prove that ∀k ≥ 2 : d Xk−1(θ(X )) = Lk (θ). We proceed by induction on k. h h Xh (k = 2) : L2 θ = (dι )2θ = dι dι θ = dι d(θ(X )) = d(X (θ(X ))) Xh Xh Xh Xh Xh h h h

(k =⇒ k + 1) :

Lk+1θ = (dι + ι d)d Xk−1(θ(X )) = dι d Xk−1(θ(X )) = d Xk(θ(X )) Xh Xh Xh h h Xh h h h h

τdιX Part 5: we prove that dατ = e h θ − θ.

∞ ! ∞ ! X τ k X τ k dα = d Xk−1(θ(X )) = τd(θ(X )) + d Xk−1(θ(X )) τ k! h h h k! h h k=1 k=2 ∞ τ k ∞ τ k X k X k τdιX = τd(θ(Xh)) + L (θ) = (dιX ) (θ) = e h θ − θ k! Xh k! h k=2 k=1

τLX Part 6: we prove that e h θ = θ − τdh + dατ .

∞ τ k ∞ τ k τLX X k X k e h θ = L (θ) = θ + τLX (θ) + L (θ) k! Xh h k! Xh k=0 k=2 ∞ X τ k = θ + τ(dι + ι d)θ + (dι )k(θ) Xh Xh k! Xh k=2 ∞ X τ k = θ − τdh + τdι θ + (dι )k(θ) Xh k! Xh k=2 ∞ X τ k = θ − τdh + (dι )k(θ) = θ − τdh + dα k! Xh τ k=1

22 (0,1)τ Part 7: we prove that θ = ∂¯ ψ . This is equivalent to θ(X j ) = dψ (X j ) ∀j = 1, ..., n. τ τ zτ τ zτ

i τXh dψτ (X j ) = − d(e · κ0)X j + τdh(X j ) − dατ (X j ) zτ 2 zτ zτ zτ i τXh τLX = − d(e · κ0)X j + θ(X j ) − (e h θ)(X j ), 2 zτ zτ zτ

which is equal to θ(X j ), since zτ

i τLX τLX τLX τXh τXh ¯ (e h θ)(X j ) = (e h θ)(e h Xzj ) = e (θ(Xzj )) = − e (∂0κ0(Xzj )) zτ 2 i i i τXh τLX τLX τLX = − e (dκ0(Xzj )) = − (e h (dκ0))(e h Xzj ) = − d(e h κ0)X j . 2 2 2 zτ

Part 8: we prove the result.

¯ ¯ ¯ ¯ ¯ i∂τ ∂τ κτ = −i∂τ ∂τ (2 Im ψτ ) = ∂τ ∂τ ψτ − ∂τ ∂τ ψτ

¯ ¯ ¯ ¯ (0,1)τ (0,1)τ = −∂τ ∂τ ψτ − ∂τ ∂τ ψτ = −∂τ θ − ∂τ θ

¯ (1,0)τ (0,1)τ (1,0)τ (0,1)τ ¯ (1,0)τ ¯ (0,1)τ = −∂τ θ − ∂τ θ = −∂τ θ − ∂τ θ − ∂τ θ − ∂τ θ

¯ (1,0)τ (0,1)τ = −(∂τ + ∂τ )(θ + θ ) = −dθ = ω0

Proposition 2.4.2. Let T be as in the previous theorem, and let τ ∈ B(0,T ). Let s ∈ R be such that ∗ τ + s ∈ B(0,T ). Then ϕsκτ = κτ+s.

∗ Proof. We start by proving that ϕsαt = αt+s − αs.

Z t ∗ sXh sXh uXh ϕsαt = e · αt = e e (θ(Xh))du 0 Z t Z t+s (s+u)Xh uXh = e (θ(Xh))du = e (θ(Xh))du 0 s Z t+s Z s uXh uXh = e (θ(Xh))du − e (θ(Xh))du = αt+s − αs 0 0

sX Therefore, by complex analytic continuation, e h · ατ = ατ+s − αs.

To prove the result, we now use the deﬁnition of κτ in terms of ατ given by theorem 2.4.1.

1 κ = i(ψ − ψ¯ ) = (eτXh − eτX¯ h ) · κ + i(τ − τ¯)h − i(α − α ) τ τ τ 2 0 τ τ¯

Therefore

1 sXh (s+τ)Xh (s+¯τ)Xh sXh sXh sXh e · κτ = e − e · κ0 + i (τ + s) − (¯τ + s) e · h −i( e ατ − e ατ¯ ) 2 | {z } | {z } | {z } =h =ατ+s−αs =ατ¯+s−αs 1 = e(s+τ)Xh − e(s+¯τ)Xh · κ + i (τ + s) − (¯τ + s) h − i(α − α ) = κ . 2 0 τ+s τ¯+s τ+s

23 2.5 Geodesics on the Space of Kahler¨ metrics

In this subsection we deﬁne and describe the space of Kahler¨ metrics. We also explain what are curves in this space, deﬁne the Mabuchi metric and give the geodesic equation for this metric.

Deﬁnition 2.5.1. The space of Kahler¨ metrics on M in the cohomology class [ω0] is

∗ ∗ ∗ H(ω0,J0) := {ϕ ω0 | ϕ ∈ Diﬀ(M), [ϕ ω0] = [ω0], (M, ϕ ω, J0) is Kahler¨ }. (2.24)

The space of Kahler¨ potentials on M with base point [ω0] is

∞ ¯ K(ω0,J0) := {φ ∈ C (M) | g˜ := (ω0 + i∂0∂0φ)(·,J0·) is positive deﬁnite}. (2.25)

Remark 2.5.2.

¯ • As stated in [5], by the ∂∂-lemma any other Kahler¨ metric that is cohomologous to ω0 can be

written using a global Kahler¨ potential. This implies that H(ω0,J0) can be identiﬁed with the space of Kahler¨ potentials, modulo constants:

∼ H(ω0,J0) = K(ω0,J0)/R (2.26a) Z ∼ ∞ ¯ n = φ ∈ C (M) | g˜ := (ω0 + i∂0∂0φ)(·,J0·) 0, φω0 = 0 . (2.26b) M

We give a proof that this bijection exists below.

•H (ω0,J0) can be regarded as an inﬁnite dimensional manifold, where a tangent vector δφ0 to ∞ H(ω0,J0) at φ0 is a function on M. To see this, consider a curve in H(ω0,J0), t 7→ φt ∈ C (M) R n (where we have chosen a family of representatives for the equivalence classes such that M φtω0 =

0, so that the map t 7→ φt is smooth). The derivative with respect to t is another function:

d ∞ δφ0 := φt ∈ C (M). (2.27) dt t=0

•H (ω0,J0) can be equipped with a Riemannian metric called the Mabuchi metric, which is given by

Z 1 hδ1φ, δ2φi = (δ1φ · δ2φ) ωφ ∧ ... ∧ ωφ, (2.28) M n! ¯ where ωφ = ω0 + i∂0∂0φ. As proven in [5], the Riemannian manifold H(ω0,J0) admits a unique

Levi-Civita connection. A curve {φt}t∈I is a geodesic for this connection if and only if

¨ 1 ˙ 2 φt = k∇g˜ φtk , (2.29) 2 t g˜t

where ∇g˜t is the gradient, and k · kg˜t is the norm, both with respect to the metric g˜t = (ω0 + ¯ i∂0∂0φt)(·,J0·).

24 We now prove the statement of 2.5.2. Note that the proof of the following theorem is just a description of the mentioned bijection.

Theorem 2.5.3. There exists a bijection

∼ H(ω0,J0) = K(ω0,J0)/R.

∗ ∗ Proof. Part 1: We give a map H(ω0,J0) −→ K(ω0,J0)/R. Let ϕ ω0 ∈ H(ω0,J0). Since [ϕ ω0] = [ω0], ¯ ∞ ∗ ¯ ∗ by the ∂∂-Lemma there exists a φ ∈ C (M) such that ϕ ω0 = ω0 + i∂0∂0φ. Since (M, ϕ ω, J0) is ¯ ∗ Kahler,¨ (ω0 + i∂0∂0φ)(·,J0·) = (ϕ ω0)(·,J0·) is positive definite, and φ belongs to K(ω0,J0)/R. Define ∗ the image of ϕ ω0 to be the equivalence class of φ in K(ω0,J0)/R. We still have to show that this map 0 H(ω0,J0) −→ K(ω0,J0)/R is well defined. Specifically, we have to show that if φ is another function in ∗ ¯ 0 0 ¯ 0 M such that ϕ ω0 = ω0 + i∂0∂0φ , then φ and φ differ by a constant. This is true, since i∂0∂0(φ − φ ) = 0 and M is compact (by lemma 1.3.22). ¯ Part 2: We give a map K(ω0,J0)/R −→ H(ω0,J0) Let φ ∈ K(ω0,J0)/R. Note that i∂0∂0φ does not ¯ depend on the chosen representative for this equivalence class. Since ω0 + i∂0∂0φ is of type (1, 1), g˜ is ¯ symmetric. By hypothesis it is positive definite, hence it is a Riemannian metric. ω0 + i∂0∂0φ is closed. ¯ Since g˜ is nondegenerate and J0 is an isomorphism, ω0 +i∂0∂0φ is symplectic. We have just proven that ¯ ¯ ¯ ¯ (M, ω0 + i∂0∂0φ, J0) is Kahler.¨ Since i∂0∂0φ = d(i∂0φ) is exact, [ω0 + i∂0∂0φ] = [ω0]. We now prove that ¯ ∗ ¯ ω0 + i∂0∂0φ is of the form ϕ ω0, for some ϕ. If this is true, then ω0 + i∂0∂0φ belongs to H(ω0,J0), so we ¯ can define the image of φ to be ω0 + i∂0∂0φ. Define

¯ ωt = (1 − t)ω0 + t(ω0 + i∂0∂0φ).

Then ωt is closed. Also,

¯ ωt(·,J0·) = (1 − t)ω0(·,J0·) + t(ω0 + i∂0∂0φ)(·,J0·) is positive deﬁnite, and again since J0 is an isomorphism, ωt is nondegenerate. So, ωt is symplectic, ∞ ∗ ¯ and using Moser’s theorem we conclude that there exists a ϕ ∈ C (M) such that ϕ ω0 = ω0 + i∂0∂0φ. It is easily seen that the given functions are the inverse of the other.

2.6 Symplectic picture and complex picture

Consider the Kahler¨ manifold (M, ω0,J0, g0). We compare two different, but as we will prove equivalent, ways of changing the Kahler¨ structure:

• Symplectic picture: Fixed ω0, varying Jτ .

an Let h ∈ C (M). Using the procedure described in the previous sections with the vector ﬁeld Xh

we get a new Kahler¨ structure (ω0,Jτ ).

• Complex picture: Fixed J0, varying ωτ .

25 ¯ Let φτ ∈ H(ω0,J0). Consider the new symplectic form ωτ := ω0 + i∂0∂0φτ . We get a new Kahler¨

structure (ωτ ,J0).

Let (M, ω, J) be a Kahler¨ manifold and φ: M −→ M be a diffeomorphism. It is possible to prove that:

(i) φ∗ω is a symplectic form on M;

(ii) φ∗J is a complex structure on M;

(iii) (M, φ∗ω, φ∗J) is a Kahler¨ manifold.

This fact can be used to write the symplectic picture in terms of the complex picture, in the following way: consider a Kahler¨ structure (ω0,Jτ ) obtained from (ω0,J0) by applying the method of the previous sections (symplectic picture). The new Kahler¨ structure comes with a biholomorphism

ϕτ :(M,Jτ ) −→ (M,J0). (2.30)

−1 ∗ Deﬁne ωτ = (ϕτ ) ω0, so that ϕτ is a Kahler¨ isomorphism:

ϕτ :(M, ω0,Jτ ) −→ (M, ωτ ,J0). (2.31)

Then, (ωτ ,J0) is a Kahler¨ structure, isomorphic to (ω0,Jτ ), and with the same complex structure as the initial Kahler¨ structure.

2.7 The path of Kahler¨ metrics (generated by a complex ﬂow of an Hamiltonian vector ﬁeld) is a geodesic

an Let (M, ω0,J0) be Kahler.¨ Let h ∈ C (M), and consider its Hamiltonian vector field Xh. Let T be as in theorem 2.4.1. For all τ = it ∈ B(0,T ), where t ∈ R, by acting with the complexified flow of Xh we obtain new Kahler¨ structures (ω0,Jit) on M. We consider the path {(ω0,Jit)}t∈(−T,T ). Consider the path −1 ∗ −1 ∗ of isomorphic Kahler¨ structures {((ϕit ) ω0, (ϕit ) Jit)}t∈(−T,T ) = {(ωit,J0)}t∈(−T,T ). In short:

itX ϕ e h ∼it (M, ω0,J0, g0) (M, ω0,Jit, git) = (M, ωit,J0, g˜it). (2.32)

Theorem 2.7.1. ωit ∈ H(ω0,J0), for all t ∈ (−T,T ).

∗ Proof. Deﬁne Φt := ϕit. Then Φt ωit = ω0. Since Φt is homotopic to the identity, we have that:

∗ ∗ [ω0] = [Φt ωit] = Φt [ωit] = [ωit].

Therefore ωit ∈ H(ω0,J0).

¯ For each t ∈ (−T,T ), ωit ∈ H(ω0,J0), so we can ﬁnd a φt such that ωit = ω0 + i∂0∂0φt. Writing ω0 in

26 terms of its local Kahler¨ potentials:

¯ ω0 = i∂0∂0κ0, ¯ ω0 = i∂it∂itκit.

∗ Also, recall that ω0 = Φt ωit. Using these formulas,

¯ ∗ ∗ ¯ ¯ i∂it∂itκit = ω0 = Φt ωit = iΦt ∂0∂0(κ0 + φt) = i∂it∂it (κ0 + φt) ◦ Φt .

This suggests that we deﬁne −1 φt = κit ◦ Φt − κ0. (2.33)

0 0 However, this definition only works if when we choose a different Kahler¨ potential κ0 (that defines κit), −1 0 −1 0 the functions κit ◦ Φt − κ0 and κit ◦ Φt − κ0 agree on the intersection of their domains of definition.

Deﬁnition 2.7.2. For each t ∈] − T,T [, deﬁne a function φt on M in the following way. Let p ∈ M. −1 Let U, V be neighborhoods of ϕτ (p) as in theorem 2.4.1, with local Kahler¨ potentials κ0 : U −→ R and

κτ : V −→ R. Then, in the neighborhood ϕτ (V ) of p

−1 φt|ϕτ (V ) = κτ ◦ ϕτ − κ0. (2.34)

Proposition 2.7.3.

(i) φt is well deﬁned.

¯ (ii) ωit = ω0 + i∂0∂0φt.

Proof. (i): To prove this, one must use equations (2.22a) to (2.22e) given in theorem 2.4.1, that deﬁne 0 0 0 κτ in terms of κ0, for both κτ and κτ , and then, check that κτ − κτ = (κ0 − κ0) ◦ ϕτ . Performing the computations:

∞ ! X τ k κ0 − κ = Im ieτXh · (κ0 − κ ) + i Xk−1(∂ (κ0 − κ )(X ) − ∂¯ (κ0 − κ )(X )) τ τ 0 0 k! h 0 0 0 h 0 0 0 h k=1 ∞ ! X τ k = Im i(κ0 − κ ) + 2i Xk−1(∂ (κ0 − κ )(X )) 0 0 k! h 0 0 0 h k=1 ∞ X τ k τ¯k = κ0 − κ + Xk−1(∂ (κ0 − κ )(X )) + Xk−1(∂¯ (κ0 − κ )(X )) 0 0 k! h 0 0 0 h k! h 0 0 0 h k=1

0 ¯ 0 κ0 − κ0 is a real analytic function deﬁned on an open subset, such that ∂0∂0(κ0 − κ0) = 0. By lemma 0 ¯ ¯ 1.3.23, κ0 − κ0 is locally of the form f + f, where f is holomorphic and f is anti-holomorphic. On each 0 ¯ smaller open set where κ0 − κ0 = f + f:

∞ ∞ X τ k X τ¯k κ0 − κ = f + Xk−1(df(X )) + f¯+ Xk−1(df¯(X )) τ τ k! h h k! h h k=1 k=1 τXh τX¯ h ¯ 0 = e · f + e · f = (κ0 − κ0) ◦ ϕτ .

27 0 0 Therefore κτ − κτ = (κ0 − κ0) ◦ ϕτ on the whole domain where this expression is deﬁned. (ii): The proof is the following computation:

¯ ¯ ¯ −1 ¯ ω0 + i∂0∂0φt = i∂0∂0κ0 + i∂0∂0(κit ◦ Φt ) − i∂0∂0κ0 ¯ −1 ∗ −1 ∗ ¯ = i∂0∂0((Φt ) κit) = (Φt ) i∂it∂it(κit)

−1 ∗ = (Φt ) ω0 = ωit

The following diagram illustrates the construction and main formulas so far:

eitXh Φ∼t (M, ω0,J0, g0) (M, ω0,Jit, git) = (M, ωit,J0, g˜it) ¯ ¯ ¯ ω0 = i∂0∂0κ0 ω0 = i∂it∂itκit ωit = ω0 + i∂0∂0φt ∈ H(ω0,J0) (2.35)

∗ Φt ωit = ω0

Also, recall the formulas for κit (given in theorem 2.4.1):

i θ := (∂ − ∂¯ )κ (2.36a) 2 0 0 0 ∞ X (it)k α := Xk−1(θ(X )) (2.36b) it k! h h k=1 i ψ := − eitXh · κ + ith − α (2.36c) it 2 0 it

κit := −2 Im ψit (2.36d)

Theorem 2.7.4 (Mourao˜ and Nunes, [14]). φt satisﬁes:

¨ 1 ˙ 2 φt = k∇g˜ φtk . (2.37) 2 it g˜it

Therefore, the path of Kahler¨ metrics {(ω0,Jit)}t∈(−T,T ) is a geodesic.

Proof. We divide the proof into parts. d Part 1: We show that dt αit = iXh(αit) + iθ(Xh).

∞ ! ∞ d d X (it)k X (it)k−1 α = Xk−1(θ(X )) = i Xk−1(θ(X )) dt it dt k! h h (k − 1)! h h k=1 k=1 ∞ ! X (it)k = iX Xk−1(θ(X )) + iθ(X ) = iX (α ) + iθ(X ) h k! h h h h it h k=1

d ¯ Part 2: We show that dt κit = −Xh(ψit + ψit) − 2h + 2θ(Xh). Using (2.36c) and (2.36d), we conclude that

1 1 κ = eitXh · κ − e−itXh · κ − 2th − iα + iα . it 2 0 2 0 it −it

28 Therefore

d 1 1 d d κ = X ieitXh · κ + X (ie−itXh · κ ) − 2h − i α + i α dt it 2 h 0 2 h 0 dt it dt −it i i = −X − eitXh · κ − e−itXh · κ − α − α − 2h + 2θ(X ) h 2 0 2 0 it −it h ¯ = −Xh(ψit + ψit) − 2h + 2θ(Xh).

Part 3: Let f ∈ C∞(M). We show that:

d (f ◦ ϕ−1) = ∂ f(−iX ) + ∂¯ f(iX ) ◦ ϕ−1. (2.38) dt it it h it h it

We want to compute

d d d d (f ◦ ϕ−1) = df ϕ−1 = ∂ f ϕ−1 + ∂¯ f ϕ−1 . (2.39) dt it dt it it dt it it dt it

j −1 j −itXh j Recall that ϕit is the unique biholomorphism satisfying zit ◦ ϕit = z = e · zit.

d d d dzj ϕ−1 = zj ◦ ϕ−1 = e−itXh · zj = −iX e−itXh · zj it dt it dt it it dt it h it j −1 −1 ∗ j −1 ∗ j = −iXh zit ◦ ϕit = −iXh (ϕit ) zit = d (ϕit ) zit (−iXh) j −1 = dzit d(ϕit )(−iXh)

Performing complex conjugation, we conclude that:

d dzj ϕ−1 = dzj d(ϕ−1)(−iX ) , (2.40) it dt it it it h d dz¯j ϕ−1 = dz¯j d(ϕ−1)(iX ) . (2.41) it dt it it it h

j ¯ j ∂itf is a linear combination of the dzit, and ∂itf is a linear combination of the dz¯it. Because of this, equations (2.39), (2.40) and (2.41), imply that:

d (f ◦ ϕ−1) = ∂ f d(ϕ−1)(−iX ) + ∂¯ f d(ϕ−1)(iX ) dt it it it h it it h −1 ∗ ¯ = (ϕit ) ∂itf(−iXh) + ∂itf(iXh) ¯ −1 = ∂itf(−iXh) + ∂itf(iXh) ◦ ϕit .

˙ d ¯ −1 Part 4: We show that φt = dt κit + ∂itκit(−iXh) + ∂itκit(iXh) ◦ Φt .

˙ d d −1 d −1 d −1 rClφt = φt = κit ◦ Φt = κis ◦ Φt + κit ◦ Φs dt dt ds s=t ds s=t

Now the result follows form using equation (2.38).

29 ˙ −1 Part 5: We show that φt = −2h ◦ Φt .

d φ˙ = κ + ∂ κ (−iX ) + ∂¯ κ (iX ) ◦ Φ−1 t dt it it it h it it h t ¯ ¯ −1 = −Xh(ψit + ψit) − 2h + 2θ(Xh) + ∂itκit(−iXh) + ∂itκit(iXh) ◦ Φt making the following substitutions:

¯ ¯ Xh(ψit) = dψit(Xh) = ∂itψit(Xh) + ∂itψit(Xh) (and analogously for ψit)

(1,0)it (0,1)it ¯ ¯ θ = θ + θ = ∂itψit + ∂itψit

κit = −2 Im ψit

˙ −1 we obtain that φt = −2h ◦ Φt . Part 6: We show that

φ¨ ◦ Φ = 2k∇ hk2 t t git git (2.42)

˙ −1 Using the fact that φt = −2h ◦ Φt ,

d d φ¨ = φ˙ = −2 (h ◦ Φ−1) = −2 ∂ h(−iX ) + ∂¯ h(iX ) ◦ Φ−1, t dt t dt t it h it h t where again we used (2.38). Therefore

¨ ¯ φt ◦ Φt = 2i∂ith(Xh) − 2i∂ith(Xh)

= 2dh(JitXh) = 2ω(Xh,JitXh)

= 2g (X ,X ) = 2k∇ hk2 . it h h git git

Part 7: We show that k∇ hk2 = 1 k∇ φ˙ k2 ◦ Φ . From this and (2.42), the result follows. We git git 4 g˜it t g˜it t note the following equivalences:

2 1 ˙ 2 k∇g hk = k∇g˜ φtk ◦ Φt it git 4 it g˜it 1 ˙ ˙ ⇐⇒ git (∇git h, ∇git h)p = g˜it ∇g˜it φt, ∇g˜it φt 4 Φt(p) ∗ 1 ˙ ˙ ⇐⇒ Φt g˜it (∇git h, ∇git h)p = g˜it ∇g˜it φt, ∇g˜it φt 4 Φt(p) 1 ⇐⇒ g˜ (dΦ ∇ h, dΦ ∇ h) = g˜ ∇ φ˙ , ∇ φ˙ it t git t git Φt(p) it g˜it t g˜it t 4 Φt(p)

˙ Therefore it sufﬁces to prove that ∇g˜it φt = 2dΦt∇git h:

˙ ˙ −1 −1 g˜it ∇g˜it φt, · = dφt = −2dh ◦ dΦt = −2git(∇git h, ·) ◦ dΦt

−1 ∗ −1 = −2git(∇git h, dΦt ·) = −2Φt g˜it(∇git h, dΦt ·)

=g ˜it(−2dΦt∇git h, ·)

30 Chapter 3

Kahler¨ structures on the Cotangent Bundle of a Lie Group

Let G be a compact Lie group, with a left and right invariant Riemannian metric h·, ·i. In this chapter we study possible Kahler¨ structures on T ∗G. We start by giving an overview of known Kahler¨ structures on T ∗G. These are the standard Kahler¨ structure [12], the Kahler¨ structure of Hall and Kirwin [10], and the Kahler¨ structure of Kirwin, Mourao˜ and Nunes [12]. We study if it is possible to strengthen the result of Hall and Kirwin to R = ∞ if the manifold M is a Lie group, and if the condition of h being Ad-invariant is necessary in the theorem of Kirwin, Mourao˜ and Nunes. We also compare the various Kahler¨ structures.

3.1 T ∗G is diffeomorphic to G × g

Each element of T ∗G can be described in terms of its base point in G and the correspondent covector by left translation in g∗. In more precise terms, the following map is a diffeomorphism:

T ∗G −→ G × g∗ (3.1)

∗ α 7−→ (π(α), (dLπ(α)) α).

The metric deﬁnes a correspondence between vectors and covectors:

∗ [: g g :# (3.2) Y 7→ hY, ·i.

These two diffeomorphisms can be composed into a diffeomorphism

φ: T ∗G −→ G × g (3.3)

∗ # α 7−→ (π(α), ((dLπ(α)) α) ).

31 3.2 Kahler¨ structures on T ∗G

3.2.1 Standard Kahler¨ structure on T ∗G

We start by explaining some necessary theory about the complexiﬁcation of a Lie group.

Remark 3.2.1 (Complexiﬁcation of a Lie group).

• Let G be a Lie group. A universal complexiﬁcation of G is a complex Lie group GC with a Lie

group homomorphism ι : G −→ GC such that for any Lie group homomorphism ρ: G −→ H (where

H is a complex Lie group), there exists an unique Lie group homomorphism ρC : GC −→ H such

that ρC ◦ ι = ρ:

ι G GC ∃!ρ ρ C H

It is a fact that for any Lie group G the universal complexiﬁcation exists and is unique up to isomorphism.

• In the case where G is compact, the universal complexiﬁcation has a simpler description. If G is compact, as a consequence of representation theory, G can be regarded as a matrix Lie group. iX Then, the matrix Lie group GC = ge : g ∈ G, X ∈ g with product, differential structure and complex structure given by those of the matrices with complex entries, and with inclusion

ι: G −→ GC (3.4) g 7−→ gei0

is the universal complexiﬁcation of G.

iY • The complex structure of G at the point ge is a map JG : T iY (G ) −→ T iY (G ). By left C C ge C ge C ∼ translation, T iY (G ) = g , which in turn is isomorphic to g × g. Therefore, JG can be regarded ge C C C

as a map JG : g × g −→ g × g, which is given as a block matrix as (see [12], page 1468): C

  0 −I JG =   . (3.5) C I 0

• If we deﬁne

ψτ : G × g −→ GC (3.6) (g, Y ) 7−→ geτY

then for Im τ 6= 0, ψτ are diffeomorphisms (see [12], page 1467).

32 Recall that from example 1.1.6, T ∗G is a symplectic manifold with symplectic form ω. Consider the following diffeomorphism:

∗ φ ψi T G −→ G × g −→ GC. (3.7)

∗ Deﬁne a complex structure on T G by pulling back the complex structure of GC:

∗ JST = (ψi ◦ φ) JG . (3.8) C

∗ ∗ Then, (T G, ω, JST ) is a Kahler¨ manifold, and (ω, JST ) is the standard Kahler¨ structure on T G.

3.2.2 The Kahler¨ structure of Hall and Kirwin

In [10], Hall and Kirwin describe a Kahler¨ structure on a tubular open set of T ∗M, where M is a real analytic compact Riemannian manifold.

Theorem 3.2.2 (Hall and Kirwin, [10]). Let M be a compact, real analytic manifold equipped with a real analytic Riemannian metric g. Let T ∗,RM denote the tubular neighborhood of radius R of the zero section:

T ∗,RM := α ∈ T ∗M|g(α#, α#) < R2 . (3.9)

Let

∗ h: T M −→ R (3.10) 1 α 7−→ g(α#, α#). 2

Let β ∈ Ω2(M) be closed, and deﬁne

∗ ωβ = ωST − π β, (3.11)

∗ where ωST denotes the standard symplectic form on T M. Then, ωβ is a symplectic form, and there exists an R such that:

∗,R (i) There exists an unique complex structure on T M, JMCS, (called the magnetic complex structure), such that

∗,R −iLX T1,0(T M) = e h (ker dπ ⊗ C), (3.12)

where Xh is the Hamiltonian vector ﬁeld of h with respect to ωβ.

∗,R (ii) (T M, ωβ,JMCS) is Kahler.¨

33 3.2.3 The Kahler¨ structure of Kirwin, Mourao˜ and Nunes

Let G be a compact Lie Group, equipped with a bi-invariant Riemannian metric h·, ·i. In [12], Kirwin, Mourao˜ and Nunes describe a Kahler¨ structure on T ∗G, that can be obtained by changing the complex structure by means of a Thiemann complexiﬁer function. We present some necessary deﬁnitions.

Deﬁnition 3.2.3. Let h: T ∗G −→ R be left invariant:

∗ ∗ h((dLg) α) = h(α) ∀α ∈ T G, g ∈ G. (3.13)

We refer to h as the Thiemann complexiﬁer function. We deﬁne:

• u, the gradient of h:

u: g −→ g (3.14)

Y 7−→ (dh)Y [ ,

∗ ∗ ∗ where (dh)Y [ ∈ g in the following sense. (dh)Y [ is a map TY [ (T G) −→ R. Note that g = Te G ⊂ ∗ ∗ ∗ TY [ (T G). Restrict (dh)Y [ to g , so that it becomes a linear functional g −→ R:

∗ ∗ ∗ (dh)Y [ : g = Te G ⊂ TY [ (T G) −→ R.

∗∗ Then (dh)Y [ is an element of g , which is naturally isomorphic to g via the double dual isomorphism.

• H, the jacobian of h:

H : g −→ End(g) (3.15)   H(Y ): g −→ g Y 7−→   . W 7−→ (du)Y (W )

• ϕh, the map associated to h:

∗ ∗ ϕh : T G −→ T G (3.16)   ϕh(α): Tπ(α)G −→ R α 7−→   , [ Y 7−→ (dh)α(Y )

[ ∗ ∗ where Y ∈ Tπ(α)G ⊂ Tα(T G).

Theorem 3.2.4 (Kirwin, Mourao˜ and Nunes, [12]). Let G be a compact Lie Group, equipped with a bi-invariant Riemannian metric h·, ·i. Let h: T ∗G −→ R satisfy:

∗ ∗ • h is left invariant: h((dLg) α) = h(α) ∀α ∈ T G, g ∈ G;

34 ∗ ∗ • h is right invariant: h((dRg) α) = h(α) ∀α ∈ T G, g ∈ G;

• H is positive deﬁnite everywhere: hW, H(Y )W i > 0 ∀Y,W ∈ g; kH(Y )Zk • The norm of H has a greater than 0 lower bound: inf sup kZk > 0. Y ∈g Z∈g

Then,

(i) ϕh is a diffeomorphism.

τ ∗ ∗ τ (ii) Deﬁne J = (ψτ ◦ φ ◦ ϕh) JG . If Im τ > 0 then (T G, ω, J ) is Kahler.¨ C

3.2.4 Problems to study

We now pose the questions that we will investigate in this chapter. Question 1: In the case of a compact Lie group with bi-invariant Riemannian metric, what is the relation between the Kahler¨ structures of theorem 3.2.2 (which may only be deﬁned on a tube) and of theorem 3.2.4? Question 2: In the case where M = G is a Lie group and the Riemannian metric is bi-invariant, can we strengthen the result to R = ∞? In theorem 3.2.4, the condition that h is left invariant means that when we consider h to be a function h: G × g −→ R, it does not depend on the point of G, that is, it is a function h: g −→ R. The condition that h is right invariant means that h: g −→ R is Ad-invariant:

h(AdgY ) = h(Y ) ∀g ∈ G, Y ∈ g. (3.17)

Question 3: In theorem 3.2.4, is the condition of Ad-invariance necessary?

3.3 Useful coordinates based on the diffeomorphism T ∗G ∼= G × g

In this section we introduce a basis of vector ﬁelds and of forms in G × g, that we will use to study the questions posed in the previous section. Let:

1 n •{ T1, ..., Tn} be a h·, ·i-orthonormal basis of g, with associated coordinates {y , ..., y };

•{ X1, ..., Xn} be the left invariant vector ﬁelds on G such that (Xj)e = Tj;

1 n j j •{ ω , ..., ω } be the basis of differential forms in G that is dual to {X1, ..., Xn}, that is ω (Xk) = δk.

Then,

∂ ∂ T (G × g) = span{X , ..., X } ⊕ span , ..., , (3.18) (g,Y ) 1 n ∂y1 ∂yn ∗ 1 n 1 n T(g,Y )(G × g) = span{ω , ..., ω } ⊕ span dy , ..., dy . (3.19)

35 Proposition 3.3.1. In the coordinates that we have just described,

n X θ = yjωj, (3.20) j=1 n  n  X 1 X ω = = −dθ = ωj ∧ dyj + Cj yjωk ∧ ωl . (3.21)  2 kl  j=1 k,l=1

Proof. Recall that from example 1.1.6, θ is given by:

n ! n !! X ∂ X ∂ θ akX + bk = (dL Y )[ dπ akX + bk . (3.22) (x,Y ) k ∂yk x k ∂yk k=1 k=1 As a consequence of the metric being left invariant,

[ ∗ [ (dLxY ) = (dLx−1 ) Y . (3.23)

Therefore,

n ! n !! X ∂ X ∂ θ akX + bk = (dL Y )[ dπ akX + bk (x,Y ) k ∂yk x k ∂yk k=1 k=1 n ! n ! ∗ [ X k [ X k = (dLx−1 ) Y a Xk = Y a Tk k=1 k=1 * n + n X k X j k = Y, a Tk = y a hTj,Tki k=1 j,k=1 n n n ! X X X ∂ = yjaj = yjωj akX + bk , k ∂yk j=1 j=1 k=1

n n X X ω = −dθ = −d yjωj = − dyj ∧ ωj + yjdωj j=1 j=1 n  n  X 1 X = ωj ∧ dyj + Cj yjωk ∧ ωl .  2 kl  j=1 k,l=1

The metric h·, ·i being left invariant means that is is deﬁned by left translation from the metric at the identity.

Proposition 3.3.2. Given that h·, ·i is left invariant, it being right invariant is equivalent to the fact that the metric at the identity is Ad-invariant:

hAdgY, AdgZi = hY,Zi ∀g ∈ G, ∀Y,Z ∈ g. (3.24)

The metric at the identity being Ad-invariant implies that it is ad-invariant:

hadX Y,Zi + hY, adX Zi = 0 ∀X,Y,Z ∈ g. (3.25)

36 Proof. The ﬁrst statement is a consequence of the fact that Adg = dLgdRg−1 = dRg−1 dLg. Let X,Y,Z ∈ g. Equation (3.24) implies that

hAdexp(tX)Y, Adexp(tX)Zi = hY,Zi.

Differentiating both sides with respect to t and setting t = 0 we obtain equation (3.25).

Proposition 3.3.3. The components of adY in the basis {Tj}j=1,...,n are given by:

j n h i X l j adY = y Clk. (3.26) k l=1

Also, adY is anti-symmetric:

h ij h ik adY = − adY . (3.27) k j

Proof. To prove the ﬁrst statement, notice that ad is given by Lie brackets, which in turn are determined by the structure constants:

n j n n n ! X h i X l X X l j adY Tj = adY (Tk) = [Y,Tk] = y [Tl,Tk] = y Clk Tj. k j=1 l=1 j=1 l=1

This proves the ﬁrst statement. The second one is a consequence of h·, ·i being ad-invariant:

h ij h ik adY = hadY Tj,Tki = −hTj, adY Tki = − adY . k j

Consider h, u, H and ϕh as in deﬁnition 3.2.3. We now explain how these maps look like in the coordinates and basis that we set in the beginning of this section.

Proposition 3.3.4. Let h, u, H and ϕh be as in deﬁnition 3.2.3. Then:

• u is given in coordinates by

∂h ∂h u(y1, ..., yn) = , ..., , (3.28) ∂y1 Y ∂yn Y

which means that hu(Y ), ·i = (dh)Y , where h is seen as a function h: g −→ R.

• The matrix that represents H(Y ) in the basis {T1, ..., Tn} is:

 2 2  ∂ h ∂ h ∂y1∂y1 ··· ∂y1∂yn  Y Y   . . .  H(Y ) =  . .. .  . (3.29)   2 2  ∂ h ∂ h  ∂yn∂y1 ··· ∂yn∂yn Y Y

• When seen as a map G × g −→ G × g, ϕh is given by:

ϕh(x, Y ) = (x, u(Y )). (3.30)

37 dL ×id ∼ x−∼1 • Because of the diffeomorphism T(x,Y )(G × g) = TxG ⊕ TY g = g ⊕ g, the derivative of ϕh can be seen as an endomorphism of g ⊕ g. Under this identiﬁcation, it is given by:

  I 0 (dϕh)(x,Y ) =   . (3.31) 0 H(Y )

Proof. Proof of the coordinate expression for u: In the basis {T1, ..., Tn},

n X k u(Y ) = (u(Y )) Tk. (3.32) k=1

k ∂h We want to prove that (u(Y )) = k . ∂y Y

k k j (u(Y )) = dy (u(Y )) (y are coord. in the basis {Tj})

k = (dh)Y [ (dy ) (def. of u + double dual iso.) d = h(Y [ + tdyk) dt t=0 d ∞ ∗ ∞ = h(Y + tTk) (h ∈ C (T G) to h ∈ C (g)) dt t=0 ∂h = . ∂yk Y

Proof of the matrix expression for H:

n j n 2 X ∂u k X ∂ h k (du)Y (W ) = dy (W )Tj = w Tj (3.33) ∂yk Y ∂yj∂yk Y j,k=1 j,k=1

∗ Proof of the expression for ϕh : G × g −→ G × g: Let α ∈ T G.

∗ # φ(α) = π(α), ((dLπ(α)) α) =: (x, Y ) (3.34)

∗ α = (dLx−1 ) Y (3.35)

−1 We want to prove that φ ◦ ϕh ◦ φ (x, Y ) = (x, u(Y )). Since from the deﬁnition of ϕh we have that # ∗ π(ϕh(α)) = π(α) = x, we have to prove that (dLx) ϕh(α) = u(Y ).

D ∗ # E ∗ ((dLx) ϕh(α)) ,Z = (dLx) ϕh(α) (Z) = ϕh(α) (dLx)Z

[ ∗ [ = (dh)α ((dLx)Z) = (dh)α (dLx−1 ) Z d d ∗ [ ∗ [ = h α + t(dLx−1 ) Z = h (dLx−1 ) α +tZ dt t=0 dt t=0 | {z } =Y [ [ [ = (dh)Y [ (Z ) = Z (u(Y )) = hu(Y ),Zi, where we used the fact that h is left invariant. This proves the result since the metric is nondegenerate.

Proof of the expression for (dϕh)(x,Y ) in matrix form: Let (V,W ) ∈ g ⊕ g. We want to compute

38   V (dϕh)(x,Y )   . W

Under the isomorphism

dLx−1 ×id TxG ⊕ TY g −→ g ⊕ g, (3.36)

d (V,W ) ∈ g ⊕ g corresponds to dt (x exp(tV ),Y + tW ) ∈ T(x,Y )(G × g). t=0

d d (dϕh)(x,Y ) (x exp(tV ),Y + tW ) = ϕh(x exp(tV ),Y + tW ) dt t=0 dt t=0 d = (x exp(tV ), u(Y + tW )) dt t=0

= (dLxV, (du)Y (W ))

= (dLxV,H(Y )W ) ∈ TxG ⊕ TY g, which corresponds to (V,H(Y )W ) ∈ g ⊕ g. Therefore, as an endomorphism of g ⊕ g,

    V V (dϕh)(x,Y )   =   (3.37) W H(Y )W which completes the proof.

Lemma 3.3.5. Let h: T ∗G −→ R be left invariant. Consider the following conditions:

(i) h: g −→ R is Ad-invariant;

(ii) [Y, u(Y )] = 0 ∀Y ∈ G;

(iii) adu(Y ) = adY H(Y ) ∀Y ∈ G;

(iv) adu(Y ) = H(Y )adY ∀Y ∈ G.

Then, (i) =⇒ (ii) ⇐⇒ (iii) ⇐⇒ (iv). If G is compact and connected, (ii) =⇒ (i).

Proof. (i) =⇒ (ii):

h(AdgY ) = h(Y ) ∀g ∈ G, Y ∈ g

=⇒ h(Adexp(tV )Y ) = h(Y ) ∀Y,V ∈ g, t ∈ R d ⇐⇒ h(Ad Y ) = 0 ∀Y,V ∈ g, t ∈ dt exp(tV ) R

On the other hand,

d d d h(Adexp(tV )Y ) = h(Adexp((t+s)V )Y ) = h(Adexp(tV )Adexp(sV )Y ) dt ds s=0 ds s=0 d D E = (dh) d Ad exp(sV ) (Ad Y ) = u(Ad Y ), ad (Ad Y ) Adexp(tV )Y e exp(tV ) exp(tV ) V exp(tV ) | {z } ds s=0 =ad | {z } =V

39 D E D E = − u(Adexp(tV )Y ), adAdexp(tV )Y (V ) = adAdexp(tV )Y (u(Adexp(tV )Y )),V D E = − [Adexp(tV )Y, u(Adexp(tV )Y )],V .

So, we have just proven that

h(AdgY ) = h(Y ) ∀g ∈ G, Y ∈ g

=⇒ h(Adexp(tV )Y ) = h(Y ) ∀Y,V ∈ g, t ∈ R D E ⇐⇒ [Adexp(tV )Y, u(Adexp(tV )Y )],V = 0 ∀Y,V ∈ g, t ∈ R ⇐⇒ [Y, u(Y )] = 0 ∀Y ∈ g.

If G is compact and connected, (ii) =⇒ (i): Recall that on a compact connected Lie group, the exponential is surjective (for a statement and proof of this fact see [2], page 165). Using this fact:

h(AdgY ) = h(Y ) ∀g ∈ G, Y ∈ g

⇐⇒ h(Adexp(tV )Y ) = h(Y ) ∀Y,V ∈ g, t ∈ R D E ⇐⇒ [Adexp(tV )Y, u(Adexp(tV )Y )],V = 0 ∀Y,V ∈ g, t ∈ R ⇐⇒ [Y, u(Y )] = 0 ∀Y ∈ g.

(ii) ⇐⇒ (iii): This proof consists in writing [Y, u(Y )] in the coordinates {y1, ..., yn} and differentiating with respect to the yj.

n X ∂h [Y, u(Y )] = Cl yj T jk ∂yk l j,k,l=1 Therefore,

[Y, u(Y )] = 0 n X ∂h ⇐⇒ Cl yj = 0 ∀l jk ∂yk j,k=1  n  ∂ X ∂h ⇐⇒ Cl yj = 0 ∀l, m ∂ym  jk ∂yk  j,k=1 n n X ∂h X ∂2h ⇐⇒ Cl + Cl yj = 0 ∀l, m mk ∂yk jk ∂ym∂yk k=1 j,k=1 n h il X h il ⇐⇒ − adu(Y ) + adY Hmk = 0 ∀l, m m k k=1 h il ⇐⇒ − adu(Y ) + adY H(Y ) = 0 ∀l, m m

⇐⇒ − adu(Y ) + adY H(Y ) = 0.

40 (iii) ⇐⇒ (iv):

T T T T T adu(Y ) = adY H ⇐⇒ adu(Y ) = (adY H(Y )) ⇐⇒ adu(Y ) = H(Y ) adY

T ⇐⇒ −adu(Y ) = H(Y )(−adY ) ⇐⇒ adu(Y ) = H(Y )adY

3.4 Relation between theorems 3.2.2 and 3.2.4

In section 3.2, we presented two theorems that give Kahler¨ structures on cotangent bundles:

2 ∗,R • Hall Kirwin: β ∈ Ω (M) deﬁnes a Kahler¨ structure (T M, ωβ,JMCS).

∗ ∗ τ • Kirwin Mourao˜ Nunes: τ ∈ C and h: T G −→ R deﬁne a Kahler¨ structure (T G, ωST ,J ).

Let G be a compact real analytic Lie group with a real analytic bi-invariant Riemannian metric. In this case, given the relevant structures (β in one case, τ, h in the other) both theorems give a Kahler¨ structure on an open subset of T ∗G. An interesting question is whether the theorems ”intersect”, this is, if the Kahler¨ structure of Hall and Kirwin for some β is isomorphic to the Kahler¨ structure of Kirwin, Mourao˜ and Nunes for some τ, h. We pose this question, which is Question 1 from section 3.2.4, in the following precise way: Question 1: Let G be a compact real analytic Lie group with a real analytic bi-invariant Riemannian metric. Do there exist

• β ∈ Ω2(G) closed,

• τ ∈ C : Im τ > 0,

• h: T ∗G −→ R, satisfying the conditions of theorem 3.2.4,

∗ ∗,R τ and an open set U ⊂ T G such that the Kahler¨ manifolds (T G, ωβ,JMCS) and (U, ωST ,J ) are isomorphic? Assume that such β, τ, h exist. Let Φ: T ∗,RG −→ U ⊂ T ∗G be a Kahler¨ isomorphism. Since Φ is a Kahler¨ isomorphism, we have that:

∗ ∗ ωβ = Φ ωST = ωST − π β, (3.38)

∗ τ JMCS = Φ J . (3.39)

∗ τ The condition that JMCS = Φ J can be written in terms of the (1,0) tangent spaces as:

MCS ∗,R τ ∗,R (dΦ)T1,0 (T G) = T1,0(Φ(T G)). (3.40)

From the statement of the theorem of Hall and Kirwin:

−iL β MCS ∗,R X T1,0 (T G) = e E (ker dπ ⊗ C), (3.41)

41 1 # # β where E(α) = 2 g(α , α ), and XE is the Hamiltonian vector ﬁeld of E with respect to ωβ. From theorem 3.10 in [12]:

τ ∗,R τL¯ XST T1,0(Φ(T G)) = e h (ker dπ ⊗ C). (3.42)

Its easy to see that

−iL β X −iLXST (dΦ)e E (ker dπ) ⊗ C = e E◦Φ ◦ dΦ(ker π ⊗ C). (3.43)

Therefore, equation 3.40 can be written as:

τL¯ XST −iLXST e h (ker dπ ⊗ C) = e E◦Φ ◦ dΦ(ker dπ ⊗ C), (3.44) which is then a necessary condition for the two theorems ”intersecting”.

Example 3.4.1. Let β = 0, h = E, and τ = i. Then, the Kahler¨ structure of Hall and Kirwin is deﬁned for R = ∞. In this case, the Kahler¨ structure of Hall and Kirwin and the Kahler¨ structure of Kirwin, Mourao˜ and Nunes are both equal to the standard Kahler¨ structure of T ∗G.

Example 3.4.2. This example is an attempt at generalizing the previous example based on equation

3.44. Let ψ : g −→ g be a diffeomorphism satisfying ψ ◦ Adg = Adg ◦ ψ. Deﬁne

Φ(x, Y ) = (x, ψ(Y )), (3.45)  n  X j j j β = d  (ψ (Y ) − y )ω  , (3.46) j=1 τ = i, (3.47)

h = E ◦ Φ. (3.48)

Then, β is closed, h is left invariant by definition, and Ad-invariant because ψ and h·, ·i are Ad-invariant. ∗ Also, ωβ = Φ ωST and equation (3.44) is satisfied. Unfortunately, in this case the Hessian of h does not need to satisfy the requirements needed to ϕh be a diffeomorphism, so this h cannot be used to define the complex structure J τ .

Conclusions about question 1 of subsection 3.2.4: Unfortunately it is hard to ﬁnd examples of Kahler¨ structures that can be obtained using both theorems.

42 3.5 The theorem of Hall and Kirwin in the case of a Lie group

Conjecture 3.5.1. Let G be a compact, real analytic Lie group equipped with a real analytic, left invariant Riemannian metric g. Let

∗ h: T G −→ R (3.49) 1 α 7−→ g(α#, α#). 2

Let β ∈ Ω2(G) be closed, and deﬁne ∗ ωβ = ωST − π β. (3.50)

Then,

(i) ωβ is a symplectic form.

∗ (ii) There exists an unique complex structure on T G, JMCS, (called the magnetic complex structure), such that

∗ −iLX T1,0(T G) = e h (ker dπ) ⊗ C, (3.51)

where Xh is the Hamiltonian vector ﬁeld of h with respect to ωβ.

∗ (iii) (T G, ωβ,JMCS) is Kahler.¨

3.5.1 Attempt at proving conjecture 3.5.1

∗ Proposition 3.5.2. Let M be a manifold, ωST be the standard symplectic form on T M, and β be a ∗ closed 2-form on M. Then ωβ = ωST − π β is symplectic.

∗ ∗ Proof. ωβ is closed: dωβ = dωST − dπ β = dωST − π dβ = 0.

ωβ is nondegenerate: Let X ∈ X(M) such that ωβ(X, ·) = 0. We want to prove that X = 0. Let 1 n ∗ {x , ..., x , p1, ..., pn} be the usual coordinates on T M, as described in example 1.1.6. In these coordinates,

n X i ωST = dx ∧ dpi, (3.52) i=1 n X i j β = βijdx ⊗ dx , (3.53) i,j=1 n X k ∂ ∂ X = a k + bk . (3.54) ∂x ∂pk k=1

Therefore,

∗ 0 = ωβ(X, ·) = ωST (X, ·) − π β(X, ·) n  n  n X X j i X i = −bi − βjia  dx + a dpi, i=1 j=1 i=1

43 which implies that the ai and bi are zero, and hence that X is zero.

Lemma 3.5.3. When seen as a function in G × g, h depends only on the point of g and is given in the coordinates {y1, ..., yn} by:

n 1 X h(y1, ..., yn) = (yj)2. (3.55) 2 j=1

∗ ∗ Proof. Let (x, Y ) ∈ G×g. The corresponding element in T G under the diffeomorphism 3.3 is (dLx−1 ) ge(Y, ·), which by left invariance of g is equal to gx(dLxY, ·). Therefore,

1 1 h(x, Y ) = h(g (dL Y, ·)) = g (dL Y, dL Y ) = g (Y,Y ). x x 2 x x x 2 e

1 n 1 n 1 Pn j 2 So, h(x, Y ) does not depend on x, and is given in the coordinates {y , ..., y } by h(y , ..., y ) = 2 j=1(y ) .

To try to prove conjecture 3.5.1, in principle we could use the following method:

1. Using the diffeomorphism φ we treat T ∗G as G × g. We use the basis of vector ﬁelds/forms introduced in section 3.3.

Pn k l 2. Writing β as k,l=1 βklω ∧ ω , ωβ is

n n  n  X X X 1 ω = ωj ∧ dyj + Cj yj − β ωk ∧ ωl. (3.56) β  2 kl kl j=1 k,l=1 j=1

From this, we can compute Xh.

3. e−iXh (ker dπ) ⊗ C can be written as:

−iL −iL ∂ iL ∂ e Xh (ker dπ) ⊗ = span e Xh , ..., e Xh . (3.57) C C ∂y1 ∂yn

τLXh ∂ Compute e ∂yj , for τ ∈ C.

4. Deﬁne JMCS to be the map

∗ ∗ JMCS : T (T G) ⊗ C −→ T (T G) ⊗ C (3.58)

−iL ∂ −iL ∂ e Xh 7−→ ie Xh ∂yj ∂yj

iL ∂ iL ∂ e Xh 7−→ −ie Xh ∂yj ∂yj

∗ n −iL ∂ −iL ∂ iL ∂ iL ∂ o This deﬁnition works if T (T G) ⊗ = span e Xh , ..., e Xh , e Xh , ..., e Xh . C C ∂y1 ∂yn ∂y1 ∂yn 2 JMCS = −id, so it is an almost complex structure. Check that it is integrable, for example by proving that its Nijenhuis tensor is zero.

44 ¯ ¯ ∗ ¯ 5. If {Z1, ..., Zn, Z1, ..., Zn} is a basis of T (T G) ⊗ C, (where the Zj are J-holomorphic and the Zj are J-anti-holomorphic), and if ω is a symplectic form on T ∗G, then it is easy to prove that

  ω(·,J·) is symmetric  ω(Zj,Zk) = 0 ∀j, k ⇐⇒ , (3.59) ω(·,J·) is positive deﬁnite [iω(Z¯j,Zk)] 0

where [iω(Z¯j,Zk)] 0 means that the matrix with entries iω(Z¯j,Zk) is positive deﬁnite. Use this

fact to prove that ωβ and JMCS are compatible.

τLXh ∂ 2 Unfortunately, it is very hard to compute e ∂yj , even in the case where β ∈ Ω (G) is exact and left invariant. In this case, β can be written as

n n X X 1 β = ajdωj = − a Cj ωk ∧ ωl, (3.60) 2 j kl j=1 j,k,l=1 where the aj are constants.

3.5.2 ωβ as a pullback of ωST by a diffeomorphism

Consider the case where in addition to being compact, G is semisimple and connected. We use results from Chevalley and Eilenberg about the cohomology of Lie groups to show that β must be exact. We refer to [3]. Theorem 21.1. of this article implies that H2(g) = 0, where H2(g) is the cohomology of degree 2 of g. Theorem 15.2. of [3] implies that H2(G, R) =∼ H2(g), where H2(G, R) is the singular cohomology with real coefﬁcients. And De Rham’s theorem states that on any manifold the singular cohomology and the De Rham cohomology are isomorphic. In short:

2 ∼ 2 ∼ 2 0 = H (g) = H (G, R) = HdR(G). (3.61)

2 Since HdR(G) = 0 and β is closed, it is exact. Let α be such that β = dα. α can be written in the basis {ω1, ..., ωn}:

n X j α = ajω , (3.62) j=1

∞ where the aj belong to C (G).

ωβ is exact:

∗ ∗ ∗ ∗ ωβ = ωST − π β = dθ − π dα = dθ − dπ α = d(θ − π α), (3.63) and can be written more explicitly as:

n X j j ωβ = d (y − aj)ω (3.64) j=1 n  n  X 1 X = ωj ∧ dyj + Cj (yj − a )ωk ∧ ωl . (3.65)  2 kl j  j=1 k,l=1

45 j Consider the map φα : G × g −→ G × g that consists in translating the y by the aj:

φα : G × g −→ G × g (3.66)  n   n  X j X j p, y Tj7−→ p, (y − aj(p))Tj . j=1 j=1

Proposition 3.5.4. φα is a diffeomorphism.

Proof. φα is injective:

 n   n   n   n  X j X j X j X j φα p, y Tj = φα q, z Tj ⇐⇒ p, (y − aj(p))Tj = q, (z − aj(q))Tj j=1 j=1 j=1 j=1    p = q  p = q ⇐⇒ ⇐⇒ . j j j j  y − aj(p) = z − aj(q)  y = z

Pn j j j φα is surjective: Let p, j=1 z Tj ∈ G × g. Deﬁne y = z + aj(p). Then

 n   n  X j X j φα p, y Tj = p, z Tj . j=1 j=1

1 n 1 n 1 n φα is a local diffeomorphism: Let {x , ..., x } be local coordinates on G. In the coordinates {x , ..., x , y , ..., y },

φα is given by:

1 n 1 n 1 n 1 n n φα(x , ..., x , y , ..., y ) = (x , ..., x , y − a1(x), ..., y − a (x)). (3.67)

Therefore, its Jacobian matrix in these coordinates is:

  I 0 dφα =   , (3.68) −dα I whose determinant is 1. Then, since dφα is an isomorphism at all points of G × g, the inverse function theorem implies that φα is a local diffeomorphism at all points of G × g.

φα is a bijective local diffeomorphism, so it is a diffeomorphism.

Using φα, ωβ can be written as a pullback of the standard symplectic form:

Proposition 3.5.5. ∗ ωβ = φαωST (3.69)

Proof.

n ∗ ∗ ∗ ∗ X j j φαωST = φαdθ = dφαθ = dφα y ω j=1 n n X ∗ j ∗ j X j j = d (φαy )(φαω ) = d (y − aj)ω = ωβ, j=1 j=1

46 ∗ j j j where we used the fact that φαω = ω . In informal terms, this is because ω is zero on the part of vectors that is tangent to g, and dφα is the identity on the part of vectors tangent to G.

Conclusions about question 2 of subsection 3.2.4: Very hard computations stop us from pro- gressing in the proof. Using the result of this section, we could try to prove the equivalent statement that ∗ −1 ∗ (T G, ωST , (φα ) JMCS) is Kahler.¨ However, this does not make the computations any easier, because

−LXST ∗ −1 ∗ ∗ h◦φ T1,0(T G, ωST , (φα ) JMCS) = dφαT1,0(T G, ωβ,JMCS) = e α (ker dπ ⊗ C), (3.70)

−LXST h◦φα ∂ and again e ∂yj is hard to compute.

3.6 The theorem of Kirwin, Mourao˜ and Nunes in the case of non Ad-invariant h

In this section we will study theorem 3.2.4 in the case where we drop the condition of h being Ad- invariant. We will try to see if this condition is necessary, or if we can ﬁnd examples of non Ad-invariant complexiﬁer functions for which the theorem is true. So, for the remainder of this section, assume the hypothesis of theorem 3.2.4 except Ad-invariance of h.

τ 3.6.1 ϕh is a diffeomorphism. The complex structure J

In [12] it is proven in the form of a lemma that ϕh is a diffeomorphism. The proof given there uses the conditions of H being positive deﬁnite and having greater that zero lower bound for the norm, but it does not use the condition of h being Ad-invariant. Therefore, the same proof holds in our case, and ϕh is a diffeomorphism. Recall that the pullback of a complex structure by a diffeomorphism is still a complex structure. τ ∗ Therefore, J = (ψτ ◦ φ ◦ ϕh) JG is a complex structure. As stated in [12], dψτ is given by: C

 −τ1adY  −τ1adY I−e cos(τ2adY ) e cos(τ2adY ) (dψ ) = adY , (3.71) τ (x,Y )  −τ1adY  −τ1adY e sin(τ2adY ) −e sin(τ2adY ) adY where the blocks are with respect to the identiﬁcations

dL ×id ∼ x−∼1 T(x,Y )(G × g) = TxG ⊕ TY g = g ⊕ g, (3.72)

dL τY −1 (xe∼ ) ∼ TxeτY GC = gC = g ⊕ g. (3.73)

In the following commutative diagram, we present all relevant diffeomorphisms and their expressions as block matrices:

∗ ϕh ∗ φ ψτ T G −→ T G −→ G × g −→ GC (3.74)

47 ∗ (dϕh)α ∗ Tα(T G) Tϕh(α)(T G)

(dφ) (dφ) α ϕh(α)

(dψτ )(x,u) JG C T(x,Y )(G × g) T(x,u(Y ))(G × g) Txeτu(Y ) GC Txeτu(Y ) GC

−1 −1 ∼= ∼= dL dL xeτu(Y ) xeτu(Y ) (3.75)

TxG ⊕ TY g TxG ⊕ Tu(Y )g gC gC dL ×id ∼ ∼ x−1 dLx−1 ×id = = g ⊕ g g ⊕ g g ⊕ g g ⊕ g M1 M2 M3

where M1, M2, M3 are the following matrices:

  I 0 M1 =   , (3.76) 0 H(Y )

 I−e−τ1adu(Y ) cos(τ ad )  −τ1adu(Y ) 2 u(Y ) e cos(τ2adu(Y )) adu(Y ) M2 = −τ ad , (3.77)  e 1 u(Y ) sin(τ ad )  −τ1adu(Y ) 2 u(Y ) −e sin(τ2adu(Y )) adu(Y )   0 −I M3 =   . (3.78) I 0

Then, as an isomorphism of g ⊕ g, J τ is given by:

τ −1 J = (M2M1) M3M2M1. (3.79)

Now we have T ∗G equipped with the symplectic form ω and the complex structure J τ . We want to see if these structures are compatible.

∗ 3.6.2 A basis for T1,0(T G)

∗ τ We will now ﬁnd a basis of T1,0(T G) by computing n linearly independent (+i)-eigenvectors of J . Since expression (3.79) describes J τ as a 2 × 2 matrix of linear operators from g to g, we write the (candidates to) eigenvectors as 2 × 1 columns where each entry belongs to g. We try to ﬁnd eigenvectors of the form

  M(Y )Tj   , −1 H(Y ) Tj where M(Y ) is a linear endomorphism of g. So, we solve the following equation for M(Y ):

 −1  −1           1 0 0 −I 1 0 M(Y )Tj M(Y )Tj    M2     M2      = i   , −1 −1 0 H(Y ) I 0 0 H(Y ) H(Y ) Tj H(Y ) Tj and obtain that

48 1 M(Y ) = 1 − eτ¯adu(Y ) . adu(Y ) So, for all j = 1, ..., n

  1 τ¯adu(Y ) ad 1 − e Tj  u(Y )  −1 H(Y ) Tj

τ is a (+i)-eigenvector of J . This can be written as a vector in T(g,Y )(G × g), in the basis of equation 3.18:

k n " τ¯ad # n X 1 − e u(Y ) X k ∂ τ : −1 Zj = Xk + H(Y ) j k . (3.80) adu(Y ) ∂y k=1 j k=1

τ τ Also, since H(Y ) is invertible, {Z1 , ..., Zn} is a linearly independent set. Considering the dimension on ∗ T1,0(T G) we conclude that it is a basis:

T (T ∗G) = span {Zτ , ..., Zτ }. (3.81) 1,0 C 1 n

3.6.3 Equations for T ∗G being Kahler¨ in terms of h

Recall that ω and J τ are compatible if and only if

  ω(Zj,Zk) = 0 ∀j, k . (3.82) [iω(Z¯j,Zk)] 0

Using equations (3.21) and (3.80), we can compute ω(Zj,Zk) and iω(Z¯j,Zk):

" −τ¯adu(Y ) τ¯adu(Y ) I − e −1 −1 I − e ω(Zj,Zk) = − H(Y ) − H(Y ) adu(Y ) adu(Y ) #j I − e−τ¯adu(Y ) I − eτ¯adu(Y ) + adY , (3.83) adu(Y ) adu(Y ) k

" −τadu(Y ) τ¯adu(Y ) I − e −1 −1 I − e iω(Z¯j,Zk) = i − H(Y ) − H(Y ) adu(Y ) adu(Y ) #j I − e−τadu(Y ) I − eτ¯adu(Y ) + adY . (3.84) adu(Y ) adu(Y ) k

We conclude that in the case of h not necessarily Ad-invariant, theorem 3.2.4 can be written:

Theorem 3.6.1. Let G be a compact Lie Group, equipped with a bi-invariant Riemannian metric h·, ·i.

Let h: T ∗G −→ R satisfy:

∗ ∗ • h is left invariant: h((dLg) α) = h(α) ∀α ∈ T G, g ∈ G;

49 • H is positive deﬁnite everywhere: hW, H(Y )W i > 0 ∀Y,W ∈ g; kH(Y )Zk • The norm of H has a greater than 0 lower bound: inf sup kZk > 0. Y ∈g Z∈g

τ ∗ ∗ τ Then, ϕh is a diffeomorphism. Deﬁne J = (ψτ ◦ φ ◦ ϕh) JG . For Im τ > 0, (T G, ω, J ) is Kahler¨ if and C only if h: g −→ R satisﬁes the following equations:

−τ¯adu(Y ) τ¯adu(Y ) −τ¯adu(Y ) τ¯adu(Y ) I − e −1 −1 I − e I − e I − e − H(Y ) − H(Y ) + adY = 0, (3.85) adu(Y ) adu(Y ) adu(Y ) adu(Y )

−τadu(Y ) τ¯adu(Y ) −τadu(Y ) τ¯adu(Y ) I − e −1 −1 I − e I − e I − e − i H(Y ) − iH(Y ) + i adY 0. (3.86) adu(Y ) adu(Y ) adu(Y ) adu(Y )

Example 3.6.2 (Ad-invariant h). If h is Ad-invariant, then from lemma 3.3.5, [Y, u(Y )] = 0 and adu(Y ) = −1 adY H(Y ) = H(Y )adY , and therefore adY , adu(Y ) and H(Y ) commute with each other. Using these properties we can show that h satisﬁes equations (3.85) and (3.86).

−τ¯adu(Y ) τ¯adu(Y ) −τ¯adu(Y ) τ¯adu(Y ) I − e −1 −1 I − e I − e I − e − H(Y ) − H(Y ) + adY adu(Y ) adu(Y ) adu(Y ) adu(Y ) ! −τ¯adu(Y ) τ¯adu(Y ) −τ¯adu(Y ) τ¯adu(Y ) −1 −I + e − I + e I − e + I − e = H(Y ) + adY 2 adu(Y ) adu(Y ) ! −τ¯adu(Y ) τ¯adu(Y ) −τ¯adu(Y ) τ¯adu(Y ) −1 −I + e − I + e −1 I − e + I − e = H(Y ) + H(Y ) adu(Y ) 2 adu(Y ) adu(Y ) = 0

Performing an analogous computation,

−τadu(Y ) τ¯adu(Y ) −τadu(Y ) τ¯adu(Y ) I − e −1 −1 I − e I − e I − e − i H(Y ) − iH(Y ) + i adY adu(Y ) adu(Y ) adu(Y ) adu(Y ) e−2iτ2adu(Y ) − I = iH(Y )−1 . adu(Y )

−2iτ2ad To check that iH(Y )−1 e u(Y ) −I is positive deﬁnite, we consider its eigenvalues. Since H(Y )−1 and adu(Y ) iadu(Y ) are hermitian, they are diagonalizable, and since they commute they are simultaneously diago- −1 nalizable. Then, H(Y ) and adu(Y ) are also simultaneously diagonalizable. Let {v1, ..., vn} be a basis −1 of eigenvectors for both matrices, for which H(Y ) has eigenvalues λj, and adu(Y ) has eigenvalues iµj:

−1 H(Y ) vj = λjvj, (3.87)

adu(Y )vj = (iµj)vj. (3.88)

Note that the λj are real and positive, because H(Y ) is symmetric and positive deﬁnite, and that the µj

50 −2iτ ad −1 e 2 u(Y ) −I are real because adu(Y ) is skew-symmetric. {v1, ..., vn} are also eigenvectors of iH(Y ) : adu(Y )

∞ −2iτ2ad k e u(Y ) − I X (−2iτ2) iH(Y )−1 v = iH(Y )−1 adk−1 v ad j k! u(Y ) j u(Y ) k=1 ∞ k X (−2iτ2) = iλ (iµ )k−1v j k! j j k=1 2τ2µj λj e − 1 = vj. µj

−2iτ ad 2τ2µj 2 u(Y ) λj (e −1) So, we conclude that the eigenvalues of iH(Y )−1 e −I are , which are all positive adu(Y ) µj −2iτ ad −1 e 2 u(Y ) −I for τ2 > 0. Therefore, for τ2 > 0 we have that iH(Y ) is a positive deﬁnite matrix. Then, adu(Y ) from theorem 3.6.1 we conclude that (T ∗G, ω, J τ ) is a Kahler¨ Manifold, which is the content of theorem 3.2.4.

Example 3.6.3. Consider the following complexiﬁer function:

h: g −→ R (3.89) 1 Y 7−→ hY,Y i + hY,Zi, 2 where Z ∈ g. Note that if Z 6= 0, then h is not Ad-invariant. We prove that h satisﬁes equations (3.85) and (3.86) if and only if adZ = 0. For this h:

H(Y ) = I,

u(Y ) = Y + Z.

Therefore, using adY = adu(Y ) − adZ , equations (3.85) and (3.86) are equivalent to

 I − e−τ¯adu(Y ) I − eτ¯adu(Y )   − adZ = 0  adu(Y ) adu(Y ) . −2iτ ad −τad τ¯ad  e 2 u(Y ) I − e u(Y ) I − e u(Y ) i − i adZ 0  adu(Y ) adu(Y ) adu(Y )

Suppose that these equations are true. Then, at Y = −Z, the ﬁrst equation is adZ = 0. On the other hand, suppose that adZ = 0. Then,

I − e−τ¯adu(Y ) I − eτ¯adu(Y ) − adZ = 0 adu(Y ) adu(Y ) and e−2iτ2adu(Y ) I − e−τadu(Y ) I − eτ¯adu(Y ) e−2iτ2adu(Y ) i − i adZ = i . adu(Y ) adu(Y ) adu(Y ) adu(Y )

−2iτ2ad Following the same reasoning as in the previous example, i e u(Y ) is positive deﬁnite. adu(Y ) Recall that the center of g is the kernel of ad. Then, if the center of g is not {0}, it is possible to

ﬁnd a non zero Z ∈ g such that adZ . However, in this case there exists an orthogonal decomposition LieG = LieH ⊕ LieK, where H is abelian and Z ∈ H. In LieH, h will be bi-invariant because H is

51 abelian, and in LieK the term hY,Zi is zero. So, h will be Ad-invariant anyway.

Conclusions about question 3 of subsection 3.2.4: The hypothesis of Ad-invariance can be replaced by equations 3.85 and 3.86. Unfortunately it is hard to display examples of an h that is not Ad-invariant but still satisﬁes equations 3.85 and 3.86.

52 Chapter 4

A Short Overview of Geometric Quantization

4.1 Introduction and motivation

Classical and quantum mechanics are two distinct theories that model nature. On table 4.1 we present a short comparison between the mathematical formulation of the two theories. There exist many examples of classical systems that have a corresponding quantum system. An interesting question is: Given a classical mechanical system, is there a corresponding quantum mechanical system? We state this question in mathematical terms: Given a symplectic manifold (M, ω), does there exist a Hilbert space (H, h·, ·i) and a linear map q : C∞(M) −→ Op(H) such that:

(i) q(1) = idH;

(ii) q(f, g) = i[q(f), q(g)];

(iii) q(f) = (q(f))†;

(iv) when seen as a representation, q is irreducible.

The map q is a correspondence between the observables of the classical and quantum systems. Unfortunately, in general it is not possible to ﬁnd such a map q satisfying all the requirements (see [15]).

Table 4.1: Comparison between classical and quantum mechanics Classical Quantum Space of states A symplectic manifold (M, ω) A Hilbert space (H, h·, ·i)

Functions on M, Linear operators on H, Observables (physical quantities of interest) i.e. elements of C∞(M) i.e. elements of Op(H)

53 4.2 General set up

In this section we give the general ”recipe” for geometric quantization. Given a symplectic manifold (M, ω) admitting certain structures, we explain an approach on how to produce a Hilbert space (H, h·, ·i).

Deﬁnition 4.2.1. Let (M, ω) be a symplectic manifold. A prequantum line bundle on M is given by a complex line bundle π : L −→ M, a connection ∇ on L, and an hermitian inner product h on L, that are subject to the following:

• h and ∇ are compatible:

X(h(s, r)) = h(∇X s, r) + h(s, ∇X s) ∀X ∈ X(M), s, r ∈ Γ(L). (4.1)

• The curvature of ∇ is −iω: R(X, Y, s) = −iω(X,Y )s. (4.2)

Deﬁnition 4.2.2. Let (M, ω) be a symplectic manifold of dimension 2n.A polarization is a distribution

P in TM ⊗ C that satisﬁes:

•P is involutive: X,Y ∈ X(P) =⇒ [X,Y ] ∈ X(P). (4.3)

•P is Lagrangian:

dimC P = n, (4.4a) ω(X,Y ) = 0 ∀X,Y ∈ P. (4.4b)

There exist several types of polarizations:

P is real ⇐⇒ P = P¯;

∞ P is Kahler¨ ⇐⇒ P ∩ P¯ = 0 and iω(·, ·): P¯ × P −→ C (M; C) is positive deﬁnite.

We now explain how one can approach the question of deﬁning the Hilbert space H. Start by considering the space of P¯-polarized sections of L:

s ∈ Γ(L): ∇X s = 0 ∀X ∈ X(P¯) , and deﬁne an inner product h·, ·i on this space by integration. Then, H can be deﬁned as

H = s ∈ Γ(L): ∇X s = 0 ∀X ∈ X(P¯), ksk < ∞ , (4.5) where the closure is with respect to h·, ·i.

54 Chapter 5

An Example of Quantization of the Cotangent Bundle of a Lie Group

In this chapter we perform geometric quantization on the cotangent bundle of a Lie group. Given a Lie group G, we introduce a prequantum line bundle L, and for each τ with Im τ > 0 we introduce a mixed polarization Pτ . This polarization is deﬁned using a new example of a complexiﬁer function h, which is not Ad-invariant. We then compute its associated Hilbert space Hτ , and show that all these Hilbert spaces {Hτ }Im τ>0 are naturally isomorphic.

5.1 General setup

Let:

• G be a compact Lie group with Lie algebra g, equipped with a bi-invariant Riemannian metric h·, ·i.

• h be a maximal abelian subalgebra. h being abelian means that [X,Y ] = 0 for all X,Y ∈ h. h being maximal means that if it is contained in another abelian subalgebra k, then k = h. Denote the dimension of h by r.

j j •{ T1, ..., Tn} be an orthonormal basis of g, such that {T1, ..., Tr} is a basis of h, and y ,Xj, ω be deﬁned as in section 3.3.

• h: g −→ R, the complexiﬁer function, be deﬁned as

n 1 X h(y1, ..., yn) = H yjyk, (5.1) 2 jk j,k=1

where H is symmetric, Hjk = 0 if j or k ≥ r + 1, and the r × r matrix with entries Hjk, j, k = 1, ..., r is positive deﬁnite.

l Remark 5.1.1. [Tj,Tk] = 0 ∀j, k = 1, ..., r because Tj and Tk are in h. Since Cjk is totally antisymmetric, l we conclude that Cjk = 0 if any two of the indices are in {1, ..., r}.

55 5.2 A prequantum line bundle for T ∗G

We consider the line bundle whose total space is L = T ∗G × C, and whose projection map is:

∗ ∗ π : T G × C −→ T G. (5.2)

Since this is a trivial line bundle, the sections are functions from T ∗G to C:

∗ ∞ ∗ Γ(T G × C) = C (T G; C). (5.3)

We now introduce the required structures on L. We deﬁne h to be the hermitian structure that on the ﬁber of L above α ∈ T ∗G is the following hermitian inner product:

hα : C × C −→ C (5.4) (z, w) 7−→ zw.¯

Proposition 5.2.1. The connections on L that are compatible with h and have curvature −iω are of the form

∇: X(T ∗G) × Γ(L) −→ Γ(L) (5.5)

(X, s) 7−→ ∇X s = X(s) + iα(X)s, where α is a 1-form on T ∗G such that dα = dθ.

Proof. Let ∇ be a connection on L that is compatible with h with curvature −iω. Compatibility condition:

h(∇X s, r) + h(s, ∇X r) = X(h(s, r)) ⇐⇒ r¯∇X s + s∇X r =rX ¯ (s) + sX(¯r) (5.6)

r = 1 =⇒ ∇X s + s∇X 1 = X(s)

r = 1, s = 1 =⇒ ∇X 1 + ∇X 1 = 0

From which we conclude that

∇X s = X(s) + (∇X 1)s. (5.7)

∇1 is a linear map from vector ﬁelds to complex numbers. Since ∇X 1 + ∇X 1 = 0, we can say that

∇1 = iα, (5.8)

56 where α is a real valued form on T ∗G. Therefore,

∇X s = X(s) + α(X)s. (5.9)

Curvature condition:

R(X, Y, s) = −iω(X,Y )s (5.10)

⇐⇒ ∇X (∇Y s) − ∇Y (∇X s) − ∇[X,Y ]s = idθ(X,Y )s (5.11)

Using equation 5.9, we can show that

∇X (∇Y s) − ∇Y (∇X s) − ∇[X,Y ]s = idα(X,Y )s. (5.12)

Therefore, the curvature of ∇ is −iω if and only if dα = dθ.

For the converse, assume that ∇X s = X(s)+α(X)s, where α is a real valued form such that dα = dθ. Then one can show that ∇ is a connection, compatible with h, with curvature −iω.

We deﬁne a connection ∇ on L by:

∇X s = X(s) + iθ(X)s. (5.13)

5.3 A polarization for T ∗G

Consider the vertical polarization of T ∗G:

∂ ∂ P := ker dπ ⊗ = span , ..., ⊂ T (T ∗G) ⊗ . (5.14) 0 C C ∂y1 ∂yn C

Its immediate from deﬁnition 4.2.2 that P0 is a polarization. Our objective in this section is to compute the following distribution: τL¯ ∂ τL¯ ∂ P = span e Xh , ..., e Xh , (5.15) τ C ∂y1 ∂yn

τL¯ X which is essentially e h applied to P0:

τL¯ X X ∈ X(Pτ ) ⇐⇒ ∃V ∈ X(P0): X = e h V. (5.16)

In fact, we will prove that Pτ is a mixed polarization (in the sense that is is a direct sum of a real distribution and an holomorphic distribution). The computation of Pτ is lengthy, so we split it into propositions

5.3.1- 5.3.6. Proposition 5.3.8 is an alternative proof of equations (5.28) for Pτ obtained in propositions 5.3.2 and 5.3.6, that assumes previous knowledge of the equations.

57 Proposition 5.3.1 (Hamiltonian vector ﬁeld).

r n r X X X ∂ X = H yjX + ykCk H yl (5.17) h ij i ji jl ∂yi i,j=1 i,k=r+1 j,l=1

Proof. Using equation 3.21 for ω, and that

n ∂h X = H yk, (5.18) ∂yj jk k=1

j or k ≥ r + 1 =⇒ Hjk= 0,

k two indices in {1, ..., r} =⇒ Cij = 0, we can compute that

 r n r  n X X X ∂ X ∂h ω H yjX + ykCk H yl , · = dyj = dh. (5.19)  ij i ji jl ∂yi  ∂yj i,j=1 i,k=r+1 j,l=1 j=1

Proposition 5.3.2.

 Pr Pn j ∂ ∂ − j=1 HjiXj − j=r+1 [adY H]i ∂yj if i = 1, ..., r LXh i = (5.20) ∂y Pn j ∂ j=r+1 adH(Y ) i ∂yj if i = r + 1, ..., n

Also, if i = r + 1, ..., n,

n ∂ j ∂ τL¯ X X τ¯ad e h = e H(Y ) (5.21) ∂yi i ∂yj j=r+1 Proof.

 r n r  ∂ X X X ∂ ∂ L = H yjX + ykCk H yl , Xh ∂yi  kj i jm jl ∂ym ∂yi  k,j=1 m,k=r+1 j,l=1 r n r X ∂ X X ∂ ∂ = − H yj X − ykCk H yl ∂yi kj k ∂yi jm jl ∂ym k,j=1 m,k=r+1 j,l=1 r n r n r X X X ∂ X X ∂ = − H X − Ci H yl − ykCk H ki k jm jl ∂ym jm ji ∂ym k=1 m=r+1 j,l=1 m,k=r+1 j,l=1 r n n X X m ∂ X m ∂ = − H X + ad − [ad H] ki k H(Y ) i ∂ym Y i ∂ym k=1 m=r+1 m=r+1

m m If i = 1, ..., r, then adH(Y ) i = 0. If i = r + 1, ..., n, then Hki = 0 and [adY H]i = 0. This completes the proof of equation (5.20). We now prove equation (5.21):

n ∂ X j ∂ L = ad if i = r + 1, ...n Xh ∂yi H(Y ) i ∂yj j=r+1

58 n ∂ X n j ∂ =⇒ Ln = ad if i = r + 1, ...n Xh ∂yi H(Y ) i ∂yj j=r+1 n ∂ j ∂ τL¯ X X τ¯ad =⇒e h = e H(Y ) if i = r + 1, ...n. ∂yi i ∂yj j=r+1

Proposition 5.3.3. τL¯ −τ¯ad e Xh (Y ) = e H(Y ) (Y ) (5.22)

τL¯ where e Xh acts componentwise on Y .

Proof. n r m X X k k l m Xh(y ) = y CjmHjly = − adH(Y )(Y ) k=r+1 j,l=1

∂ ∂ 1 r So, Xh(Y ) = −adH(Y )Y . Xh only has terms in ∂yr+1 , ..., ∂yn , and H(Y ) only depends on y , ..., y .

Therefore, Xh(H(Y )) = 0. We conclude that

n n Xh (Y ) = −adH(Y ) (Y ), which implies that

τL¯ −τ¯ad e Xh (Y ) = e H(Y ) (Y ).

Proposition 5.3.4.

τL¯ Xh e (adY ) = ade−τ¯adH(Y ) (Y ) (5.23)

Proof.

τL¯ X e h (ad ) = ad τL¯ = ad −τ¯ad Y e Xh Y e H(Y ) (Y )

Proposition 5.3.5. If i = 1, ..., r,

r n τL¯ ∂ X X k ∂ e Xh L = − H X − [ad H] (5.24) Xh ∂yi ji j Y i ∂yk j=1 k=r+1

Proof. Since [Xi,Xj] = 0 for i, j = 1, ..., r, then LXh Xj = 0 for j = 1, ..., r. This implies that

 r  r τL¯ X X X e h − HjiXj = − HjiXj. (5.25) j=1 j=1

Using this fact and the property of the exponential of a product (see proposition 2.1.5), we can compute τL¯ Xh ∂ e LXh ∂yi :

  r n τL¯ ∂ τL¯ X X j ∂ e Xh L = e Xh − H X − [ad H] Xh ∂yi  ji j Y i ∂yj  j=1 j=r+1 r n j ∂ X X τL¯ X τL¯ X = − H X − e h ad H e h ji j Y i ∂yj j=1 j=r+1

59 r n h ij k ∂ X X τ¯adH(Y ) = − HjiXj − ad −τ¯adH(Y ) H e e (Y ) i j ∂yk j=1 k,j=r+1 r n h ik ∂ X X τ¯adH(Y ) = − HjiXj − e ad −τ¯adH(Y ) H e (Y ) i ∂yk j=1 k=r+1

τ¯adH(Y ) τ¯adH(Y ) We claim that e ade−τ¯adH(Y ) (Y ) = adY e . Recall that, as a special case of remark 1.4.8, exp ◦ad = Ad ◦ exp.

τ¯adH(Y ) e ad −τ¯adH(Y ) (Z) = Ad τH¯ (Y ) adAd (Z) e (Y ) e e−τH¯ (Y )

τ¯adH(Y ) = AdeτH¯ (Y ) [Ade−τH¯ (Y ) (Y ),Z] = [Y, AdeτH¯ (Y ) Z] = adY e (Z)

Where in the ﬁrst equality we used the fact that exp ◦ad = Ad ◦ exp, which is a special case of remark

1.4.8, and in the third equality we used the fact that AdeτH¯ (Y ) is a Lie algebra homomorphism (see proposition 1.4.7).

τ¯adH(Y ) It remains to be proven that adY e H = adY H. This is true because adH(Y )H = 0:

r k X m j adH(Y )H j = y HmlClkHki = 0. k,l,m

Proposition 5.3.6. If i = 1, ..., r,

r n τL¯ ∂ X ∂ X k ∂ e Xh = − τH¯ X + − τ¯ [ad H] (5.26) ∂yi ji j ∂yi Y i ∂yk j=1 k=r+1

Proof. We start by proving that the equation is true for τ¯ = t ∈ R.

r n d tL ∂ tL ∂ X X k ∂ e Xh = e Xh L = − H X − [ad H] , dt ∂yi Xh ∂yi ji j Y i ∂yk j=1 k=r+1 which implies that

  Z t r n tL ∂ X X k ∂ ∂ e Xh = − H X − [ad H] ds + ∂yi  ji j Y i ∂yk  ∂yi 0 j=1 k=r+1 r n X ∂ X k ∂ = − tH X + − t [ad H] . ji j ∂yi Y i ∂yk j=1 k=r+1

This proves the result for τ¯ ∈ R. Considering the complex analytic continuation of each side of equation

r n tL ∂ X ∂ X k ∂ e Xh = − tH X + − t [ad H] ∂yi ji j ∂yi Y i ∂yk j=1 k=r+1 we conclude that the result is true for τ¯ ∈ C.

We have the following formulas for Pτ :

60 τL¯ ∂ τL¯ ∂ τL¯ ∂ τL¯ ∂ P = span e Xh , ..., e Xh , e Xh , ..., e Xh , (5.27) τ C ∂y1 ∂yr ∂yr+1 ∂yn where

 Pr ∂ Pn j ∂ − τH¯ jiXj + i − τ¯ [adY H] j if i = 1, ..., r τL¯ ∂  j=1 ∂y j=r+1 i ∂y e Xh = . (5.28) ∂yi n j P τ¯adH(Y ) ∂ j=r+1 e i ∂yj if i = r + 1, ..., n

It is possible to give a more direct proof of equation (5.28) if we already know the result of the previous computation. Basically we check that each side is the solution of a differential equation with unique solution.

Lemma 5.3.7. Let Y0 be a vector ﬁeld. Then, there exist an unique family of vector ﬁelds {Yt} that is smooth in t such that

 d  Yt = LX Yt dt h (5.29)  Yt|t=0 = Y0, which is given by tL t ∗ Y = e Xh Y = (φ ) Y , t 0 Xh 0 (5.30)

tLX if e h Y0 is well deﬁned.

tLX Proof. existence: The family of vector ﬁelds Yt = e h Y0 satisﬁes equation (5.29).

d −tLXh uniqueness: Let Yt be a family of vector ﬁelds satisfying (5.29). We start by proving that dt e Yt = 0:

d d d −tLX −tLX −tLX −tLX e h Y = −L e h Y + e h Y = e h −L + Y = 0. dt t Xh t dt t Xh dt t

−tLX This implies that e h Yt does not depend on t, in particular it is equal to its value in t = 0:

−tLX −tLX e h Yt = e h Yt = Y0. t=0

Therefore,

tLX Yt = e h Y0.

Proposition 5.3.8. If i = 1, ..., r:

 r n  d X ∂ X k ∂ − tH X + − t [ad H] dt  ji j ∂yi Y i ∂yk  j=1 k=r+1

 r n  X ∂ X k ∂ = L − tH X + − t [ad H] (5.31) Xh  ji j ∂yi Y i ∂yk  j=1 k=r+1

61   r ∂ n ∂ ∂ X X k − tHjiXj + − t [adY H]i  = (5.32) ∂yi ∂yk t=0 ∂yi j=1 k=r+1

If i = r + 1, ..., n:  n   n  d X j ∂ X j ∂ etadH(Y ) = L etadH(Y ) (5.33) dt  i ∂yj  Xh  i ∂yj  j=r+1 j=r+1

 n  ∂ j ∂ ∂ tLX X tad e h = e H(Y )  = (5.34) ∂yi i ∂yj t=0 ∂yi j=r+1

This (together with lemma 5.3.7) implies that equation (5.28) is true for τ¯ ∈ R, which in turn implies that it is true for τ¯ ∈ C by performing complex analytic continuation.

Proof. Equations (5.32) and (5.34) are true. Proof of equation (5.31):

 r n  r n d X ∂ X k ∂ X X k ∂ ∂ − tH X + − t [ad H] = − H X − [ad H] = L dt  ji j ∂yi Y i ∂yk  ji j Y i ∂yk Xh ∂yi j=1 k=r+1 j=1 k=r+1

 r n  X ∂ X k ∂ L − tH X + − t [ad H] Xh  ji j ∂yi Y i ∂yk  j=1 k=r+1

 r n n  ∂ X X k ∂ X k ∂ = L − t H L X + t [L ad H] + t [ad H] L Xh ∂yi  ji Xh j Xh Y i ∂yk Y i Xh ∂yk  j=1 | {z } =0 k=r+1 k=r+1  n n  ∂ X k ∂ X k l ∂ = L − t t ad H + t [ad H] ad ad H Xh ∂yi  −adH(Y )Y i ∂yk Y i H(Y ) Y k ∂yk  k=r+1 l,k=r+1 n ∂ X k ∂ = L − t ad H + ad ad H Xh ∂yi −adH(Y )Y H(Y ) Y i ∂yk k=r+1 k n   ∂ X ∂ ∂ = L − t ad ad H −ad ad H + ad ad H = L Xh ∂yi  Y H(Y ) H(Y ) Y H(Y ) Y  ∂yk Xh ∂yi k=r+1 | {z } =0 i

Proof of equation (5.33):

 n  n X j ∂ X j ∂ L etadH(Y ) = etadH(Y ) L Xh  i ∂yj  i Xh ∂yj j=r+1 j=r+1

n  n  X j ∂ d X j ∂ = etadH(Y ) ad = etadH(Y ) H(Y ) i ∂yj dt  i ∂yj  j=r+1 j=r+1

τL¯ Xh ∂ ∗ Notice that the vector ﬁelds e ∂yi are linearly independent at every point of T G.

We now prove that Pτ is a polarization. As we will see, this is a consequence of P0 being a polariza-

τL¯ X tion, and that Pτ = e h P0.

Theorem 5.3.9. Pτ is a polarization.

τL¯ Proof. We start by proving that e Xh ω = ω:

62 τL¯ X LX ω = dιX ω + ιX dω = ddh = 0 =⇒ e h ω = ω. h h h |{z} |{z} =0 =0

The dimension of Pτ is n, as we have seen. It sufﬁces to check equations (4.3) and (4.4b) for a basis of Pτ .

Pτ is Lagrangian:

∂ ∂ ∂ ∂ τL¯ X τL¯ X τL¯ X τL¯ X τL¯ X ω e h , e h = e h ω e h , e h ∂yj ∂yk ∂yj ∂yk ! τL¯ ∂ ∂ = e Xh ω , ) = 0 ∂yj ∂yk | {z } =0

Pτ is involutive: ! τL¯ ∂ τL¯ ∂ τL¯ ∂ ∂ e Xh , e Xh = e Xh , = 0 ∂yj ∂yj ∂yj ∂yj | {z } =0

5.4 T ∗G is foliated by Kahler¨ manifolds

For j = 1, ..., r, deﬁne:

r n τ τL¯ ∂ X ∂ X k ∂ Z = e Xh = − τH¯ X + − τ¯ [ad H] , (5.35) j ∂yj kj k ∂yj Y j ∂yk k=1 k=r+1 1 Aτ = (Zτ + Z¯τ ) = Re Zτ , (5.36) j 2 j j j i Bτ = (Z¯τ − Zτ ) = Im Zτ . (5.37) j 2 j j j

Consider the following distribution on T ∗G:

Σ = span {Aτ , ..., Aτ ,Bτ , ..., Bτ } . (5.38) R 1 r 1 r

Proposition 5.4.1. Σ is involutive.

Proof. Using

r n σL ∂ X ∂ X j ∂ e Xh = − σH X + − σ [ad H] ∀σ ∈ , i = 1, ..., r (5.39) ∂yi ji j ∂yi Y i ∂yj C j=1 j=r+1 one can prove that σL ∂ λL ∂ e Xh , e Xh = 0, ∀σ, λ ∈ , ∀j, k = 1, ..., r. (5.40) ∂yj ∂yj C

τ τ τLXh ∂ τL¯ Xh ∂ Since the Aj and the Bj are linear combinations of e ∂yj and e ∂yj , then equation 5.40 implies that Σ is involutive:

τ τ Aj ,Ak = 0, (5.41a) τ τ Aj ,Bk = 0, (5.41b)

63 τ τ Bj ,Bk = 0. (5.41c)

Using Frobenius’ theorem, we conclude that Σ is integrable by a foliation F. Denote by Lα the leaf ∗ of the foliation that contains α ∈ T G. Then, F = {Lα}α∈I .

Theorem 5.4.2. Let α ∈ T ∗G, and consider the leaf of F that contains α:

∗ ι: Lα −→ T G. (5.42)

Then:

∗ (i) ι ω is a symplectic form on Lα.

(ii) J : TLα −→ TLα deﬁned by

τ τ J(Zj ) = iZj (5.43a) ¯τ ¯τ J(Zj ) = −iZj (5.43b)

is a complex structure on Lα.

∗ (iii) If τ2 > 0, (Lα, ι ω, J) is a Kahler¨ manifold.

Proof. (i): ι∗ω is closed:

dι∗ω = ι∗dω = 0

∗ τ τ ¯τ ¯τ j ι ω is nondegenerate: {Z1 , ..., Zr , Z1 , ..., Zr } is a basis of TLα ⊗ C. We can deﬁne the forms Ωτ by

j τ j Ωτ (Zk ) = δk, (5.44a) j ¯τ Ωτ (Zk ) = 0. (5.44b)

As a consequence of this deﬁnition,

¯ j τ Ωτ (Zk ) = 0, (5.45a) ¯ j ¯τ j Ωτ (Zk ) = δk. (5.45b)

1 r ¯ 1 ¯ r τ τ ¯τ ¯τ ∗ {Ωτ , ..., Ωτ , Ωτ , ..., Ωτ } is the dual frame to {Z1 , ..., Zr , Z1 , ..., Zr }, and it is a basis of T Lα ⊗ C. There- fore, we can write ι∗ω as:

r ∗ X j k j ¯ k ¯ j ¯ k ι ω = MjkΩτ ∧ Ωτ + NjkΩτ ∧ Ωτ + PjkΩτ ∧ Ωτ , (5.46) j,k=1

64 where the Mjk and the Pjk are antisymmetric. The Mjk, Njk and Pjk are given by:

τ τ 2Mjk = ω(Zj ,Zk ), (5.47a) τ ¯τ Njk = ω(Zj , Zk ), (5.47b) ¯τ ¯τ 2Pjk = ω(Zj , Zk ), (5.47c)

τ and using equation (3.21) for ω and equation (5.35) for Zj , we conclude that

τ τ ω(Zj ,Zk ) = 0, (5.48a) τ ¯τ ω(Zj , Zk ) = 2iτ2Hjk, (5.48b) ¯τ ¯τ ω(Zj , Zk ) = 0. (5.48c)

Therefore

r ∗ X j ¯ k ι ω = 2iτ2HjkΩτ ∧ Ωτ . (5.49) j,k=1

Since the components of ι∗ω form a nondegenerate matrix (recall that the restriction of H to j, k = 1, ..., r is positive deﬁnite), ι∗ω is nondegenerate. (ii): J is almost complex:

2 τ τ τ J (Zj ) = iJ(Zj ) = Zj (5.50a) 2 ¯τ ¯τ ¯τ J (Zj ) = −iJ(Zj ) = −Zj (5.50b)

J is integrable: Consider the Nijenhuis tensor associated to J:

N (X,Y ) = [JX,JY ] − J[X,JY ] − J[JX,Y ] − [X,Y ].

Equations (5.41a)-(5.41c) imply that:

τ τ Zj ,Zk = 0, (5.51a) τ ¯τ Zj , Zk = 0, (5.51b) ¯τ ¯τ Zj , Zk = 0. (5.51c)

From these equations, and the deﬁnition of the Nijenhuis tensor it follows easily that

τ τ N (Zj ,Zk ) = 0, (5.52) τ ¯τ N (Zj , Zk ) = 0, (5.53) ¯τ τ N (Zj ,Zk ) = 0, (5.54) ¯τ ¯τ N (Zj , Zk ) = 0. (5.55)

65 From which we conclude that N = 0. So, J is integrable. (iii): We have to prove that

∗ τ τ ι ω(Zj ,Zk ) = 0, (5.56) ∗ ¯τ τ i(ι ω)(Zj ,Zk ) 0. (5.57)

∗ τ τ We have already seen that ι ω(Zj ,Zk ) = 0. Also,

∗ ¯τ τ ∗ τ ¯τ i(ι ω)(Zj ,Zk ) = −i(ι ω)(Zk , Zj ) = −i(2iτ2Hkj) = 2τ2Hjk, (5.58) which is positive deﬁnite if and only if τ2 > 0.

Remark 5.4.3. If we write Pτ as

( ) ( ) ∂ ∂ P = span Zτ , ..., Zτ ⊕ span , ..., , (5.59) τ C 1 r C ∂yr+1 ∂yn | {z } | {z } =Σ =Σ C R then we can see that Pτ is a mixed polarization, in the sense that it is Kahler¨ in some directions and real and the rest. Note that the distributions ΣC and ΣR satisfy:

• ΣC ∩ ΣC = 0,

iω : ΣC × ΣC −→ C is positive deﬁnite.

• ΣR = ΣR.

Proposition 5.4.4. For all α ∈ T ∗G

κ = 2τ2h (5.60)

∗ is a Kahler¨ potential for (Lα, i ω, J).

Proof. Note that we have the complex structure J speciﬁed not it terms of complex coordinates, but in terms of a basis of complex vector ﬁelds. We start by writing the condition for κ to be a potential

ω = i∂∂κ¯ (5.61)

τ ¯τ in terms of the vector ﬁelds Zj and Zj .

ω = i∂∂κ¯ ⇐⇒ −dθ = d(i∂κ¯ ) = 0

⇐⇒ d(θ + i∂κ¯ ) = 0

¯ τ ¯τ ⇐⇒ d(θ + i∂κ)(Zj , Zk ) = 0 τ ¯ ¯τ ¯τ ¯ τ ¯ τ ¯τ ⇐⇒ Zj (θ + i∂κ)(Zk ) − Zk (θ + i∂κ)(Zj ) − (θ + i∂κ)([Zj , Zk ]) = 0 | {z } =0 τ ¯τ ¯τ ¯τ τ ⇐⇒ Zj θ(Zk ) + iZk (κ) = Zk θ(Zj )

66 In the third equivalence we used the fact that θ + i∂κ¯ is of type (1,1). So, κ is a Kahler¨ potential if and only if it satisﬁes

τ ¯τ ¯τ τ τ ¯τ iZj (Zk (κ)) = Zk θ(Zj ) − Zj θ(Zk )) ∀j, k = 1, ..., r. (5.62)

Pn j j τ Using the fact that θ = j=1 y ω , and expression 5.35 for Zj we can perform the following computations:

r τ X k ¯τ τ θ(Zj ) = −τ¯ Hkjy =⇒ Zk θ(Zj ) = −τH¯ jk, k=1 r ¯τ X k τ ¯τ θ(Zj ) = −τ Hkjy =⇒ Zj θ(Zk )) = −τHjk. k=1

Equation 5.62 takes the form:

τ ¯τ iZj (Zk (κ)) = (τ − τ¯)Hjk, (5.63) and κ = 2τ2h satisﬁes this equation:

∂2κ iZτ (Z¯τ (κ)) = i = (τ − τ¯)H . (5.64) j k ∂yj∂yk jk

5.5 Computation of the P¯τ -polarized sections

On section 4.2 we explained the approach on how to deﬁne Hτ . Following that reasoning, in this section we compute the set of sections that are polarized along P¯τ .

Deﬁnition 5.5.1. Let f be a real or complex function on T ∗G. The Kostant–Souriau operator of f acts on sections of L and is deﬁned by

ˆ f = i∇Xf + f. (5.65)

Proposition 5.5.2. ˆ h = iXh − h (5.66)

Proof. n  n  n X j j X l X j k θ(Xh) = y ω  Hkly Xk = Hjky y = 2h j=i k,l=1 j,k=1

ˆ h(s) = i∇Xh s + hs = i Xh(s) + i θ(Xh) s + hs = iXh(s) − hs | {z } =2h

Lemma 5.5.3. Let s ∈ Γ(T ∗G × C), X ∈ X(T ∗G), σ ∈ C. Then

h i −σLX iσh iσhˆ e h ∇ σL s = e ∇ e s . (5.67) e Xh X X

67 Proof. Since

LXh θ = dιXh θ + ιXh dθ = d(2h) − ιXh ω = 2dh − dh = dh L2 θ = d ι dh +ι ddh = 0 Xh Xh Xh | {z } |{z} =0 =0

−σL we conclude that e Xh θ = θ − σdh.

iσh h iσhˆ i e ∇X e s h i = eiσh X e−σ(Xh+ih)s + iθ(X) e−σ(Xh+ih)s

= eiσh X e−σXh se−iσh + iθ(X) e−σXh s e−iσh

= eiσh X e−σXh s e−iσh − iσ(e−σXh s)e−iσhX(h) + iθ(X) e−σXh s e−iσh

= X e−σXh s + i(θ − σdh)(X) e−σXh s

−σX −σLX −σX = X e h s + i e h θ (X) e h s

−σLX σLX σLX = e h e h X (s) + iθ e h X (s) −σLX = e h ∇ σL s e Xh X

Proposition 5.5.4.

∂ ˆ ∇ s = 0 ∀X ∈ X(P¯ ) ⇐⇒ eiτhs = 0 ∀j = 1, ..., n (5.68) X τ ∂yj

Proof.

∇X s = 0 ∀X ∈ X(P¯τ )

⇐⇒ ∇ τL s = 0 ∀j = 1, ..., n e Xh ∂ ∂yj −τLX ⇐⇒ e h ∇ τL s = 0 ∀j = 1, ..., n e Xh ∂ ∂yj iτh h iτhˆ i ⇐⇒ e ∇ ∂ e s = 0 ∀j = 1, ..., n ∂yj iτhˆ ⇐⇒ ∇ ∂ e s = 0 ∀j = 1, ..., n ∂yj ∂ ˆ ⇐⇒ eiτhs = 0 ∀j = 1, ..., n ∂yj where in the third equivalence we used lemma 5.5.3.

As a consequence of the previous proposition, we conclude that

−iτhˆ ∇X s = 0 ∀X ∈ X(P¯τ ) ⇐⇒ s = e f, (5.69) for some f ∈ C∞(T ∗G) depending only on the base point of T ∗G.

68 5.6 The inner product of Hτ . An unitary isomorphism of Hilbert spaces

The approach given on section 4.2 suggests that the Hilbert space associated to the vertical polarization

P0 can be deﬁned simply as

2 H0 = L (G, dx), (5.70) where dx is the Haar measure of G. Its inner product is

Z hf, gi0 = hf, giL2(G,dx) = fgdx.¯ (5.71) G Remark 5.6.1. We state some relevant facts about L2(G, dx). Consider the set of all irreducible unitary representations of G:

{λα : G −→ GL(V )}α∈I .

Also, consider its partition into equivalence classes:

{[λα]}α∈J .

For each class [λ ], choose a representative λ and a basis {vα, ..., vα } of V . Deﬁne the represen- α α 1 dα α jk jk tative functions λα : G −→ C the following way: λα (g) is the (j, k) entry of the matrix that represents λ (g) in the basis {vα, ..., vα }. α 1 dα Then, it is a fact that the space spanned by the representative functions of a given class of representations does not depend on the chosen basis. Also, it is a consequence of the Peter-Weyl theorem that jk 2 {λα }α∈J,j,k∈1,...,dα is an orthogonal basis of L (G, dx):

L2(G, dx) = span {λjk} . (5.72) C α α∈J,j,k∈1,...,dα

We also have the Weyl orthogonality relations:

jk lm 1 hλα , λβ iL2(G,dx) = δjlδkmδαβ . (5.73) dim Vα

Remark 5.6.2. Each representation λα induces a representation of the Lie algebra:

λα G GL(Vα)

exp exp (5.74)

deλα g End(Vα)

The matrices λα(g), with g ∈ G, are unitary. The matrices deλα(X), with X ∈ g, are then derivatives of curves of unitary matrices, hence anti-hermitian. It is a fact from linear algebra that anti-hermitian matrices are diagonalizable. The matrices {deλα(X)}X∈h commute, because h is an abelian subalgebra:

69 [deλα(X), deλα(Y )] = deλα [X,Y ] = 0. (5.75) | {z } =0

{deλα(X)}X∈h is then a set of diagonalizable matrices that commute. Therefore, it is a set of simultaneously diagonalizable matrices. So, for each α, we can choose a basis {vα, ..., vα } of V such that for all 1 dα α

X ∈ h deλα(X) is diagonal:

 α  iλ1 (X) ··· 0    . .. .  deλα(X) =  . . .  . (5.76)   0 ··· iλα (X) dα

iλα : g −→ are linear functionals, which give the eigenvalue of the matrix in the basis {vα, ..., vα }: j C 1 dα

α α α [deλα(X)]vj = iλj (X)vj . (5.77)

α Since the matrices {deλα(X)}X∈h are anti-hermitian, the λj are real valued.

As we computed in the previous section, the set of P¯τ -polarized sections is

¯ −iτhˆ ∞ ∞ ∗ s ∈ Γ(L): ∇X s = 0 ∀X ∈ X(Pτ ) = {e f : f ∈ C (G; C) ⊂ C (T G; C)}. (5.78)

It is possible to compute that

τL 1 r τL¯ 1 r r 1 r 1 r e Xh (ω ∧ ... ∧ ω ) ∧ e Xh (ω ∧ ... ∧ ω ) = (¯τ − τ) det Hω ∧ ... ∧ ω ∧ dy ∧ ... ∧ dy . (5.79)

This equation, in conjunction with the half form correction, suggests that we deﬁne the following inner product on this set of P¯τ -polarized sections:

r r Z −iτhˆ −iτhˆ τ −iτhˆ −iτhˆ he f, e giτ = det H e f e g dxdy. (5.80) π G×h Choose a set of λ as in remark 5.6.1 and for each α an associated basis {vα, ..., vα } as in remark α 1 dα 5.6.2.

Proposition 5.6.3. τXh jk jk τu(Y ) e λα (x, Y ) = λα xe (5.81)

Proof.

jk d jk t d jk Xl · λα (x) = λα (φX (x)) = λα (x exp(tTl)) dt t=0 l dt t=0 dα dα d X jm mk X jm d mk = λα (x)λα (exp(tTl)) = λα (x) λα (exp(tTl)) dt t=0 dt t=0 m=1 m=1

dα X jm mk = λα (x)deλα (Tl) m=1

70 r r dα jk X l jk X l X jm mk Xh · λα (x, Y ) = (u(Y )) Xlλα (x) = (u(Y )) λα (x)deλα (Tl) l=1 l=1 m=1

dα r ! dα X jm mk X l X jm mk = λα (x)deλα (u(Y )) Tl = λα (x)deλα (u(Y )) m=1 l=1 m=1

∂ ∂ 1 r Note that Xh has terms in ∂yr+1 , ..., ∂yn , and u depends only on y , ..., y . Therefore,

d α mk l jk X jm h mk li Xh · λα (x, Y ) = λα (x) deλα (u(Y )) (5.82) m=1

∞ l ∞ l dα X τ X τ X h limk eτXh · λjk (x, Y ) = Xl λjk(x, Y ) = λjm(x) d λmk(u(Y )) α l! h α l! α e α l=0 l=0 m=1 mk dα " ∞ l # dα X X τ l X = λjm(x) (d λ (u(Y ))) = λjm(x)λmk eτu(Y ) α l! e α α α m=1 l=0 m=1 jk τu(Y ) = λα xe

Proposition 5.6.4.

ˆ α −iτh jk jk iτλj (u) e λα = λα e (5.83) jk α −iτhˆ ¯jk −iτλ¯ j (u) e λα = λα e (5.84)

Proof.

ˆ −iτh jk τ(Xh+ih) jk iτh τXh jk e λα (x, Y ) = e λα (x, Y ) = e e λα (x, Y )

dα iτh jk τu(Y ) iτh X jl lk τu(Y ) = e λα xe = e λα (x)λα e l=1

dα dα lk α iτh X jl deλα(τu(Y )) iτh X jl iτλk (u(Y )) = e λα (x) e = e λα (x)δlke l=1 l=1 α iτh jk iτλj (u) = e λα (x)e

Which proves the ﬁrst equation. The second one follows from performing complex conjugation.

Proposition 5.6.5. ˆ ˆ δjlδkmδαβ α [ −iτh jk −iτh lm 2τ2h((λk ) ) he λα , e λβ iτ = e (5.85) dim Vα Proof.

−iτhˆ jk −iτhˆ lm he λα , e λβ iτ r r Z τ −iτhˆ jk −iτhˆ lm = det H e λα e λβ dxdy π G×h r τ r Z Z α β jk ¯lm iτh −iτh¯ iτλj (u) −iτλ¯ m(u) = det H λα λβ dx e e e e dy π G h

71 r r Z δjlδkmδαβ τ −2τ h−2τ λα(u) = det H e 2 2 k dy dim Vα π h r r Z δjlδkmδαβ τ 1 −τ uT H−1u−2τ ((λα)[)T u = e 2 2 k du dim Vα π det H h

The integral over h can be computed using the Gaussian integral formula:

Z − 1 xT Ax+BT x r/2 −1/2 1 BT A−1B e 2 dx = (2π) (det A) e 2 , (5.86) r R r −1 T α [ T where A is a positive deﬁnite r × r matrix, and B is a vector in R . A = 2τ2H and B = −τ2((λk ) ) yields that

ˆ ˆ δjlδkmδαβ α [ −iτh jk −iτh lm 2τ2h((λk ) ) he λα , e λβ iτ = e dim Vα Corollary 5.6.6. −iτhˆ −iτhˆ limhe f, e giτ = hf, gi0 (5.87) τ→0

Proof.

ˆ ˆ δjlδkmδαβ α [ δjlδkmδαβ −iτh jk −iτh lm 2τ2h((λk ) ) jk lm limhe λα , e λβ iτ = lim e = = hλα , λβ i0 (5.88) τ→0 τ→0 dim Vα dim Vα

2 Hτ should be the Hilbert space that contains Pτ -polarized sections of L that are L with respect to 5.80. From proposition 5.6.5, we can conclude that

Hτ = s ∈ Γ(L): ∇X s = 0 ∀X ∈ X(P¯τ ), kskτ < ∞ (5.89) n ˆ o = e−iτhf | f ∈ L2(G, dx) (5.90) n o = span e−iτhˆ λjk . (5.91) C dα α∈J,j,k=1,...,dα

−iτhˆ We want to deﬁne a unitary isomorphism H0 −→ Hτ . The obvious attempt is e : H0 −→ Hτ . However, proposition 5.6.5 shows that e−iτhˆ preserves orthogonality but not the norm. Because of this, jk we introduce the linear map Qh, which is the unique linear map that acts on the basis {λα }α∈J,j,k∈1,...,dα of H0 as:

Qh : H0 −→ H0 (5.92)

jk α [ jk λα 7−→ h((λk ) )λα

Theorem 5.6.7. The map ˆ −iτh iτQh Uτ := e ◦ e : H0 −→ Hτ (5.93) is a unitary isomorphism of Hilbert spaces.

Proof. Uτ is bijective, because it has inverse

72 ˆ −1 −iτQh −iτh Uτ = e ◦ e . (5.94)

ˆ Uτ is linear because h and Qh are linear. We have to show that Uτ preserves the inner product. It sufﬁces to show that it preserves the norm of the elements of a basis.

ˆ ˆ α [ α [ ˆ jk −iτh iτQh jk −iτh iτh((λk ) ) jk iτh((λk ) ) −iτh jk Uτ (λα ) = e e (λα ) = e e λα = e e (λα )

α [ ˆ jk 2 iτh((λk ) ) −iτh jk 2 kUτ (λα )kτ = ke e (λα )kτ α [ ˆ iτh((λk ) ) 2 −iτh jk 2 = |e | ke (λα )kτ α [ α [ ˆ iτ1h((λk ) ) 2 −τ2h((λk ) ) 2 −iτh jk 2 = |e | |e | ke (λα )kτ | {z } =1 −2τ h((λα)[) 1 2τ h((λα)[) = e 2 k e 2 k dim Vα 1 = dim Vα jk 2 = kλα k0

73 74 Bibliography

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76 Appendix A

Moser’s Theorem

In this appendix we give a proof of Moser’s Theorem. We start with some useful propositions.

A.1 Lie derivative of time dependent vector ﬁeld in terms of an isotopy

The Lie derivative along a time dependent vector field Xt is obtained by considering the Lie derivative along the vector field Xt for each t. The next formula gives a way to write the Lie derivative of a form along a time dependent vector field in terms of the isotopy that defines it.

Proposition A.1.1. The Lie derivative of a form along a time dependent vector ﬁeld Xt with (local) isotopy ρt satisﬁes

d ρ∗ω = ρ∗L ω. (A.1) dt t t Xt

Proof. Part 1: The formula is true if ω = f ∈ Ω0(M) = C∞(M):

d d ρ∗f (p) = f (ρ (p)) = (X · f)(ρ (p)) dt t dt t t t

d =⇒ (ρ∗f) = ρ∗ (X · f) = ρ∗L ω dt t t t t Xt

Part 2: Both sides of the equation commute with d:

∗ ∗ ∗ ∗ ∗ ρt LXt dω = ρt (ιXt ddω +dιXt dω) = ρt (ddιXt ω +dιXt dω) = ρt dLXt ω = dρt LXt ω | {z } | {z } =0 =0

d d d ρ∗dω = d(ρ∗ω) = d ρ∗ω dt t dt t dt t

Since both sides of the equation commute with d, we conclude that if the formula is true for ω then it is true for dω.

77 Part 3: If the formula is true for ω and η, then it is true for ω ∧ η:

d d (ρ∗(ω ∧ η)) = (ρ∗ω ∧ ρ∗η) dt t dt t t d d = (ρ∗ω) ∧ ρ∗η + (ρ∗ω) ∧ (ρ∗η) dt t t t dt t ∗ ∗ ∗ ∗ = (ρt LXt ω) ∧ ρt η + (ρt ω) ∧ (ρt LXt η) (by hypothesis)

∗ = ρt ((LXt ω) ∧ η + ω ∧ (LXt η))

∗ = ρt (LXt (ω ∧ η))

Part 4: If (U, x1, ..., xn) is a coordinate chart, and ω ∈ Ωk(U), then

X i1 ik ω = ωi1...ik dx ∧ ... ∧ dx .

1≤i1<...

Proposition A.1.2. If ωt is a family of forms that is smooth in t then

d dω ρ∗ω = ρ∗ L ω + t (A.2) dt t t t Xt t dt

Proof.

d ∗ d ∗ ρt ωt = ρsωs dt ds s=t d ∗ ∗ d = ρsωt + ρt ωs ds s=t ds s=t dω = ρ∗L ω + ρ∗ t t Xt t t dt dω = ρ∗ L ω + t . t Xt t dt

A.2 Proof of Moser’s Theorem

2 Theorem A.2.1 (Moser). Let M be a compact manifold, ω0, ω1 ∈ Ω (M) be symplectic forms, such that

2 (i) [ω0] = [ω1] ∈ H (M, R);

(ii) ∀t ∈ [0, 1] ωt := (1 − t)ω0 + tω1 is symplectic.

∗ Then there exists an isotopy ρt : M −→ M such that ρt ωt = ω0 ∀t ∈ [0, 1]

Proof. Let ρ: M × R −→ M be an isotopy, with time dependent vector ﬁeld Xt. We claim that

dω ρ∗ω = ω ∀t ∈ [0, 1] ⇐⇒ dι ω + t = 0 ∀t ∈ [0, 1]. t t 0 Xt t dt

To see that the claim is true, note that

78 d ρ∗ω = ω ∀t ∈ [0, 1] ⇐⇒ ρ∗ω = ρ∗ω ∀t ∈ [0, 1] ⇐⇒ ρ∗ω = 0 ∀t ∈ [0, 1]. t t 0 t t 0 0 dt t t

Also

  d ∗ ∗ dωt ∗ dωt ∗ dωt ρt ωt = ρt LXt ωt + = ρt dιXt ωt + ιXt dωt +  = ρt dιXt ωt + . dt dt |{z} dt dt =0 Therefore

d dω ρ∗ω = ω ⇐⇒ ρ∗ω = 0 ⇐⇒ dι ω + t = 0, t t 0 dt t t Xt t dt which proves the claim.

dωt 1 Since [ω0] = [ω1], dt = ω1 − ω0 = dµ, for some µ ∈ Ω (M). For all t ∈ R, Moser’s equation, which is

ιXt ωt + µ = 0, (A.3) deﬁnes a unique vector ﬁeld Xt, that is smooth in t. Since M is compact, there exists an isotopy

ρ: M × R −→ M that integrates Xt.

dωt Since 0 = dιXt ωt + dµ = dιXt ωt + dt , ρ is the pretended isotopy.

79 80