complexified symplectomorphisms and geometric quantization · the analogous to darboux’s theorem...
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Complexified Symplectomorphisms and GeometricQuantization
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 AbreuSupervisor: Prof. João Pimentel Nunes
Member of the Committee: Prof. José Manuel Vergueiro Monteiro Cidade MourãoMember 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.
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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. Sılvia 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.
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Resumo
Nesta tese estudamos geometria complexa, simplectica e Kahler.
O capıtulo 1 e introdutorio e apresentamos os preliminares teoricos relevantes para o resto da tese.
No capıtulo 2 apresentamos um resumo de [14]. Partindo de uma variedade Kahler, 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 capıtulo 3 estudamos estruturas Kahler 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 possıvel definir estruturas Kahler em T ∗G usando funcoes
complexificadoras Ad-invariantes. Provamos que a condicao de Ad-invariancia pode ser substituıda por
um conjunto de equacoes mais geral.
No capıtulo 5, damos um exemplo de quantizacao geometrica de T ∗G. Introduzimos um fibrado de
linha L em T ∗G e uma polarizacao Pτ definida por fazer um pullback da polarizacao vertical pelo fluxo
complexificado de um campo vetorial Hamiltoniano. A partir destas estruturas, definimos um espaco de
Hilbert cujos elementos sao as seccoes integraveis e Pτ -polarizadas de L.
Palavras-chave: geometria Kahler, quantizacao geometrica, metrica de Mabuchi, grupo de
Lie, fibrado cotangente, difeomorfismos complexificados
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Abstract
In this thesis, we study complex, symplectic and Kahler geometry.
Chapter 1 is introductory and presents the relevant theoretical background for the rest of the thesis.
In chapter 2 we present an overview of [14]. Starting with a Kahler 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 chapter 3 we study Kahler 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 chapter 5, 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
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Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Theoretical Background 1
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
3.6.1 ϕh is a diffeomorphism. The complex structure Jτ . . . . . . . . . . . . . . . . . . 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 Pτ -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
φtX 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 〈·, ·〉 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
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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
ω =
n∑i=1
dxi ∧ dyi. (1.1)
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︸ ︷︷ ︸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
ω =
n∑i=1
dxi ∧ dyi.
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 ,
the structure of a symplectic manifold. If (U, x1, ..., xn) is a coordinate chart on M , then (dx1)p, ..., (dxn)p
form a basis of T ∗pM . Then, any element α of T ∗pM can be given as α =∑ni=1 pi(dx
i)p. This defines a
local coordinate chart on T ∗M , (T ∗U, x1, ..., xn, p1, ..., pn), where the (x1, ..., xn) are the coordinates of
p = π(α) in M , and (p1, ..., pn) are the coefficients of α in the basis dx1, ..., dxn.
Define θ ∈ Ω1(T ∗M) pointwise by:
θα : Tα(T ∗M) −→ R (1.2)
v 7−→ α(dπ · v).
θ is the canonical symplectic potential. If v =∑ni=1 a
i ∂∂xi +
∑ni=1 bi
∂∂pi
, then
θα(v) = α(dπ · v) = α
dπ n∑j=1
aj∂
∂xj+
n∑j=1
bj∂
∂pj
= α
n∑j=1
aj∂
∂xj
=
n∑i=1
pi(dxi)p︸ ︷︷ ︸
∈ T∗pM
n∑j=1
aj∂
∂xj
=
n∑i=1
piai =
n∑i=1
pi(dxi)α︸ ︷︷ ︸
∈ Tα(T∗M)
(n∑j=1
aj∂
∂xj+
n∑j=1
bj∂
∂pj︸ ︷︷ ︸v
).
Therefore, in local coordinates θ can be written as θ =∑ni=1 pidx
i.
Define the canonical symplectic form ω ∈ Ω2(T ∗M) by ω = −dθ. The local expression for ω is then
ω = −d
(n∑i=1
pidxi
)=
n∑i=1
dxi ∧ dpi.
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
dρtdt
= Xt ρt. (1.3)
In other words, Xt is given at a point p by (Xt)p = dds
∣∣∣s=t
ρs(ρ−1t (p)
).
• Conversely, ifXt is a time dependent vector field, and ifM is compact (orXt 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)?
Theorem 1.2.4 (Moser). Let M be a compact manifold, ω0, ω1 ∈ Ω2(M) be symplectic forms, such that
(i) [ω0] = [ω1] ∈ H2(M,R);
(ii) ∀t ∈ [0, 1] ωt := (1− t)ω0 + tω1 is symplectic.
Then there exists an isotopy ρt : M −→M such that ρ∗tωt = ω0 ∀t ∈ [0, 1]
Proof. See appendix A.
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
J : TM −→ TM such that J2 = −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),
• A = ϕα : Uα ⊂M −→ Vα ⊂ Cnα∈I a complex atlas,
satisfying:
• M = ∪α∈IUα ;
• ∀α ∈ I, ϕα is an homeomorphism;
• ∀α, β ∈ I such that Uα ∩ Uβ 6= ∅,
ϕβ ϕ−1α : ϕα(Uα ∩ Uβ) ⊂ Cn −→ ϕβ(Uα ∩ Uβ) ⊂ Cn
is biholomorphic (this means that both ϕβ ϕ−1α and its inverse are holomorphic);
• A is maximal: if ψ : U ⊂ M −→ V ⊂ Cn is an homeomorphism such that ∀α ∈ I ψ ϕ−1α 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 ,
TpM = spanR
∂
∂xj,∂
∂yj: j = 1, ..., n
.
Define
Jp : TpM −→ TpM
∂
∂xj7−→ ∂
∂yj
∂
∂yj7−→ − ∂
∂xj.
Then clearly J2p = −id.
Now, we prove that J is a globally well defined object. Let (U, z1, ..., zn), (V,w1, ..., wn) be coordinate
neighborhoods such that U ∩ V 6= ∅. We will show that J |U = J |V on U ∩ V . If zj = xj + iyj and wj =
uj + ivj , since the transition functions are biholomorphic, they satisfy the Cauchy-Riemann equations:∂uk
∂xj=∂vk
∂yj
∂uk
∂yj= −∂v
k
∂xj.
(1.4)
4
We will now show that J |U = J |V .
J |V(
∂
∂xj
)= J |V
(n∑k=1
∂uk
∂xj∂
∂uk+∂vk
∂xj∂
∂vk
)
=
n∑k=1
∂uk
∂xj∂
∂vk− ∂vk
∂xj∂
∂uk
=
n∑k=1
∂vk
∂yj∂
∂vk+∂uk
∂yj∂
∂uk=
∂
∂yj
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 bun-
dle 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)
T 1,0M := (T1,0M)∗ = complex-linear cotangent vectors, (1.5c)
T 0,1M := (T0,1M)∗ = complex-antilinear cotangent vectors. (1.5d)
Also, define
Ωk(M ;C) :=⊕
l+m=k
Ωl,m(M) = k-forms on M with complex values, (1.6)
where
Ωl,m(M) := sections of the vector bundle
(l∧T 1,0M
)∧
(m∧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:
TpM = spanR
∂
∂xj,∂
∂yj: j = 1, ..., n
,
TpM ⊗ C = spanC
∂
∂xj,∂
∂yj: j = 1, ..., n
,
J
(∂
∂xj
)=
∂
∂yjand J
(∂
∂yj
)= − ∂
∂xj.
5
Using these equations, the spaces in definition 1.3.4 become:
T1,0M := (+i)− eigenspace of J = spanC
1
2
∂
∂xj− i
2
∂
∂yj︸ ︷︷ ︸:= ∂
∂zj
j=1,...,n
, (1.9a)
T0,1M := (−i)− eigenspace of J = spanC
1
2
∂
∂xj+i
2
∂
∂yj︸ ︷︷ ︸:= ∂
∂zj
j=1,...,n
, (1.9b)
T 1,0M := (T1,0M)∗ = spanCdxj + idyj︸ ︷︷ ︸=dzj
j=1,...,n, (1.9c)
T 0,1M := (T0,1M)∗ = spanCdxj − idyj︸ ︷︷ ︸=dzj
j=1,...,n. (1.9d)
And if (U, z1, ..., zn) is a complex coordinate neighborhood,
Ωl,m(U) =
∑|J|=l,|K|=m
bJ,KdzJ ∧ dzK : bJ,K ∈ C∞(U ;C)
. (1.10)
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;
(v) π2,0 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
(in particular using that J2 = −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.
A Kahler 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
ω =i
2
n∑j,k=1
hjkdzj ∧ dzk, (1.11)
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.
Lemma 1.3.13. Let (M,ω, J, g) be a Kahler manifold. Let h ∈ C∞(M). Then, ‖Xh‖2 = ‖∇h‖2.
Proof.
〈∇h,∇h〉 = dh(∇h) = ω(Xh,∇h) = 〈JXh,∇h〉
= dh(JXh) = ω(Xh, JXh) = 〈Xh, Xh〉
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:
〈·, ·〉 : Ωk(M)× Ωk(M) −→ R (1.12)
(α, β) 7−→∫M
α ∧ ∗β.
• Adjoints with respect to the inner product: the adjoint of d, d∗, is uniquely specified by the property
〈dα, β〉 = 〈α, d∗β〉. 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 = 〈∆ω, ω〉 = 〈dd∗ω + d∗dω, ω〉
= 〈dd∗ω, ω〉+ 〈d∗dω, ω〉 = 〈d∗ω, d∗ω〉+ 〈dω, dω〉
= ‖d∗ω‖2 + ‖dω‖2.
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). LetM 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 〈·, ·〉.
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 +
∂α1,0 + ∂α0,1 + ∂α0,1. Since ω is of type (1, 1), ∂α1,0 and ∂α0,1 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 = ∂α0,1 = ∂γ︸︷︷︸=0
+ ∂∂η︸︷︷︸=0
+∂∂∗ξ,
0 = 〈∂α0,1, ξ〉 = 〈∂∂∗ξ, ξ〉 = 〈∂∗ξ, ∂∗ξ〉 =⇒ ∂∗ξ = 0.
So, α0,1 = γ + ∂η, and therefore
α = γ + ∂η + γ + ∂η,
ω = dα = dγ︸︷︷︸=0
+d∂η + dγ︸︷︷︸=0
+d∂η
= ∂∂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 ∂α0,1 are zero. The ∂-Lemma applied to α0,1 implies that
locally, α0,1 = ∂ψ (hence α1,0 = ∂ψ). Then, 2 Imψ is a local Kahler potential:
ω = dα = ∂α1,0 + ∂α0,1
= ∂∂ψ + ∂∂ψ = −∂∂ψ + ∂∂ψ
= ∂∂(ψ − ψ) = i∂∂(2 Imψ).
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 = gi + hi.
Therefore,
gi − hi = gi − 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 =
gi − hi = ci, where ci is a real locally constant function. Define fi = hi + ci2 . Then, fi is holomorphic, fi
10
is anti-holomorphic, and
fi + fi = hi +ci2
+ hi +ci2
= hi + hi + ci = hi + gi = φ|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 homomor-
phism 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)
g 7−→
Ψg : G −→ G
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)
X 7−→
adX : g −→ g
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;
• Ψg is a Lie group isomorphism: Ψg(a)Ψg(b) = Ψg(ab) and (Ψg)−1 = Ψg−1 ;
• Ad is a Lie group homomorphism: Adgh = AdgAdh;
• Adg is a Lie algebra isomorphism: Adg[X,Y ] = [AdgX,Adg, Y ] and (Adg)−1 = Adg−1 ;
• ad is a Lie algebra homomorphism: ad[X,Y ] = [adX , adY ];
• adX is a Lie algebra derivation: adX [Y,Z] = [adXY,Z] + [Y, adXZ].
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:
g h
G H
φ=deΦ
exp exp
Φ
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 homomor-
phism 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 ].
deAd(X)(Y ) =d
dt
∣∣∣t=0
(deψexp(tX)Y
)=
d
dt
∣∣∣t=0
d
ds
∣∣∣s=0
(ψexp(tX) exp(sY )
)=
d
dt
∣∣∣t=0
d
ds
∣∣∣s=0
(exp(tX) exp(sY ) exp(−tX)
)=
d
dt
∣∣∣t=0
d
ds
∣∣∣s=0
(exp(tX) exp(sY )
)− d
dt
∣∣∣t=0
d
ds
∣∣∣s=0
(exp(sY ) exp(tX)
)= V.
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
V · f =d
dt
∣∣∣t=0
d
ds
∣∣∣s=0
f(
exp(tX) exp(sY ))− d
dt
∣∣∣t=0
d
ds
∣∣∣s=0
f(
exp(sY ) exp(tX))
=d
dt
∣∣∣t=0
d
ds
∣∣∣s=0
f(
exp(tX) exp(sY ))− d
ds
∣∣∣s=0
d
dt
∣∣∣t=0
f(
exp(sY ) exp(tX))
=d
dt
∣∣∣t=0
(Y · f
)(exp(tX))− d
ds
∣∣∣s=0
(X · f
)(exp(sY ))
=(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
set of M . The structure functions of this frame, Ckij , are defined by the equation:
[Xi, Xj ] =
n∑k=1
CkijXk. (1.24)
Note that since the Lie bracket is antisymmetric, so are the structure functions: Ckij = −Ckji. We now
give some properties of these functions.
13
Proposition 1.5.2. If ω1, ..., ωn is the field of coframes dual to X1, ..., Xn, then
dωi +1
2
n∑j,k=1
Cijkωj ∧ ωk = 0. (1.25)
Proof.
dωi(Xl, Xm) = Xl(ωi(Xm))−Xm(ωi(Xl))− ωi([Xl, Xm])
= −ωi([Xl, Xm]) = −ωi n∑j=1
CjlmXj
= −Cilm = −1
2
n∑j,k=1
Cijkωj ∧ ωk(Xl, Xm).
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.
Proof. We prove that Ckij(g) = Ckij(e) for all g ∈ G.
n∑k=1
Ckij(g)(Xk)g = [Xi, Xj ]g = [dLgXi, dLgXj ]g
= dLg[Xi, Xj ]e = dLg
n∑k=1
Ckij(e)(Xk)e
=
n∑k=1
Ckij(e)(Xk)g,
which implies that Ckij(g) = Ckij(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:
eτLXS :=
∞∑k=0
τk
k!LkX(S). (2.1)
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 etLXS converges, and
etLXS = (φtX)∗S, (2.2)
where φtX is the flow of X.
15
Proof. We start by showing by induction that dk
dtk((φtX)∗S) = LkX((φtX)∗S).
(k = 1) :d
dt(φtX)∗S =
d
ds
∣∣∣s=0
(φt+sX )∗S =d
ds
∣∣∣s=0
(φsX)∗(φtX)∗S = LX((φtX)∗S) (2.3)
(k =⇒ k + 1) : Note that (φtX)∗X = X, because the curve φtX is tangent to X and φtX φsX = φt+sX .
Therefore, LX and (φtX)∗ commute:
(φtX)∗LXS = L(φtX)∗X(φtX)∗S = LX(φtX)∗S. (2.4)
dk+1
dtk+1((φtX)∗S) =
d
dt
dk
dtk((φtX)∗S) =
d
dtLkX((φtX)∗S)
=d
dt(φtX)∗LkXS = LX(φtX)∗LkXS = (φtX)∗Lk+1
X S,
Where we used the induction hypothesis. This proves that dk
dtk((φtX)∗S) = LkX((φtX)∗S). In particular,
dk
dtk
∣∣∣t=0
(φtX)∗S = LkXS. (2.5)
Since X is real analytic, then φtX is real analytic in t and as a map on M . Since S is also real analytic,
(φtX)∗S is real analytic in t and as a tensor field on M . Let p ∈ M . We can expand (φtX)∗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
(φtX)∗S =
∞∑k=0
tk
k!Sk (2.6)
in Vp, where the Sk are some tensors in Vp whose coefficients are power series in x1, ..., xn 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 = minTp1 , ..., TpN . Let t be such that |t| < T . In each Vpj , it is true that
LkXS =dk
dtk
∣∣∣t=0
(φtX)∗S =dk
dtk
∣∣∣t=0
∞∑k=0
tk
k!Sk = Sk (2.7)
which implies that∑∞k=0
tk
k!LkX(S) converges to (φtX)∗S in each Vpj . We conclude that if |t| < T then∑∞
k=0tk
k!LkX(S) converges to (φtX)∗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 eτLXS converges.
Proof. Let T be as in theorem 2.1.2. We start by noticing that∑∞k=0
τk
k! LkX(S) converges if and only
if∑∞k=0
τk
k! LkX(S)(X1, ..., Xl, ω
1, ..., ωm) converges for all p ∈ M , X1, ..., Xl ∈ TpM and ω1, ..., ωm ∈
T ∗pM . Let p ∈ M , X1, ..., Xl ∈ TpM and ω1, ..., ωm ∈ T ∗pM . For ease of notation, denote ak =
1k!L
kX(S)(X1, ..., Xl, ω
1, ..., ωm). As a consequence of the last theorem,
|τ | < T and τ ∈ R =⇒∞∑k=0
akτk converges. (2.8)
16
From complex analysis, the power series∑∞k=0 akτ
k has a radius of convergence R, that satisfies:
|τ | < R =⇒∞∑k=0
akτk converges, (2.9)
|τ | > R =⇒∞∑k=0
akτk diverges. (2.10)
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
|τ | < T =⇒∞∑k=0
τk
k!LkX(S)(X1, ..., Xl, ω
1, ..., ωm) converges, (2.11)
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) = eτLXS ⊗ eτLXR; (2.12)
• if S is a tensor field of type (l,m),
eτLX (S(X1, ..., Xl, ω1, ..., ωm)) = eτLXS(eτLXX1, ..., e
τLXXl, eτLXω1, ..., eτLXωm); (2.13)
• if Y,Z are vector fields,
eτLX [Y,Z] =[eτLXY, eτLXZ
]. (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:
(φtX)∗(S ⊗R) = (φtX)∗S ⊗ (φtX)∗R, (2.15)
(φtX)∗(S(X1, ..., Xl, ω1, ..., ωm)) = (φtX)∗S((φtX)∗X1, ..., (φ
tX)∗Xl, (φ
tX)∗ω1, ..., (φtX)∗ωm), (2.16)
(φtX)∗[Y,Z] =[(φtX)∗Y, (φtX)∗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:
(i) (Definition of Jτ ) Let p ∈ M , and let (Uα, z10 , ..., z
n0 ) 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;
• the series zjτ := eτX · zj0 are uniformly convergent on Vα,p;
• (Vα,p, z1τ , ..., z
nτ ) is a Jτ -holomorphic coordinate neighborhood of p.
(ii) There exists a unique biholomorphism ϕτ : (M,Jτ ) −→ (M,J0) that satisfies the following property:
for all p ∈M , there exists (Uα, z10 , ..., z
n0 ) a J0-holomorphic coordinate neighborhood of p and there
exists (Vα,p, z1τ , ..., z
nτ ) a Jτ -holomorphic coordinate neighborhood satisfying the conditions of (i),
such that ϕτ (Vα,p) ⊂ Uα and zjτ = zj0 ϕτ .
Proof. See theorems 2.5 and 2.6. in [14].
Remark 2.2.2.
• ϕτ being a biholomorphism means that Jτ = dϕ−1τ 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.
• Performing complex conjugation on equation zj0 ϕτ = zjτ , we obtain that zj0 ϕτ = zjτ = eτX · zj0.
• 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)
such that zj ϕτ,J = eτX · zj (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:
(M,Jτ+σ) (M,Jσ)
(M,Jτ ) (M,J0)
ϕσ,Jτ
ϕτ,Jσ
ϕτ+σ,J0 ϕσ,J0
ϕτ,J0
We now note that:
If t ∈ R, since (φtX)∗zj = etX · zj , then ϕt is the flow of X:
ϕt = φtX , (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
real analytic. Let h ∈ Can(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:
XeτXh ·f = eτLXh ·Xf . (2.20)
Proof. We start by proving by induction on k that LkXhXf = XXkh(f).
(k = 1) : LXhXf = [Xh, Xf ] = Xh,f = XXh(f)
(k =⇒ k + 1) : Lk+1Xh
Xf = LXhLkXhXf = LXhXXkh(f) = [Xh, XXkh(f)] = Xh,Xkh(f) = XXk+1
h (f)
Now we prove the result.
XeτXh ·f = X∑∞k=0
τk
k! Xkh(f)
=
∞∑k=0
τk
k!XXkhf
=
∞∑k=0
τk
k!LkXhXf = eτLXh ·Xf .
Note that in the case τ ∈ R, equation 2.20 is the known statement that symplectomorphisms preserve
Hamiltonian vector fields.
Proposition 2.3.2. Let (U, z1, ..., zn) be a J0-complex coordinate chart on M and let (V, z1τ , ..., z
nτ ) be a
Jτ coordinate chart defined by (U, z1, ..., zn). Then on V we have that
eτLXhXzj = Xzjτ, (2.21a)
eτLXhXzj = Xzjτ. (2.21b)
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,
g0 real analytic. Let h ∈ Can(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) (Kahler potential for (M,ω0, Jτ , gτ )) For all p ∈M there exists:
• (Uα, z10 , ..., z
n0 ) 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
θ :=i
2(∂0 − ∂0)κ0 (2.22a)
αt :=
∫ t
0
esXh(θ(Xh))ds (2.22b)
ατ := unique complex analytic continuation of αt (2.22c)
ψτ := − i2eτXh · κ0 + τh− ατ (2.22d)
κτ := −2 Imψτ (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 ′ be as in theorem 2.2.1. Along the proof, we will restrict T ′ as we need to.
(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 :
ω0
(∂
∂zjτ,∂
∂zkτ
)= eτLXh
(ω0
(∂
∂zj0,∂
∂zk0
))= 0. (2.23)
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
(ii): Let p ∈ M . There exists (Uα, z10 , ..., z
n0 ) 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
• (Uα, z10 , ..., z
n0 ) 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.
Let Vαj ,pjj=1,...,N be a finite open cover of M . Take T = minTα1,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.
θ = − i2
(∂0 − ∂0)κ0 = θ
θ =i
2∂0κ0︸ ︷︷ ︸
∈Ω(1,0)(M)
− i
2∂0κ0︸ ︷︷ ︸
∈Ω(0,1)(M)
=⇒ θ(1,0) =i
2∂0κ0 and θ(0,1) = − i
2∂0κ0
dθ = − i2
(∂0 + ∂0)(∂0 − ∂0)κ0 = − i2
(∂0∂0 − ∂0∂0)κ0 = −i∂0∂0κ0 = −ω0
Part 2: we prove that ατ =∑∞k=1
τk
k! Xk−1h (θ(Xh)).
αt =
∫ t
0
euXh(θ(Xh))du =
∫ t
0
∞∑k=0
uk
k!Xkh(θ(Xh))du
=
∞∑k=0
1
k!
(∫ t
0
ukdu
)Xkh(θ(Xh)) =
∞∑k=0
1
k!
tk+1
k + 1Xkh(θ(Xh))
=
∞∑k=1
tk
k!Xk−1h (θ(Xh))
Therefore, the complex analytic continuation of αt is
21
ατ =
∞∑k=1
τk
k!Xk−1h (θ(Xh)).
Part 3: we prove that ∀k ≥ 2 : LkXhθ = (dιXh)kθ. We proceed by induction on k.
(k = 2) :
L2Xhθ = (dιXh + ιXhd)2θ
= dιXhdιXhθ + dιXhιXh dθ︸︷︷︸=−ω0
+ιXh dd︸︷︷︸=0
ιXhθ + ιXhdιXh dθ︸︷︷︸=−ω0
= (dιXh)2θ − d(ω(Xh, Xh))− ιXhddh
= (dιXh)2θ
(k =⇒ k + 1) :
Lk+1Xh
θ = (dιXh + ιXhd)LkXhθ = (dιXh + ιXhd)(dιXh)kθ
= (dιXh)k+1θ + ιXhd(dιXh)kθ = (dιXh)k+1θ
Part 4: we prove that ∀k ≥ 2 : d(Xk−1h (θ(Xh))
)= LkXh(θ). We proceed by induction on k.
(k = 2) :
L2Xhθ = (dιXh)2θ = dιXhdιXhθ = dιXhd(θ(Xh)) = d(Xh(θ(Xh)))
(k =⇒ k + 1) :
Lk+1Xh
θ = (dιXh + ιXhd)d(Xk−1h (θ(Xh))
)= dιXhd
(Xk−1h (θ(Xh))
)= d
(Xkh(θ(Xh))
)Part 5: we prove that dατ = eτdιXh θ − θ.
dατ = d
( ∞∑k=1
τk
k!Xk−1h (θ(Xh))
)= τd(θ(Xh)) + d
( ∞∑k=2
τk
k!Xk−1h (θ(Xh))
)
= τd(θ(Xh)) +
∞∑k=2
τk
k!LkXh(θ) =
∞∑k=1
τk
k!(dιXh)k(θ) = eτdιXh θ − θ
Part 6: we prove that eτLXh θ = θ − τdh+ dατ .
eτLXh θ =
∞∑k=0
τk
k!LkXh(θ) = θ + τLXh(θ) +
∞∑k=2
τk
k!LkXh(θ)
= θ + τ(dιXh + ιXhd)θ +
∞∑k=2
τk
k!(dιXh)k(θ)
= θ − τdh+ τdιXhθ +
∞∑k=2
τk
k!(dιXh)k(θ)
= θ − τdh+
∞∑k=1
τk
k!(dιXh)k(θ) = θ − τdh+ dατ
22
Part 7 : we prove that θ(0,1)τ = ∂τψτ . This is equivalent to θ(Xzjτ) = dψτ (Xzjτ
) ∀j = 1, ..., n.
dψτ (Xzjτ) = − i
2d(eτXh · κ0)Xzjτ
+ τdh(Xzjτ)− dατ (Xzjτ
)
= − i2d(eτXh · κ0)Xzjτ
+ θ(Xzjτ)− (eτLXh θ)(Xzjτ
),
which is equal to θ(Xzjτ), since
(eτLXh θ)(Xzjτ) = (eτLXh θ)(eτLXhXzj ) = eτXh(θ(Xzj )) = − i
2eτXh(∂0κ0(Xzj ))
= − i2eτXh(dκ0(Xzj )) = − i
2(eτLXh (dκ0))(eτLXhXzj ) = − i
2d(eτLXhκ0)Xzjτ
.
Part 8: we prove the result.
i∂τ ∂τκτ = −i∂τ ∂τ (2 Imψτ ) = ∂τ ∂τ ψτ − ∂τ ∂τψτ
= −∂τ ∂τψτ − ∂τ ∂τψτ = −∂τθ(0,1)τ − ∂τθ(0,1)τ
= −∂τθ(1,0)τ − ∂τθ(0,1)τ = −∂τθ(1,0)τ − ∂τθ(0,1)τ − ∂τθ(1,0)τ − ∂τθ(0,1)τ
= −(∂τ + ∂τ )(θ(1,0)τ + θ(0,1)τ ) = −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 ϕ∗sκτ = κτ+s.
Proof. We start by proving that ϕ∗sαt = αt+s − αs.
ϕ∗sαt = esXh · αt = esXh∫ t
0
euXh(θ(Xh))du
=
∫ t
0
e(s+u)Xh(θ(Xh))du =
∫ t+s
s
euXh(θ(Xh))du
=
∫ t+s
0
euXh(θ(Xh))du−∫ s
0
euXh(θ(Xh))du = αt+s − αs
Therefore, by complex analytic continuation, esXh · 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.
κτ = i(ψτ − ψτ ) =1
2(eτXh − eτXh) · κ0 + i(τ − τ)h− i(ατ − ατ )
Therefore
esXh · κτ =1
2
(e(s+τ)Xh − e(s+τ)Xh
)· κ0 + i
((τ + s)− (τ + s)
)esXh · h︸ ︷︷ ︸
=h
−i( esXhατ︸ ︷︷ ︸=ατ+s−αs
− esXhατ︸ ︷︷ ︸=ατ+s−αs
)
=1
2
(e(s+τ)Xh − e(s+τ)Xh
)· κ0 + i
((τ + s)− (τ + s)
)h− i(ατ+s − ατ+s) = κτ+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)
∼=φ ∈ C∞(M) | g := (ω0 + i∂0∂0φ)(·, J0·) 0,
∫M
φωn0 = 0
. (2.26b)
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)
(where we have chosen a family of representatives for the equivalence classes such that∫Mφtω
n0 =
0, so that the map t 7→ φt is smooth). The derivative with respect to t is another function:
δφ0 :=d
dt
∣∣∣t=0
φt ∈ C∞(M). (2.27)
• H(ω0, J0) can be equipped with a Riemannian metric called the Mabuchi metric, which is given
by
〈δ1φ, δ2φ〉 =
∫M
1
n!(δ1φ · δ2φ) ωφ ∧ ... ∧ ωφ, (2.28)
where ωφ = ω0 + i∂0∂0φ. As proven in [5], the Riemannian manifold H(ω0, J0) admits a unique
Levi-Civita connection. A curve φtt∈I is a geodesic for this connection if and only if
φt =1
2‖∇gt φt‖2gt , (2.29)
where ∇gt is the gradient, and ‖ · ‖gt is the norm, both with respect to the metric gt = (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
H(ω0, J0) −→ K(ω0, J0)/R is well defined. Specifically, we have to show that if φ′ is another function in
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 .
Let h ∈ Can(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)
Define ωτ = (ϕ−1τ )∗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
Let (M,ω0, J0) be Kahler. Let h ∈ Can(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
of isomorphic Kahler structures ((ϕ−1it )∗ω0, (ϕ
−1it )∗Jit)t∈(−T,T ) = (ωit, J0)t∈(−T,T ). In short:
(M,ω0, J0, g0)eitXh (M,ω0, Jit, git)
ϕit∼= (M,ωit, J0, git). (2.32)
Theorem 2.7.1. ωit ∈ H(ω0, J0), for all t ∈ (−T, T ).
Proof. Define Φt := ϕit. Then Φ∗tωit = ω0. Since Φt is homotopic to the identity, we have that:
[ω0] = [Φ∗tωit] = Φ∗t [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 = Φ∗tωit. Using these formulas,
i∂it∂itκit = ω0 = Φ∗tωit = iΦ∗t∂0∂0(κ0 + φt) = i∂it∂it
((κ0 + φt) Φt
).
This suggests that we define
φt = κit Φ−1t − κ0. (2.33)
However, this definition only works if when we choose a different Kahler potential κ′0 (that defines κ′it),
the functions κit Φ−1t − κ0 and κ′it Φ−1
t − κ′0 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 .
Let U, V be neighborhoods of ϕ−1τ (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
φt|ϕτ (V ) = κτ ϕ−1τ − 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
κτ in terms of κ0, for both κτ and κ′τ , and then, check that κ′τ − κτ = (κ′0 − κ0) ϕτ . Performing the
computations:
κ′τ − κτ = Im
(ieτXh · (κ′0 − κ0) + i
∞∑k=1
τk
k!Xk−1h (∂0(κ′0 − κ0)(Xh)− ∂0(κ′0 − κ0)(Xh))
)
= Im
(i(κ′0 − κ0) + 2i
∞∑k=1
τk
k!Xk−1h (∂0(κ′0 − κ0)(Xh))
)
= κ′0 − κ0 +
∞∑k=1
(τk
k!Xk−1h (∂0(κ′0 − κ0)(Xh)) +
τk
k!Xk−1h (∂0(κ′0 − κ0)(Xh))
)
κ′0 − κ0 is a real analytic function defined on an open subset, such that ∂0∂0(κ′0 − κ0) = 0. By lemma
1.3.23, κ′0 − κ0 is locally of the form f + f , where f is holomorphic and f is anti-holomorphic. On each
smaller open set where κ′0 − κ0 = f + f :
κ′τ − κτ = f +
∞∑k=1
(τk
k!Xk−1h (df(Xh))
)+ f +
∞∑k=1
(τk
k!Xk−1h (df(Xh))
)= eτXh · f + eτXh · f = (κ′0 − κ0) ϕτ .
27
Therefore κ′τ − κτ = (κ′0 − κ0) ϕτ on the whole domain where this expression is defined.
(ii): The proof is the following computation:
ω0 + i∂0∂0φt = i∂0∂0κ0 + i∂0∂0(κit Φ−1t )− i∂0∂0κ0
= i∂0∂0((Φ−1t )∗κit) = (Φ−1
t )∗i∂it∂it(κit)
= (Φ−1t )∗ω0 = ωit
The following diagram illustrates the construction and main formulas so far:
(M,ω0, J0, g0)eitXh (M,ω0, Jit, git)
Φt∼= (M,ωit, J0, git)
ω0 = i∂0∂0κ0 ω0 = i∂it∂itκit ωit = ω0 + i∂0∂0φt ∈ H(ω0, J0) (2.35)
Φ∗tωit = ω0
Also, recall the formulas for κit (given in theorem 2.4.1):
θ :=i
2(∂0 − ∂0)κ0 (2.36a)
αit :=
∞∑k=1
(it)k
k!Xk−1h (θ(Xh)) (2.36b)
ψit := − i2eitXh · κ0 + ith− αit (2.36c)
κit := −2 Imψit (2.36d)
Theorem 2.7.4 (Mourao and Nunes, [14]). φt satisfies:
φt =1
2‖∇git φt‖2git . (2.37)
Therefore, the path of Kahler metrics (ω0, Jit)t∈(−T,T ) is a geodesic.
Proof. We divide the proof into parts.
Part 1: We show that ddtαit = iXh(αit) + iθ(Xh).
d
dtαit =
d
dt
( ∞∑k=1
(it)k
k!Xk−1h (θ(Xh))
)= i
∞∑k=1
(it)k−1
(k − 1)!Xk−1h (θ(Xh))
= iXh
( ∞∑k=1
(it)k
k!Xk−1h (θ(Xh))
)+ iθ(Xh) = iXh(αit) + iθ(Xh)
Part 2: We show that ddtκit = −Xh(ψit + ψit)− 2h+ 2θ(Xh). Using (2.36c) and (2.36d), we conclude
that
κit =1
2eitXh · κ0 −
1
2e−itXh · κ0 − 2th− iαit + iα−it.
28
Therefore
d
dtκit =
1
2Xh
(ieitXh · κ0
)+
1
2Xh(ie−itXh · κ0)− 2h− i d
dtαit + i
d
dtα−it
= −Xh
(− i
2eitXh · κ0 −
i
2e−itXh · κ0 − αit − α−it
)− 2h+ 2θ(Xh)
= −Xh(ψit + ψit)− 2h+ 2θ(Xh).
Part 3: Let f ∈ C∞(M). We show that:
d
dt(f ϕ−1
it ) =(∂itf(−iXh) + ∂itf(iXh)
) ϕ−1
it . (2.38)
We want to compute
d
dt(f ϕ−1
it ) = df
(d
dtϕ−1it
)= ∂itf
(d
dtϕ−1it
)+ ∂itf
(d
dtϕ−1it
). (2.39)
Recall that ϕit is the unique biholomorphism satisfying zjit ϕ−1it = zj = e−itXh · zjit.
dzjit
(d
dtϕ−1it
)=
d
dt
(zjit ϕ
−1it
)=
d
dt
(e−itXh · zjit
)= −iXh
(e−itXh · zjit
)= −iXh
(zjit ϕ
−1it
)= −iXh
((ϕ−1it )∗zjit
)= d
((ϕ−1it )∗zjit
)(−iXh)
= dzjit
(d(ϕ−1
it )(−iXh))
Performing complex conjugation, we conclude that:
dzjit
(d
dtϕ−1it
)= dzjit
(d(ϕ−1
it )(−iXh)), (2.40)
dzjit
(d
dtϕ−1it
)= dzjit
(d(ϕ−1
it )(iXh)). (2.41)
∂itf is a linear combination of the dzjit, and ∂itf is a linear combination of the dzjit. Because of this,
equations (2.39), (2.40) and (2.41), imply that:
d
dt(f ϕ−1
it ) = ∂itf(d(ϕ−1
it )(−iXh))
+ ∂itf(d(ϕ−1
it )(iXh))
= (ϕ−1it )∗
(∂itf(−iXh) + ∂itf(iXh)
)=(∂itf(−iXh) + ∂itf(iXh)
) ϕ−1
it .
Part 4: We show that φt =(ddtκit + ∂itκit(−iXh) + ∂itκit(iXh)
) Φ−1
t .
rClφt =d
dtφt =
d
dt
(κit Φ−1
t
)=
d
ds
∣∣∣s=t
(κis Φ−1
t
)+
d
ds
∣∣∣s=t
(κit Φ−1
s
)Now the result follows form using equation (2.38).
29
Part 5: We show that φt = −2h Φ−1t .
φt =
(d
dtκit + ∂itκit(−iXh) + ∂itκit(iXh)
) Φ−1
t
=(−Xh(ψit + ψit)− 2h+ 2θ(Xh) + ∂itκit(−iXh) + ∂itκit(iXh)
) Φ−1
t
making the following substitutions:
Xh(ψit) = dψit(Xh) = ∂itψit(Xh) + ∂itψit(Xh) (and analogously for ψit)
θ = θ(1,0)it + θ(0,1)it = ∂itψit + ∂itψit
κit = −2 Imψit
we obtain that φt = −2h Φ−1t .
Part 6: We show that
φt Φt = 2‖∇gith‖2git (2.42)
Using the fact that φt = −2h Φ−1t ,
φt =d
dtφt = −2
d
dt(h Φ−1
t ) = −2(∂ith(−iXh) + ∂ith(iXh)
) Φ−1
t ,
where again we used (2.38). Therefore
φt Φt = 2i∂ith(Xh)− 2i∂ith(Xh)
= 2dh(JitXh) = 2ω(Xh, JitXh)
= 2git(Xh, Xh) = 2‖∇gith‖2git .
Part 7 : We show that ‖∇gith‖2git = 14‖∇git φt‖
2git Φt. From this and (2.42), the result follows. We
note the following equivalences:
‖∇gith‖2git =1
4‖∇git φt‖2git Φt
⇐⇒ git (∇gith,∇gith)p =1
4git
(∇git φt,∇git φt
)Φt(p)
⇐⇒ Φ∗t git (∇gith,∇gith)p =1
4git
(∇git φt,∇git φt
)Φt(p)
⇐⇒ git (dΦt∇gith, dΦt∇gith)Φt(p)=
1
4git
(∇git φt,∇git φt
)Φt(p)
Therefore it suffices to prove that ∇git φt = 2dΦt∇gith:
git
(∇git φt, ·
)= dφt = −2dh dΦ−1
t = −2git(∇gith, ·) dΦ−1t
= −2git(∇gith, dΦ−1t ·) = −2Φ∗t git(∇gith, dΦ−1
t ·)
= git(−2dΦt∇gith, ·)
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 〈·, ·〉. 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→ 〈Y, ·〉.
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:
G GC
H
ι
ρ∃!ρC
It is a fact that for any Lie group G the universal complexification exists and is unique up to isomor-
phism.
• 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.
Then, the matrix Lie group GC =geiX : 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.
• The complex structure of GC at the point geiY is a map JGC : TgeiY (GC) −→ TgeiY (GC). By left
translation, TgeiY (GC) ∼= gC, which in turn is isomorphic to g × g. Therefore, JGC can be regarded
as a map JGC : g× g −→ g× g, which is given as a block matrix as (see [12], page 1468):
JGC =
0 −I
I 0
. (3.5)
• 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:
T ∗Gφ−→ G× g
ψi−→ GC. (3.7)
Define a complex structure on T ∗G by pulling back the complex structure of GC:
JST = (ψi φ)∗JGC . (3.8)
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)
α 7−→ 1
2g(α#, α#).
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:
(i) There exists an unique complex structure on T ∗,RM , JMCS , (called the magnetic complex struc-
ture), such that
T1,0(T ∗,RM) = e−iLXh (ker dπ ⊗ C), (3.12)
where Xh is the Hamiltonian vector field of h with respect to ωβ .
(ii) (T ∗,RM,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 〈·, ·〉. 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∗ = T ∗eG ⊂
TY [(T∗G). Restrict (dh)Y [ to g∗, so that it becomes a linear functional g∗ −→ R:
(dh)Y [ : g∗ = T ∗eG ⊂ TY [(T ∗G) −→ R.
Then (dh)Y [ is an element of g∗∗, which is naturally isomorphic to g via the double dual isomor-
phism.
• H, the jacobian of h:
H : g −→ End(g) (3.15)
Y 7−→
H(Y ) : g −→ g
W 7−→ (du)Y (W )
.
• ϕh, the map associated to h:
ϕh : T ∗G −→ T ∗G (3.16)
α 7−→
ϕh(α) : Tπ(α)G −→ R
Y 7−→ (dh)α(Y [)
,
where Y [ ∈ T ∗π(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 〈·, ·〉. 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: 〈W,H(Y )W 〉 > 0 ∀Y,W ∈ g;
• The norm of H has a greater than 0 lower bound: infY ∈g
(supZ∈g
‖H(Y )Z‖‖Z‖
)> 0.
Then,
(i) ϕh is a diffeomorphism.
(ii) Define Jτ = (ψτ φ ϕh)∗JGC . If Im τ > 0 then (T ∗G,ω, Jτ ) is Kahler.
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:
• T1, ..., Tn be a 〈·, ·〉-orthonormal basis of g, with associated coordinates y1, ..., yn;
• X1, ..., Xn be the left invariant vector fields on G such that (Xj)e = Tj ;
• ω1, ..., ωn be the basis of differential forms in G that is dual to X1, ..., Xn, that is ωj(Xk) = δjk.
Then,
T(g,Y )(G× g) = spanX1, ..., Xn ⊕ span
∂
∂y1, ...,
∂
∂yn
, (3.18)
T ∗(g,Y )(G× g) = spanω1, ..., ωn ⊕ spandy1, ..., dyn
. (3.19)
35
Proposition 3.3.1. In the coordinates that we have just described,
θ =
n∑j=1
yjωj , (3.20)
ω = = −dθ =
n∑j=1
ωj ∧ dyj +1
2
n∑k,l=1
Cjklyjωk ∧ ωl
. (3.21)
Proof. Recall that from example 1.1.6, θ is given by:
θ(x,Y )
(n∑k=1
akXk + bk∂
∂yk
)= (dLxY )[
(dπ
(n∑k=1
akXk + bk∂
∂yk
)). (3.22)
As a consequence of the metric being left invariant,
(dLxY )[ = (dLx−1)∗Y [. (3.23)
Therefore,
θ(x,Y )
(n∑k=1
akXk + bk∂
∂yk
)= (dLxY )[
(dπ
(n∑k=1
akXk + bk∂
∂yk
))
= (dLx−1)∗Y [
(n∑k=1
akXk
)= Y [
(n∑k=1
akTk
)
=
⟨Y,
n∑k=1
akTk
⟩=
n∑j,k=1
yjak〈Tj , Tk〉
=
n∑j=1
yjaj =
n∑j=1
yjωj
(n∑k=1
akXk + bk∂
∂yk
),
ω = −dθ = −dn∑j=1
yjωj = −n∑j=1
(dyj ∧ ωj + yjdωj
)=
n∑j=1
ωj ∧ dyj +1
2
n∑k,l=1
Cjklyjωk ∧ ωl
.
The metric 〈·, ·〉 being left invariant means that is is defined by left translation from the metric at the
identity.
Proposition 3.3.2. Given that 〈·, ·〉 is left invariant, it being right invariant is equivalent to the fact that
the metric at the identity is Ad-invariant:
〈AdgY,AdgZ〉 = 〈Y,Z〉 ∀g ∈ G,∀Y,Z ∈ g. (3.24)
The metric at the identity being Ad-invariant implies that it is ad-invariant:
〈adXY,Z〉+ 〈Y, adXZ〉 = 0 ∀X,Y, Z ∈ g. (3.25)
36
Proof. The first statement is a consequence of the fact that Adg = dLgdRg−1 = dRg−1dLg. Let X,Y, Z ∈
g. Equation (3.24) implies that
〈Adexp(tX)Y,Adexp(tX)Z〉 = 〈Y,Z〉.
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 Tjj=1,...,n are given by:
[adY
]jk
=
n∑l=1
ylCjlk. (3.26)
Also, adY is anti-symmetric:
[adY
]jk
= −[adY
]kj. (3.27)
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=1
[adY
]jkTj = adY (Tk) = [Y, Tk] =
n∑l=1
yl[Tl, Tk] =
n∑j=1
(n∑l=1
ylCjlk
)Tj .
This proves the first statement. The second one is a consequence of 〈·, ·〉 being ad-invariant:
[adY
]jk
= 〈adY Tj , Tk〉 = −〈Tj , adY Tk〉 = −[adY
]kj.
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
u(y1, ..., yn) =
(∂h
∂y1
∣∣∣Y, ...,
∂h
∂yn
∣∣∣Y
), (3.28)
which means that 〈u(Y ), ·〉 = (dh)Y , where h is seen as a function h : g −→ R.
• The matrix that represents H(Y ) in the basis T1, ..., Tn is:
H(Y ) =
∂2h
∂y1∂y1
∣∣∣Y· · · ∂2h
∂y1∂yn
∣∣∣Y
.... . .
...∂2h
∂yn∂y1
∣∣∣Y· · · ∂2h
∂yn∂yn
∣∣∣Y
. (3.29)
• When seen as a map G× g −→ G× g, ϕh is given by:
ϕh(x, Y ) = (x, u(Y )). (3.30)
37
• Because of the diffeomorphism T(x,Y )(G× g) ∼= TxG⊕TY gdLx−1×id∼= g⊕ g, the derivative of ϕh can
be seen as an endomorphism of g⊕ g. Under this identification, it is given by:
(dϕh)(x,Y ) =
I 0
0 H(Y )
. (3.31)
Proof. Proof of the coordinate expression for u: In the basis T1, ..., Tn,
u(Y ) =
n∑k=1
(u(Y ))kTk. (3.32)
We want to prove that (u(Y ))k = ∂h∂yk
∣∣∣Y
.
(u(Y ))k = dyk(u(Y )) (yj are coord. in the basis Tj)
= (dh)Y [(dyk) (def. of u + double dual iso.)
=d
dt
∣∣∣t=0
h(Y [ + tdyk)
=d
dt
∣∣∣t=0
h(Y + tTk) (h ∈ C∞(T ∗G) to h ∈ C∞(g))
=∂h
∂yk
∣∣∣Y.
Proof of the matrix expression for H:
(du)Y (W ) =
n∑j,k=1
∂uj
∂yk
∣∣∣Ydyk(W )Tj =
n∑j,k=1
∂2h
∂yj∂yk
∣∣∣YwkTj (3.33)
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)
We want to prove that φ ϕh φ−1(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 ).
⟨((dLx)∗ϕh(α))
#, Z⟩
=(
(dLx)∗ϕh(α))
(Z) = ϕh(α)(
(dLx)Z)
= (dh)α
(((dLx)Z)[
)= (dh)α
((dLx−1)∗Z[
)=
d
dt
∣∣∣t=0
h(α+ t(dLx−1)∗Z[
)=
d
dt
∣∣∣t=0
h(
(dLx−1)∗α︸ ︷︷ ︸=Y [
+tZ[)
= (dh)Y [(Z[) = Z[(u(Y )) = 〈u(Y ), Z〉,
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
(dϕh)(x,Y )
V
W
.
Under the isomorphism
TxG⊕ TY gdLx−1×id−→ g⊕ g, (3.36)
(V,W ) ∈ g⊕ g corresponds to ddt
∣∣∣t=0
(x exp(tV ), Y + tW ) ∈ T(x,Y )(G× g).
(dϕh)(x,Y )
(d
dt
∣∣∣t=0
(x exp(tV ), Y + tW )
)=
d
dt
∣∣∣t=0
ϕh(x exp(tV ), Y + tW )
=d
dt
∣∣∣t=0
(x exp(tV ), u(Y + tW ))
= (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,
(dϕh)(x,Y )
V
W
=
V
H(Y )W
(3.37)
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 ) = adYH(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
dth(Adexp(tV )Y ) = 0 ∀Y, V ∈ g, t ∈ R
On the other hand,
d
dth(Adexp(tV )Y ) =
d
ds
∣∣∣s=0
h(Adexp((t+s)V )Y ) =d
ds
∣∣∣s=0
h(Adexp(tV )Adexp(sV )Y )
= (dh)Adexp(tV )Y deAd︸ ︷︷ ︸=ad
( d
ds
∣∣∣s=0
exp(sV )︸ ︷︷ ︸=V
)(Adexp(tV )Y ) =
⟨u(Adexp(tV )Y ), adV (Adexp(tV )Y )
⟩
39
= −⟨u(Adexp(tV )Y ), adAdexp(tV )Y (V )
⟩=⟨
adAdexp(tV )Y (u(Adexp(tV )Y )), V⟩
= −⟨
[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
⇐⇒⟨
[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
⇐⇒⟨
[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 .
[Y, u(Y )] =
n∑j,k,l=1
Cljkyj ∂h
∂ykTl
Therefore,
[Y, u(Y )] = 0
⇐⇒n∑
j,k=1
Cljkyj ∂h
∂yk= 0 ∀l
⇐⇒ ∂
∂ym
n∑j,k=1
Cljkyj ∂h
∂yk
= 0 ∀l,m
⇐⇒n∑k=1
Clmk∂h
∂yk+
n∑j,k=1
Cljkyj ∂2h
∂ym∂yk= 0 ∀l,m
⇐⇒ −[adu(Y )
]lm
+
n∑k=1
[adY
]lkHmk = 0 ∀l,m
⇐⇒[− adu(Y ) + adYH(Y )
]lm
= 0 ∀l,m
⇐⇒ − adu(Y ) + adYH(Y ) = 0.
40
(iii)⇐⇒ (iv):
adu(Y ) = adYH ⇐⇒ adTu(Y ) = (adYH(Y ))T ⇐⇒ adTu(Y ) = H(Y )T adTY
⇐⇒ −adu(Y ) = H(Y )(−adTY )⇐⇒ 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:
• Hall Kirwin: β ∈ Ω2(M) defines a Kahler structure (T ∗,RM,ωβ , JMCS).
• Kirwin Mourao Nunes: τ ∈ C and h : T ∗G −→ R define a Kahler structure (T ∗G,ωST , Jτ ).
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,
and an open set U ⊂ T ∗G such that the Kahler manifolds (T ∗,RG,ωβ , JMCS) and (U, ωST , Jτ ) 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)
JMCS = Φ∗Jτ . (3.39)
The condition that JMCS = Φ∗Jτ can be written in terms of the (1,0) tangent spaces as:
(dΦ)TMCS1,0 (T ∗,RG) = T τ1,0(Φ(T ∗,RG)). (3.40)
From the statement of the theorem of Hall and Kirwin:
TMCS1,0 (T ∗,RG) = e
−iLXβE (ker dπ ⊗ C), (3.41)
41
where E(α) = 12g(α#, α#), and Xβ
E is the Hamiltonian vector field of E with respect to ωβ . From theorem
3.10 in [12]:
T τ1,0(Φ(T ∗,RG)) = eτL
XSTh (ker dπ ⊗ C). (3.42)
Its easy to see that
(dΦ)e−iL
XβE (ker dπ)⊗ C = e
−iLXSTEΦ dΦ(kerπ ⊗ C). (3.43)
Therefore, equation 3.40 can be written as:
eτL
XSTh (ker dπ ⊗ C) = e
−iLXSTEΦ 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)
β = d
n∑j=1
(ψj(Y )− yj)ωj , (3.46)
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 〈·, ·〉 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 Jτ .
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. LetG be a compact, real analytic Lie group equipped with a real analytic, left invariant
Riemannian metric g. Let
h : T ∗G −→ R (3.49)
α 7−→ 1
2g(α#, α#).
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 struc-
ture), such that
T1,0(T ∗G) = e−iLXh (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
x1, ..., xn, p1, ..., pn be the usual coordinates on T ∗M , as described in example 1.1.6. In these coordi-
nates,
ωST =
n∑i=1
dxi ∧ dpi, (3.52)
β =
n∑i,j=1
βijdxi ⊗ dxj , (3.53)
X =
n∑k=1
ak∂
∂xk+ bk
∂
∂pk. (3.54)
Therefore,
0 = ωβ(X, ·) = ωST (X, ·)− π∗β(X, ·)
=
n∑i=1
−bi − n∑j=1
βjiaj
dxi +
n∑i=1
aidpi,
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:
h(y1, ..., yn) =1
2
n∑j=1
(yj)2. (3.55)
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,
h(x, Y ) = h(gx(dLxY, ·)) =1
2gx(dLxY, dLxY ) =
1
2ge(Y, Y ).
So, h(x, Y ) does not depend on x, and is given in the coordinates y1, ..., yn by h(y1, ..., yn) = 12
∑nj=1(yj)2.
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.
2. Writing β as∑nk,l=1 βklω
k ∧ ωl, ωβ is
ωβ =
n∑j=1
ωj ∧ dyj +
n∑k,l=1
n∑j=1
1
2Cjkly
j − βkl
ωk ∧ ωl. (3.56)
From this, we can compute Xh.
3. e−iXh(ker dπ)⊗ C can be written as:
e−iLXh (ker dπ)⊗ C = spanC
e−iLXh
∂
∂y1, ..., eiLXh
∂
∂yn
. (3.57)
Compute eτLXh ∂∂yj , for τ ∈ C.
4. Define JMCS to be the map
JMCS : T (T ∗G)⊗ C −→ T (T ∗G)⊗ C (3.58)
e−iLXh∂
∂yj7−→ ie−iLXh
∂
∂yj
eiLXh∂
∂yj7−→ −ieiLXh ∂
∂yj
This definition works if T (T ∗G) ⊗ C = spanC
e−iLXh ∂
∂y1 , ..., e−iLXh ∂
∂yn , eiLXh ∂
∂y1 , ..., eiLXh ∂
∂yn
.
J2MCS = −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(·, J ·) is positive definite⇐⇒
ω(Zj , Zk) = 0 ∀j, k
[iω(Zj , Zk)] 0, (3.59)
where [iω(Zj , Zk)] 0 means that the matrix with entries iω(Zj , Zk) is positive definite. Use this
fact to prove that ωβ and JMCS are compatible.
Unfortunately, it is very hard to compute eτLXh ∂∂yj , even in the case where β ∈ Ω2(G) is exact and
left invariant. In this case, β can be written as
β =
n∑j=1
ajdωj = −n∑
j,k,l=1
1
2ajC
jklω
k ∧ ωl, (3.60)
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:
0 = H2(g) ∼= H2(G,R) ∼= H2dR(G). (3.61)
Since H2dR(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:
α =
n∑j=1
ajωj , (3.62)
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:
ωβ = d
n∑j=1
(yj − aj)ωj (3.64)
=
n∑j=1
ωj ∧ dyj +1
2
n∑k,l=1
Cjkl(yj − aj)ωk ∧ ωl
. (3.65)
45
Consider the map φα : G× g −→ G× g that consists in translating the yj by the aj :
φα : G× g −→ G× g (3.66)p, n∑j=1
yjTj
7−→p, n∑
j=1
(yj − aj(p))Tj
.
Proposition 3.5.4. φα is a diffeomorphism.
Proof. φα is injective:
φα
p, n∑j=1
yjTj
= φα
q, n∑j=1
zjTj
⇐⇒p, n∑
j=1
(yj − aj(p))Tj
=
q, n∑j=1
(zj − aj(q))Tj
⇐⇒
p = q
yj − aj(p) = zj − aj(q)⇐⇒
p = q
yj = zj.
φα is surjective: Let(p,∑nj=1 z
jTj
)∈ G× g. Define yj = zj + aj(p). Then
φα
p, n∑j=1
yjTj
=
p, n∑j=1
zjTj
.
φα is a local diffeomorphism: Let x1, ..., xn be local coordinates onG. In the coordinates x1, ..., xn, y1, ..., yn,
φα is given by:
φα(x1, ..., xn, y1, ..., yn) = (x1, ..., xn, y1 − a1(x), ..., yn − an(x)). (3.67)
Therefore, its Jacobian matrix in these coordinates is:
dφα =
I 0
−dα I
, (3.68)
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 = φ∗αωST (3.69)
Proof.
φ∗αωST = φ∗αdθ = dφ∗αθ = dφ∗α
n∑j=1
yjωj
= d
n∑j=1
(φ∗αyj)(φ∗αω
j) = d
n∑j=1
(yj − aj)ωj = ωβ ,
46
where we used the fact that φ∗αωj = ωj . In informal terms, this is because ωj 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
(T ∗G,ωST , (φ−1α )∗JMCS) is Kahler. However, this does not make the computations any easier, because
T1,0(T ∗G,ωST , (φ−1α )∗JMCS) = dφαT1,0(T ∗G,ωβ , JMCS) = e
−LXSThφα (ker dπ ⊗ C), (3.70)
and again e−L
XSThφα ∂
∂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.
3.6.1 ϕh is a diffeomorphism. The complex structure Jτ
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.
Therefore, Jτ = (ψτ φ ϕh)∗JGC is a complex structure. As stated in [12], dψτ is given by:
(dψτ )(x,Y ) =
e−τ1adY cos(τ2adY ) I−e−τ1adY cos(τ2adY )adY
−e−τ1adY sin(τ2adY ) e−τ1adY sin(τ2adY )adY
, (3.71)
where the blocks are with respect to the identifications
T(x,Y )(G× g) ∼= TxG⊕ TY gdLx−1×id∼= g⊕ g, (3.72)
TxeτY GC
dL(xeτY )−1
∼= gC ∼= g⊕ g. (3.73)
In the following commutative diagram, we present all relevant diffeomorphisms and their expressions
as block matrices:
T ∗Gϕh−→ T ∗G
φ−→ G× gψτ−→ GC (3.74)
47
Tα(T ∗G) Tϕh(α)(T∗G)
T(x,Y )(G× g) T(x,u(Y ))(G× g) Txeτu(Y )GC Txeτu(Y )GC
TxG⊕ TY g TxG⊕ Tu(Y )g gC gC
g⊕ g g⊕ g g⊕ g g⊕ g
(dϕh)α
(dφ)α (dφ)ϕh(α)
∼= ∼=
(dψτ )(x,u) JGC
dL−1
xeτu(Y )dL−1
xeτu(Y )
dLx−1×id dLx−1×id ∼= ∼=
M1 M2 M3
(3.75)
where M1, M2, M3 are the following matrices:
M1 =
I 0
0 H(Y )
, (3.76)
M2 =
e−τ1adu(Y ) cos(τ2adu(Y ))I−e−τ1adu(Y ) cos(τ2adu(Y ))
adu(Y )
−e−τ1adu(Y ) sin(τ2adu(Y ))e−τ1adu(Y ) sin(τ2adu(Y ))
adu(Y )
, (3.77)
M3 =
0 −I
I 0
. (3.78)
Then, as an isomorphism of g⊕ g, Jτ is given by:
Jτ = (M2M1)−1M3M2M1. (3.79)
Now we have T ∗G equipped with the symplectic form ω and the complex structure Jτ . We want to
see if these structures are compatible.
3.6.2 A basis for T1,0(T∗G)
We will now find a basis of T1,0(T ∗G) by computing n linearly independent (+i)-eigenvectors of Jτ . Since
expression (3.79) describes Jτ 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
H(Y )−1Tj
,
where M(Y ) is a linear endomorphism of g. So, we solve the following equation for M(Y ):
1 0
0 H(Y )
−1 M2
−10 −I
I 0
M2
1 0
0 H(Y )
M(Y )Tj
H(Y )−1Tj
= i
M(Y )Tj
H(Y )−1Tj
,
and obtain that
48
M(Y ) =1
adu(Y )
(1− eτadu(Y )
).
So, for all j = 1, ..., n
1adu(Y )
(1− eτadu(Y )
)Tj
H(Y )−1Tj
is a (+i)-eigenvector of Jτ . This can be written as a vector in T(g,Y )(G × g), in the basis of equation
3.18:
Zτj :=
n∑k=1
[(1− eτadu(Y )
)adu(Y )
]kj
Xk +
n∑k=1
[H(Y )−1
]kj
∂
∂yk. (3.80)
Also, since H(Y ) is invertible, Zτ1 , ..., Zτn is a linearly independent set. Considering the dimension on
T1,0(T ∗G) we conclude that it is a basis:
T1,0(T ∗G) = spanCZτ1 , ..., Zτn. (3.81)
3.6.3 Equations for T ∗G being Kahler in terms of h
Recall that ω and Jτ are compatible if and only if
ω(Zj , Zk) = 0 ∀j, k
[iω(Zj , Zk)] 0. (3.82)
Using equations (3.21) and (3.80), we can compute ω(Zj , Zk) and iω(Zj , Zk):
ω(Zj , Zk) =
[−I − e
−τadu(Y )
adu(Y )H(Y )−1 −H(Y )−1 I − eτadu(Y )
adu(Y )
+I − e−τadu(Y )
adu(Y )adY
I − eτadu(Y )
adu(Y )
]jk
, (3.83)
iω(Zj , Zk) = i
[−I − e
−τadu(Y )
adu(Y )H(Y )−1 −H(Y )−1 I − eτadu(Y )
adu(Y )
+I − e−τadu(Y )
adu(Y )adY
I − eτadu(Y )
adu(Y )
]jk
. (3.84)
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 〈·, ·〉.
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: 〈W,H(Y )W 〉 > 0 ∀Y,W ∈ g;
• The norm of H has a greater than 0 lower bound: infY ∈g
(supZ∈g
‖H(Y )Z‖‖Z‖
)> 0.
Then, ϕh is a diffeomorphism. Define Jτ = (ψτ φ ϕh)∗JGC . For Im τ > 0, (T ∗G,ω, Jτ ) is Kahler if and
only if h : g −→ R satisfies the following equations:
− I − e−τadu(Y )
adu(Y )H(Y )−1 −H(Y )−1 I − eτadu(Y )
adu(Y )+I − e−τadu(Y )
adu(Y )adY
I − eτadu(Y )
adu(Y )= 0, (3.85)
− i I − e−τadu(Y )
adu(Y )H(Y )−1 − iH(Y )−1 I − eτadu(Y )
adu(Y )+ i
I − e−τadu(Y )
adu(Y )adY
I − eτadu(Y )
adu(Y ) 0. (3.86)
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 ) =
adYH(Y ) = H(Y )adY , and therefore adY , adu(Y ) and H(Y )−1 commute with each other. Using these
properties we can show that h satisfies equations (3.85) and (3.86).
− I − e−τadu(Y )
adu(Y )H(Y )−1 −H(Y )−1 I − eτadu(Y )
adu(Y )+I − e−τadu(Y )
adu(Y )adY
I − eτadu(Y )
adu(Y )
= H(Y )−1
(−I + e−τadu(Y ) − I + eτadu(Y )
adu(Y )
)+ adY
(I − e−τadu(Y ) + I − eτadu(Y )
ad2u(Y )
)
= H(Y )−1
(−I + e−τadu(Y ) − I + eτadu(Y )
adu(Y )
)+H(Y )−1adu(Y )
(I − e−τadu(Y ) + I − eτadu(Y )
ad2u(Y )
)= 0
Performing an analogous computation,
− i I − e−τadu(Y )
adu(Y )H(Y )−1 − iH(Y )−1 I − eτadu(Y )
adu(Y )+ i
I − e−τadu(Y )
adu(Y )adY
I − eτadu(Y )
adu(Y )
= iH(Y )−1 e−2iτ2adu(Y ) − I
adu(Y ).
To check that iH(Y )−1 e−2iτ2adu(Y )−I
adu(Y )is positive definite, we consider its eigenvalues. Since H(Y )−1 and
iadu(Y ) are hermitian, they are diagonalizable, and since they commute they are simultaneously diago-
nalizable. Then, H(Y )−1 and adu(Y ) are also simultaneously diagonalizable. Let v1, ..., vn be a basis
of eigenvectors for both matrices, for which H(Y )−1 has eigenvalues λj , and adu(Y ) has eigenvalues
iµj :
H(Y )−1vj = 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
are real because adu(Y ) is skew-symmetric. v1, ..., vn are also eigenvectors of iH(Y )−1 e−2iτ2adu(Y )−I
adu(Y ):
iH(Y )−1 e−2iτ2adu(Y ) − I
adu(Y )vj = iH(Y )−1
∞∑k=1
(−2iτ2)k
k!adk−1u(Y )vj
= iλj
∞∑k=1
(−2iτ2)k
k!(iµj)
k−1vj
=λj(e2τ2µj − 1
)µj
vj .
So, we conclude that the eigenvalues of iH(Y )−1 e−2iτ2adu(Y )−I
adu(Y )are
λj(e2τ2µj−1)µj
, which are all positive
for τ2 > 0. Therefore, for τ2 > 0 we have that iH(Y )−1 e−2iτ2adu(Y )−I
adu(Y )is a positive definite matrix. Then,
from theorem 3.6.1 we conclude that (T ∗G,ω, Jτ ) 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)
Y 7−→ 1
2〈Y, Y 〉+ 〈Y,Z〉,
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
−τadu(Y )
adu(Y )adZ
I − eτadu(Y )
adu(Y )= 0
ie−2iτ2adu(Y )
adu(Y )− i I − e
−τadu(Y )
adu(Y )adZ
I − eτadu(Y )
adu(Y ) 0
.
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−τadu(Y )
adu(Y )adZ
I − eτadu(Y )
adu(Y )= 0
and
ie−2iτ2adu(Y )
adu(Y )− i I − e
−τadu(Y )
adu(Y )adZ
I − eτadu(Y )
adu(Y )= i
e−2iτ2adu(Y )
adu(Y ).
Following the same reasoning as in the previous example, i e−2iτ2adu(Y )
adu(Y )is positive definite.
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 〈Y,Z〉 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 re-
placed 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 me-
chanical system? We state this question in mathematical terms:
Given a symplectic manifold (M,ω), does there exist a Hilbert space (H, 〈·, ·〉) 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 mechanicsClassical Quantum
Space of states A symplectic manifold (M,ω) A Hilbert space (H, 〈·, ·〉)
Observables (physical quantities of interest) Functions on M ,i.e. elements of C∞(M)
Linear operators on H,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, 〈·, ·〉).
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(∇Xs, r) + h(s,∇Xs) ∀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 con-
sidering the space of P-polarized sections of L:
s ∈ Γ(L) : ∇Xs = 0 ∀X ∈ X(P)
,
and define an inner product 〈·, ·〉 on this space by integration. Then, H can be defined as
H =s ∈ Γ(L) : ∇Xs = 0 ∀X ∈ X(P), ‖s‖ <∞
, (4.5)
where the closure is with respect to 〈·, ·〉.
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 HτIm 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 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.
• T1, ..., Tn be an orthonormal basis of g, such that T1, ..., Tr is a basis of h, and yj , Xj , ωj be
defined as in section 3.3.
• h : g −→ R, the complexifier function, be defined as
h(y1, ..., yn) =1
2
n∑j,k=1
Hjkyjyk, (5.1)
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.
Remark 5.1.1. [Tj , Tk] = 0 ∀j, k = 1, ..., r because Tj and Tk are in h. Since Cljk is totally antisymmetric,
we conclude that Cljk = 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−→ ∇Xs = 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(∇Xs, r) + h(s,∇Xr) = X(h(s, r))⇐⇒ r∇Xs+ s∇Xr = rX(s) + sX(r) (5.6)
r = 1 =⇒ ∇Xs+ s∇X1 = X(s)
r = 1, s = 1 =⇒ ∇X1 +∇X1 = 0
From which we conclude that
∇Xs = X(s) + (∇X1)s. (5.7)
∇1 is a linear map from vector fields to complex numbers. Since ∇X1 +∇X1 = 0, we can say that
∇1 = iα, (5.8)
56
where α is a real valued form on T ∗G. Therefore,
∇Xs = X(s) + α(X)s. (5.9)
Curvature condition:
R(X,Y, s) = −iω(X,Y )s (5.10)
⇐⇒ ∇X(∇Y s)−∇Y (∇Xs)−∇[X,Y ]s = idθ(X,Y )s (5.11)
Using equation 5.9, we can show that
∇X(∇Y s)−∇Y (∇Xs)−∇[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∇Xs = 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:
∇Xs = X(s) + iθ(X)s. (5.13)
5.3 A polarization for T ∗G
Consider the vertical polarization of T ∗G:
P0 := ker dπ ⊗ C = spanC
∂
∂y1, ...,
∂
∂yn
⊂ T (T ∗G)⊗ C. (5.14)
Its immediate from definition 4.2.2 that P0 is a polarization. Our objective in this section is to compute
the following distribution:
Pτ = spanC
eτLXh
∂
∂y1, ..., eτLXh
∂
∂yn
, (5.15)
which is essentially eτLXh applied to P0:
X ∈ X(Pτ )⇐⇒ ∃V ∈ X(P0) : X = eτLXhV. (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 distri-
bution 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).
Xh =
r∑i,j=1
HijyjXi +
n∑i,k=r+1
r∑j,l=1
ykCkjiHjlyl ∂
∂yi(5.17)
Proof. Using equation 3.21 for ω, and that
∂h
∂yj=
n∑k=1
Hjkyk, (5.18)
j or k ≥ r + 1 =⇒ Hjk= 0,
two indices in 1, ..., r =⇒ Ckij = 0,
we can compute that
ω
r∑i,j=1
HijyjXi +
n∑i,k=r+1
r∑j,l=1
ykCkjiHjlyl ∂
∂yi, ·
=n∑j=1
∂h
∂yjdyj = dh. (5.19)
Proposition 5.3.2.
LXh∂
∂yi=
−∑rj=1HjiXj −
∑nj=r+1 [adYH]
ji
∂∂yj if i = 1, ..., r∑n
j=r+1
[adH(Y )
]ji
∂∂yj if i = r + 1, ..., n
(5.20)
Also, if i = r + 1, ..., n,
eτLXh∂
∂yi=
n∑j=r+1
[eτadH(Y )
]ji
∂
∂yj(5.21)
Proof.
LXh∂
∂yi=
r∑k,j=1
HkjyjXi +
n∑m,k=r+1
r∑j,l=1
ykCkjmHjlyl ∂
∂ym,∂
∂yi
= −
r∑k,j=1
∂
∂yi(Hkjy
j)Xk −
n∑m,k=r+1
r∑j,l=1
∂
∂yi(ykCkjmHjly
l) ∂
∂ym
= −r∑
k=1
HkiXk −n∑
m=r+1
r∑j,l=1
CijmHjlyl ∂
∂ym−
n∑m,k=r+1
r∑j,l=1
ykCkjmHji∂
∂ym
= −r∑
k=1
HkiXk +
n∑m=r+1
[adH(Y )
]mi
∂
∂ym−
n∑m=r+1
[adYH]mi
∂
∂ym
If i = 1, ..., r, then[adH(Y )
]mi
= 0. If i = r + 1, ..., n, then Hki = 0 and [adYH]mi = 0. This completes the
proof of equation (5.20). We now prove equation (5.21):
LXh∂
∂yi=
n∑j=r+1
[adH(Y )
]ji
∂
∂yjif i = r + 1, ...n
58
=⇒ LnXh∂
∂yi=
n∑j=r+1
[(adH(Y )
)n]ji
∂
∂yjif i = r + 1, ...n
=⇒eτLXh ∂
∂yi=
n∑j=r+1
[eτadH(Y )
]ji
∂
∂yjif i = r + 1, ...n.
Proposition 5.3.3.
eτLXh (Y ) = e−τadH(Y )(Y ) (5.22)
where eτLXh acts componentwise on Y .
Proof.
Xh(ym) =
n∑k=r+1
r∑j,l=1
ykCkjmHj lyl = −
[adH(Y )(Y )
]mSo, Xh(Y ) = −adH(Y )Y . Xh only has terms in ∂
∂yr+1 , ...,∂∂yn , and H(Y ) only depends on y1, ..., yr.
Therefore, Xh(H(Y )) = 0. We conclude that
Xnh (Y ) =
(−adH(Y )
)n(Y ),
which implies that
eτLXh (Y ) = e−τadH(Y )(Y ).
Proposition 5.3.4.
eτLXh (adY ) = ade−τadH(Y ) (Y )
(5.23)
Proof.
eτLXh (adY ) = adeτLXh Y
= ade−τadH(Y ) (Y )
Proposition 5.3.5. If i = 1, ..., r,
eτLXh
(LXh
∂
∂yi
)= −
r∑j=1
HjiXj −n∑
k=r+1
[adYH]ki
∂
∂yk(5.24)
Proof. Since [Xi, Xj ] = 0 for i, j = 1, ..., r, then LXhXj = 0 for j = 1, ..., r. This implies that
eτLXh
− r∑j=1
HjiXj
= −r∑j=1
HjiXj . (5.25)
Using this fact and the property of the exponential of a product (see proposition 2.1.5), we can compute
eτLXh(LXh
∂∂yi
):
eτLXh
(LXh
∂
∂yi
)= eτLXh
− r∑j=1
HjiXj −n∑
j=r+1
[adYH]ji
∂
∂yj
= −
r∑j=1
HjiXj −n∑
j=r+1
[eτLXh adYH
]jieτLXh
∂
∂yj
59
= −r∑j=1
HjiXj −n∑
k,j=r+1
[ade−τadH(Y ) (Y )
H]ji
[eτadH(Y )
]kj
∂
∂yk
= −r∑j=1
HjiXj −n∑
k=r+1
[eτadH(Y )ad
e−τadH(Y ) (Y )
H]ki
∂
∂yk
We claim that eτadH(Y )ade−τadH(Y ) (Y )
= adY eτadH(Y ) . Recall that, as a special case of remark 1.4.8,
exp ad = Ad exp.
eτadH(Y )ade−τadH(Y ) (Y )
(Z) = AdeτH(Y )adAde−τH(Y )
(Z)
= AdeτH(Y ) [Ade−τH(Y )(Y ), Z] = [Y,AdeτH(Y )Z] = adY eτadH(Y )(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).
It remains to be proven that adY eτadH(Y )H = adYH. This is true because adH(Y )H = 0:
[adH(Y )H
]kj
=
r∑k,l,m
ymHmlCjlkHki = 0.
Proposition 5.3.6. If i = 1, ..., r,
eτLXh∂
∂yi= −
r∑j=1
τHjiXj +∂
∂yi−
n∑k=r+1
τ [adYH]ki
∂
∂yk(5.26)
Proof. We start by proving that the equation is true for τ = t ∈ R.
d
dtetLXh
∂
∂yi= etLXhLXh
∂
∂yi= −
r∑j=1
HjiXj −n∑
k=r+1
[adYH]ki
∂
∂yk,
which implies that
etLXh∂
∂yi=
∫ t
0
− r∑j=1
HjiXj −n∑
k=r+1
[adYH]ki
∂
∂yk
ds+∂
∂yi
= −r∑j=1
tHjiXj +∂
∂yi−
n∑k=r+1
t [adYH]ki
∂
∂yk.
This proves the result for τ ∈ R.
Considering the complex analytic continuation of each side of equation
etLXh∂
∂yi= −
r∑j=1
tHjiXj +∂
∂yi−
n∑k=r+1
t [adYH]ki
∂
∂yk
we conclude that the result is true for τ ∈ C.
We have the following formulas for Pτ :
60
Pτ = spanC
eτLXh
∂
∂y1, ..., eτLXh
∂
∂yr, eτLXh
∂
∂yr+1, ..., eτLXh
∂
∂yn
, (5.27)
where
eτLXh∂
∂yi=
−∑rj=1 τHjiXj + ∂
∂yi −∑nj=r+1 τ [adYH]
ji
∂∂yj if i = 1, ..., r∑n
j=r+1
[eτadH(Y )
]ji
∂∂yj if i = r + 1, ..., n
. (5.28)
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
ddtYt = LXhYt
Yt|t=0 = Y0,(5.29)
which is given by
Yt = etLXhY0 = (φtXh)∗Y0, (5.30)
if etLXhY0 is well defined.
Proof. existence: The family of vector fields Yt = etLXhY0 satisfies equation (5.29).
uniqueness: Let Yt be a family of vector fields satisfying (5.29). We start by proving that ddt
(e−tLXhYt
)=
0:
d
dt
(e−tLXhYt
)= −LXhe−tLXhYt + e−tLXh
d
dtYt = e−tLXh
(−LXh +
d
dtYt
)= 0.
This implies that e−tLXhYt does not depend on t, in particular it is equal to its value in t = 0:
e−tLXhYt =(e−tLXhYt
) ∣∣∣t=0
= Y0.
Therefore,
Yt = etLXhY0.
Proposition 5.3.8. If i = 1, ..., r:
d
dt
− r∑j=1
tHjiXj +∂
∂yi−
n∑k=r+1
t [adYH]ki
∂
∂yk
= LXh
− r∑j=1
tHjiXj +∂
∂yi−
n∑k=r+1
t [adYH]ki
∂
∂yk
(5.31)
61
− r∑j=1
tHjiXj +∂
∂yi−
n∑k=r+1
t [adYH]ki
∂
∂yk
∣∣∣t=0
=∂
∂yi(5.32)
If i = r + 1, ..., n:
d
dt
n∑j=r+1
[etadH(Y )
]ji
∂
∂yj
= LXh
n∑j=r+1
[etadH(Y )
]ji
∂
∂yj
(5.33)
etLXh ∂
∂yi=
n∑j=r+1
[etadH(Y )
]ji
∂
∂yj
∣∣∣t=0
=∂
∂yi(5.34)
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):
d
dt
− r∑j=1
tHjiXj +∂
∂yi−
n∑k=r+1
t [adYH]ki
∂
∂yk
= −r∑j=1
HjiXj −n∑
k=r+1
[adYH]ki
∂
∂yk= LXh
∂
∂yi
LXh
− r∑j=1
tHjiXj +∂
∂yi−
n∑k=r+1
t [adYH]ki
∂
∂yk
= LXh
∂
∂yi− t
r∑j=1
Hji LXhXj︸ ︷︷ ︸=0
+
n∑k=r+1
t [LXhadYH]ki
∂
∂yk+
n∑k=r+1
t [adYH]ki LXh
∂
∂yk
= LXh
∂
∂yi− t
n∑k=r+1
t[ad−adH(Y )YH
]ki
∂
∂yk+
n∑l,k=r+1
t [adYH]ki
[adH(Y )adYH
]lk
∂
∂yk
= LXh
∂
∂yi− t
n∑k=r+1
[ad−adH(Y )YH + adH(Y )adYH
]ki
∂
∂yk
= LXh∂
∂yi− t
n∑k=r+1
adY adH(Y )H︸ ︷︷ ︸=0
−adH(Y )adYH + adH(Y )adYH
ki
∂
∂yk= LXh
∂
∂yi
Proof of equation (5.33):
LXh
n∑j=r+1
[etadH(Y )
]ji
∂
∂yj
=
n∑j=r+1
[etadH(Y )
]jiLXh
∂
∂yj
=
n∑j=r+1
[etadH(Y )adH(Y )
]ji
∂
∂yj=
d
dt
n∑j=r+1
[etadH(Y )
]ji
∂
∂yj
Notice that the vector fields eτLXh ∂
∂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-
tion, and that Pτ = eτLXhP0.
Theorem 5.3.9. Pτ is a polarization.
Proof. We start by proving that eτLXhω = ω:
62
LXhω = dιXhω + ιXh dω︸︷︷︸=0
= ddh︸︷︷︸=0
= 0 =⇒ eτLXhω = ω.
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:
ω
(eτLXh
∂
∂yj, eτLXh
∂
∂yk
)=(eτLXhω
)(eτLXh
∂
∂yj, eτLXh
∂
∂yk
)= eτLXh
(ω
(∂
∂yj,∂
∂yk)
)︸ ︷︷ ︸
=0
)= 0
Pτ is involutive: [eτLXh
∂
∂yj, eτLXh
∂
∂yj
]= eτLXh
([∂
∂yj,∂
∂yj
]︸ ︷︷ ︸
=0
)= 0
5.4 T ∗G is foliated by Kahler manifolds
For j = 1, ..., r, define:
Zτj = eτLXh∂
∂yj= −
r∑k=1
τHkjXk +∂
∂yj−
n∑k=r+1
τ [adYH]kj
∂
∂yk, (5.35)
Aτj =1
2(Zτj + Zτj ) = ReZτj , (5.36)
Bτj =i
2(Zτj − Zτj ) = ImZτj . (5.37)
Consider the following distribution on T ∗G:
Σ = spanR Aτ1 , ..., Aτr , Bτ1 , ..., Bτr . (5.38)
Proposition 5.4.1. Σ is involutive.
Proof. Using
eσLXh∂
∂yi= −
r∑j=1
σHjiXj +∂
∂yi−
n∑j=r+1
σ [adYH]ji
∂
∂yj∀σ ∈ C, i = 1, ..., r (5.39)
one can prove that [eσLXh
∂
∂yj, eλLXh
∂
∂yj
]= 0, ∀σ, λ ∈ C,∀j, k = 1, ..., r. (5.40)
Since the Aτj and the Bτj are linear combinations of eτLXh ∂∂yj and eτLXh ∂
∂yj , then equation 5.40
implies that Σ is involutive:
[Aτj , A
τk
]= 0, (5.41a)[
Aτj , Bτk
]= 0, (5.41b)
63
[Bτj , B
τk
]= 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 = Lαα∈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
J(Zτj ) = iZτj (5.43a)
J(Zτj ) = −iZτj (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
ι∗ω is nondegenerate: Zτ1 , ..., Zτr , Zτ1 , ..., Zτr is a basis of TLα ⊗ C. We can define the forms Ωjτ by
Ωjτ (Zτk ) = δjk, (5.44a)
Ωjτ (Zτk ) = 0. (5.44b)
As a consequence of this definition,
Ωjτ (Zτk ) = 0, (5.45a)
Ωjτ (Zτk ) = δjk. (5.45b)
Ω1τ , ...,Ω
rτ , Ω
1τ , ..., Ω
rτ is the dual frame to Zτ1 , ..., Zτr , Zτ1 , ..., Zτr , and it is a basis of T ∗Lα ⊗ C. There-
fore, we can write ι∗ω as:
ι∗ω =
r∑j,k=1
MjkΩjτ ∧ Ωkτ +NjkΩjτ ∧ Ωkτ + PjkΩjτ ∧ Ωkτ , (5.46)
64
where the Mjk and the Pjk are antisymmetric. The Mjk, Njk and Pjk are given by:
2Mjk = ω(Zτj , Zτk ), (5.47a)
Njk = ω(Zτj , Zτk ), (5.47b)
2Pjk = ω(Zτj , Zτk ), (5.47c)
and using equation (3.21) for ω and equation (5.35) for Zτj , we conclude that
ω(Zτj , Zτk ) = 0, (5.48a)
ω(Zτj , Zτk ) = 2iτ2Hjk, (5.48b)
ω(Zτj , Zτk ) = 0. (5.48c)
Therefore
ι∗ω =
r∑j,k=1
2iτ2HjkΩjτ ∧ Ωkτ . (5.49)
Since the components of ι∗ω form a nondegenerate matrix (recall that the restriction ofH to j, k = 1, ..., r
is positive definite), ι∗ω is nondegenerate.
(ii): J is almost complex:
J2(Zτj ) = iJ(Zτj ) = Zτj (5.50a)
J2(Zτj ) = −iJ(Zτj ) = −Zτj (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:
[Zτj , Z
τk
]= 0, (5.51a)[
Zτj , Zτk
]= 0, (5.51b)[
Zτj , Zτk
]= 0. (5.51c)
From these equations, and the definition of the Nijenhuis tensor it follows easily that
N (Zτj , Zτk ) = 0, (5.52)
N (Zτj , Zτk ) = 0, (5.53)
N (Zτj , Zτk ) = 0, (5.54)
N (Zτj , Zτk ) = 0. (5.55)
65
From which we conclude that N = 0. So, J is integrable.
(iii): We have to prove that
ι∗ω(Zτj , Zτk ) = 0, (5.56)
i(ι∗ω)(Zτj , Zτk ) 0. (5.57)
We have already seen that ι∗ω(Zτj , Zτk ) = 0. Also,
i(ι∗ω)(Zτj , Zτk ) = −i(ι∗ω)(Zτk , Z
τj ) = −i(2iτ2Hkj) = 2τ2Hjk, (5.58)
which is positive definite if and only if τ2 > 0.
Remark 5.4.3. If we write Pτ as
Pτ = spanC
Zτ1 , ..., Z
τr
︸ ︷︷ ︸
=ΣC
⊕ spanC
∂
∂yr+1, ...,
∂
∂yn
︸ ︷︷ ︸
=ΣR
, (5.59)
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)
in terms of the vector fields Zτj and Zτj .
ω = i∂∂κ⇐⇒ −dθ = d(i∂κ) = 0
⇐⇒ d(θ + i∂κ) = 0
⇐⇒ d(θ + i∂κ)(Zτj , Zτk ) = 0
⇐⇒ Zτj((θ + i∂κ)(Zτk )
)− Zτk
((θ + i∂κ)(Zτj )
)− (θ + i∂κ)([Zτj , Z
τk ]︸ ︷︷ ︸
=0
) = 0
⇐⇒ Zτj(θ(Zτk ) + iZτk (κ)
)= Zτk
(θ(Zτj )
)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
iZτj (Zτk (κ)) = Zτk(θ(Zτj )
)− Zτj
(θ(Zτk )) ∀j, k = 1, ..., r. (5.62)
Using the fact that θ =∑nj=1 y
jωj , and expression 5.35 for Zτj we can perform the following compu-
tations:
θ(Zτj ) = −τr∑
k=1
Hkjyk =⇒ Zτk
(θ(Zτj )
)= −τHjk,
θ(Zτj ) = −τr∑
k=1
Hkjyk =⇒ Zτj
(θ(Zτk )) = −τHjk.
Equation 5.62 takes the form:
iZτj (Zτk (κ)) = (τ − τ)Hjk, (5.63)
and κ = 2τ2h satisfies this equation:
iZτj (Zτk (κ)) = i∂2κ
∂yj∂yk= (τ − τ)Hjk. (5.64)
5.5 Computation of the Pτ -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 Pτ .
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
f = i∇Xf + f. (5.65)
Proposition 5.5.2.
h = iXh − h (5.66)
Proof.
θ(Xh) =
n∑j=i
yjωj
n∑k,l=1
HklylXk
=
n∑j,k=1
Hjkyjyk = 2h
h(s) = i∇Xhs+ hs = i(Xh(s) + i θ(Xh)︸ ︷︷ ︸
=2h
s)
+ hs = iXh(s)− hs
Lemma 5.5.3. Let s ∈ Γ(T ∗G× C), X ∈ X(T ∗G), σ ∈ C. Then
e−σLXh(∇eσLXhX
s)
= eiσh[∇X
(eiσhs
)]. (5.67)
67
Proof. Since
LXhθ = dιXhθ + ιXhdθ = d(2h)− ιXhω = 2dh− dh = dh
L2Xhθ = d ιXhdh︸ ︷︷ ︸
=0
+ιXh ddh︸︷︷︸=0
= 0
we conclude that e−σLXh θ = θ − σdh.
eiσh[∇X
(eiσhs
)]= eiσh
[X(e−σ(Xh+ih)s
)+ iθ(X)
(e−σ(Xh+ih)s
)]= eiσh
[X(e−σXhse−iσh
)+ iθ(X)
(e−σXhs
)e−iσh
]= eiσh
[X(e−σXhs
)e−iσh − iσ(e−σXhs)e−iσhX(h) + iθ(X)
(e−σXhs
)e−iσh
]= X
(e−σXhs
)+ i(θ − σdh)(X)
(e−σXhs
)= X
(e−σXhs
)+ i(e−σLXh θ
)(X)
(e−σXhs
)= e−σLXh
((eσLXhX
)(s) + iθ
(eσLXhX
)(s))
= e−σLXh(∇eσLXhX
s)
Proposition 5.5.4.
∇Xs = 0 ∀X ∈ X(Pτ )⇐⇒ ∂
∂yj
(eiτhs
)= 0 ∀j = 1, ..., n (5.68)
Proof.
∇Xs = 0 ∀X ∈ X(Pτ )
⇐⇒ ∇eτLXh ∂
∂yj
s = 0 ∀j = 1, ..., n
⇐⇒ e−τLXh
(∇eτLXh ∂
∂yj
s
)= 0 ∀j = 1, ..., n
⇐⇒ eiτh[∇ ∂
∂yj
(eiτhs
)]= 0 ∀j = 1, ..., n
⇐⇒ ∇ ∂
∂yj
(eiτhs
)= 0 ∀j = 1, ..., n
⇐⇒ ∂
∂yj
(eiτhs
)= 0 ∀j = 1, ..., n
where in the third equivalence we used lemma 5.5.3.
As a consequence of the previous proposition, we conclude that
∇Xs = 0 ∀X ∈ X(Pτ )⇐⇒ s = e−iτhf, (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
H0 = L2(G, dx), (5.70)
where dx is the Haar measure of G. Its inner product is
〈f, g〉0 = 〈f, g〉L2(G,dx) =
∫G
fgdx. (5.71)
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 [λα], choose a representative λα and a basis vα1 , ..., vαdα of Vα. Define the represen-
tative functions λjkα : G −→ C the following way: λjkα (g) is the (j, k) entry of the matrix that represents
λα(g) in the basis vα1 , ..., vαdα.
Then, it is a fact that the space spanned by the representative functions of a given class of represen-
tations does not depend on the chosen basis. Also, it is a consequence of the Peter-Weyl theorem that
λjkα α∈J,j,k∈1,...,dα is an orthogonal basis of L2(G, dx):
L2(G, dx) = spanCλjkα α∈J,j,k∈1,...,dα . (5.72)
We also have the Weyl orthogonality relations:
〈λjkα , λlmβ 〉L2(G,dx) = δjlδkmδαβ1
dimVα. (5.73)
Remark 5.6.2. Each representation λα induces a representation of the Lie algebra:
G GL(Vα)
g End(Vα)
λα
exp
deλα
exp (5.74)
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
= 0. (5.75)
deλα(X)X∈h is then a set of diagonalizable matrices that commute. Therefore, it is a set of simultane-
ously diagonalizable matrices. So, for each α, we can choose a basis vα1 , ..., vαdα of Vα such that for all
X ∈ h deλα(X) is diagonal:
deλα(X) =
iλα1 (X) · · · 0
.... . .
...
0 · · · iλαdα(X)
. (5.76)
iλαj : g −→ C are linear functionals, which give the eigenvalue of the matrix in the basis vα1 , ..., vαdα:
[deλα(X)]vαj = iλαj (X)vαj . (5.77)
Since the matrices deλα(X)X∈h are anti-hermitian, the λαj are real valued.
As we computed in the previous section, the set of Pτ -polarized sections is
s ∈ Γ(L) : ∇Xs = 0 ∀X ∈ X(Pτ )
= e−iτhf : f ∈ C∞(G;C) ⊂ C∞(T ∗G;C). (5.78)
It is possible to compute that
eτLXh (ω1 ∧ ... ∧ ωr) ∧ eτLXh (ω1 ∧ ... ∧ ωr) = (τ − τ)r detHω1 ∧ ... ∧ ωr ∧ dy1 ∧ ... ∧ dyr. (5.79)
This equation, in conjunction with the half form correction, suggests that we define the following inner
product on this set of Pτ -polarized sections:
〈e−iτhf, e−iτhg〉τ =
√( τπ
)rdetH
∫G×h
(e−iτhf
)(e−iτhg
)dxdy. (5.80)
Choose a set of λα as in remark 5.6.1 and for each α an associated basis vα1 , ..., vαdα as in remark
5.6.2.
Proposition 5.6.3. (eτXhλjkα
)(x, Y ) = λjkα
(xeτu(Y )
)(5.81)
Proof.
(Xl · λjkα
)(x) =
d
dt
∣∣∣t=0
λjkα (φtXl(x)) =d
dt
∣∣∣t=0
λjkα (x exp(tTl))
=d
dt
∣∣∣t=0
dα∑m=1
λjmα (x)λmkα (exp(tTl)) =
dα∑m=1
λjmα (x)d
dt
∣∣∣t=0
λmkα (exp(tTl))
=
dα∑m=1
λjmα (x)deλmkα (Tl)
70
(Xh · λjkα
)(x, Y ) =
r∑l=1
(u(Y ))lXlλjkα (x) =
r∑l=1
(u(Y ))ldα∑m=1
λjmα (x)deλmkα (Tl)
=
dα∑m=1
λjmα (x)deλmkα
(r∑l=1
(u(Y ))lTl
)=
dα∑m=1
λjmα (x)deλmkα (u(Y ))
Note that Xh has terms in ∂∂yr+1 , ...,
∂∂yn , and u depends only on y1, ..., yr. Therefore,
(X lh · λjkα
)(x, Y ) =
dα∑m=1
λjmα (x)[(deλ
mkα (u(Y ))
)l]mk(5.82)
(eτXh · λjkα
)(x, Y ) =
∞∑l=0
τ l
l!X lhλ
jkα (x, Y ) =
∞∑l=0
τ l
l!
dα∑m=1
λjmα (x)[(deλ
mkα (u(Y ))
)l]mk=
dα∑m=1
λjmα (x)
[ ∞∑l=0
τ l
l!(deλα(u(Y )))
l
]mk=
dα∑m=1
λjmα (x)λmkα
(eτu(Y )
)= λjkα
(xeτu(Y )
)Proposition 5.6.4.
e−iτhλjkα = λjkα eiτλαj (u) (5.83)
e−iτhλjkα = λjkα e−iτλαj (u) (5.84)
Proof.
(e−iτhλjkα
)(x, Y ) =
(eτ(Xh+ih)λjkα
)(x, Y ) = eiτh
(eτXhλjkα
)(x, Y )
= eiτhλjkα
(xeτu(Y )
)= eiτh
dα∑l=1
λjlα (x)λlkα
(eτu(Y )
)= eiτh
dα∑l=1
λjlα (x)(edeλα(τu(Y ))
)lk= eiτh
dα∑l=1
λjlα (x)δlkeiτλαk (u(Y ))
= eiτhλjkα (x)eiτλαj (u)
Which proves the first equation. The second one follows from performing complex conjugation.
Proposition 5.6.5.
〈e−iτhλjkα , e−iτhλlmβ 〉τ =δjlδkmδαβ
dimVαe2τ2h((λαk )[) (5.85)
Proof.
〈e−iτhλjkα , e−iτhλlmβ 〉τ
=
√( τπ
)rdetH
∫G×h
(e−iτhλjkα
)(e−iτhλlmβ
)dxdy
=
√( τπ
)rdetH
∫G
λjkα λlmβ dx
∫h
eiτhe−iτheiτλαj (u)e−iτλ
βm(u)dy
71
=δjlδkmδαβ
dimVα
√( τπ
)rdetH
∫h
e−2τ2h−2τ2λαk (u)dy
=δjlδkmδαβ
dimVα
√( τπ
)r 1
detH
∫h
e−τ2uTH−1u−2τ2((λαk )[)Tudu
The integral over h can be computed using the Gaussian integral formula:
∫Rre−
12xTAx+BT xdx = (2π)r/2(detA)−1/2e
12B
TA−1B , (5.86)
where A is a positive definite r × r matrix, and B is a vector in Rr. A = 2τ2H−1 and BT = −τ2((λαk )[)T
yields that
〈e−iτhλjkα , e−iτhλlmβ 〉τ =δjlδkmδαβ
dimVαe2τ2h((λαk )[)
Corollary 5.6.6.
limτ→0〈e−iτhf, e−iτhg〉τ = 〈f, g〉0 (5.87)
Proof.
limτ→0〈e−iτhλjkα , e−iτhλlmβ 〉τ = lim
τ→0
δjlδkmδαβdimVα
e2τ2h((λαk )[) =δjlδkmδαβ
dimVα= 〈λjkα , λlmβ 〉0 (5.88)
Hτ should be the Hilbert space that contains Pτ -polarized sections of L that are L2 with respect to
5.80. From proposition 5.6.5, we can conclude that
Hτ =s ∈ Γ(L) : ∇Xs = 0 ∀X ∈ X(Pτ ), ‖s‖τ <∞
(5.89)
=e−iτhf | f ∈ L2(G, dx)
(5.90)
= spanC
e−iτhλjkdα
α∈J,j,k=1,...,dα
. (5.91)
We want to define a unitary isomorphism H0 −→ Hτ . The obvious attempt is e−iτh : H0 −→ Hτ .
However, proposition 5.6.5 shows that e−iτh preserves orthogonality but not the norm. Because of this,
we introduce the linear map Qh, which is the unique linear map that acts on the basis λjkα α∈J,j,k∈1,...,dα
of H0 as:
Qh : H0 −→ H0 (5.92)
λjkα 7−→ h((λαk )[)λjkα
Theorem 5.6.7. The map
Uτ := e−iτh eiτQh : H0 −→ Hτ (5.93)
is a unitary isomorphism of Hilbert spaces.
Proof. Uτ is bijective, because it has inverse
72
U−1τ = e−iτQh e−iτh. (5.94)
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.
Uτ (λjkα ) = e−iτh(eiτQh(λjkα )
)= e−iτh
(eiτh((λαk )[)λjkα
)= eiτh((λαk )[)e−iτh(λjkα )
‖Uτ (λjkα )‖2τ = ‖eiτh((λαk )[)e−iτh(λjkα )‖2τ
= |eiτh((λαk )[)|2‖e−iτh(λjkα )‖2τ
= |eiτ1h((λαk )[)|2︸ ︷︷ ︸=1
|e−τ2h((λαk )[)|2‖e−iτh(λjkα )‖2τ
= e−2τ2h((λαk )[) 1
dimVαe2τ2h((λαk )[)
=1
dimVα
= ‖λjkα ‖20
73
74
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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
dtρ∗tω = ρ∗tLXtω. (A.1)
Proof. Part 1: The formula is true if ω = f ∈ Ω0(M) = C∞(M) :
(d
dtρ∗t f
)(p) =
d
dtf (ρt(p)) = (Xt · f) (ρt(p))
=⇒ d
dt(ρ∗t f) = ρ∗t (Xt · f) = ρ∗tLXtω
Part 2: Both sides of the equation commute with d:
ρ∗tLXtdω = ρ∗t (ιXtddω︸ ︷︷ ︸=0
+dιXtdω) = ρ∗t (ddιXtω︸ ︷︷ ︸=0
+dιXtdω) = ρ∗t dLXtω = dρ∗tLXtω
d
dtρ∗t dω =
d
dtd(ρ∗tω) = d
(d
dtρ∗tω
)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
dt(ρ∗t (ω ∧ η)) =
d
dt(ρ∗tω ∧ ρ∗t η)
=
(d
dt(ρ∗tω)
)∧ ρ∗t η + (ρ∗tω) ∧
(d
dt(ρ∗t η)
)= (ρ∗tLXtω) ∧ ρ∗t η + (ρ∗tω) ∧ (ρ∗tLXtη) (by hypothesis)
= ρ∗t ((LXtω) ∧ η + ω ∧ (LXtη))
= ρ∗t (LXt(ω ∧ η))
Part 4: If (U, x1, ..., xn) is a coordinate chart, and ω ∈ Ωk(U), then
ω =∑
1≤i1<...<ik≤n
ωi1...ikdxi1 ∧ ... ∧ dxik .
Therefore, the formula is true for all ω.
Proposition A.1.2. If ωt is a family of forms that is smooth in t then
d
dtρ∗tωt = ρ∗t
(LXtωt +
dωtdt
)(A.2)
Proof.
d
dtρ∗tωt =
d
ds
∣∣∣s=t
ρ∗sωs
=d
ds
∣∣∣s=t
ρ∗sωt + ρ∗td
ds
∣∣∣s=t
ωs
= ρ∗tLXtωt + ρ∗tdωtdt
= ρ∗t
(LXtωt +
dωtdt
).
A.2 Proof of Moser’s Theorem
Theorem A.2.1 (Moser). Let M be a compact manifold, ω0, ω1 ∈ Ω2(M) be symplectic forms, such that
(i) [ω0] = [ω1] ∈ H2(M,R);
(ii) ∀t ∈ [0, 1] ωt := (1− t)ω0 + tω1 is symplectic.
Then there exists an isotopy ρt : M −→M such that ρ∗tωt = ω0 ∀t ∈ [0, 1]
Proof. Let ρ : M × R −→M be an isotopy, with time dependent vector field Xt. We claim that
ρ∗tωt = ω0 ∀t ∈ [0, 1] ⇐⇒ dιXtωt +dωtdt
= 0 ∀t ∈ [0, 1].
To see that the claim is true, note that
78
ρ∗tωt = ω0 ∀t ∈ [0, 1]⇐⇒ ρ∗tωt = ρ∗0ω0 ∀t ∈ [0, 1]⇐⇒ d
dtρ∗tωt = 0 ∀t ∈ [0, 1].
Also
d
dtρ∗tωt = ρ∗t
(LXtωt +
dωtdt
)= ρ∗t
dιXtωt + ιXt dωt︸︷︷︸=0
+dωtdt
= ρ∗t
(dιXtωt +
dωtdt
).
Therefore
ρ∗tωt = ω0 ⇐⇒d
dtρ∗tωt = 0⇐⇒ dιXtωt +
dωtdt
= 0,
which proves the claim.
Since [ω0] = [ω1], dωtdt = ω1 − ω0 = dµ, for some µ ∈ Ω1(M). For all t ∈ R, Moser’s equation, which
is
ιXtωt + 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.
Since 0 = dιXtωt + dµ = dιXtωt + dωtdt , ρ is the pretended isotopy.
79
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