Marcaccio_Tesi

UNIVERSITÀ DEGLI STUDI DI TORINO
DIPARTIMENTO DI MATEMATICA GIUSEPPE PEANO
SCUOLA DI SCIENZE DELLA NATURA
Corso di Laurea Magistrale in Matematica
Master thesis
Calabi-Yau manifolds
Supervisor: Prof.ssa Anna Fino
Cosupervisor: Prof. Joel Fine Candidate: Giulia Marcaccio
Academic Year 2015-2016

Contents
Introduction iii
1 Complex manifolds 1
1.1 Complex coordinates . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Complex forms: the (p,q)-forms . . . . . . . . . . . . . . . . . 4
1.3.1 1-forms . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 Higher degree forms . . . . . . . . . . . . . . . . . . . 5
1.4 Canonical bundle . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.1 Line bundles over M . . . . . . . . . . . . . . . . . . . 10
1.5 Cohomologies . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.1 The de Rham cohomology . . . . . . . . . . . . . . . . 13
1.5.2 Dolbeault cohomology . . . . . . . . . . . . . . . . . . 15
2 Some Algebraic Geometry tools 17
2.1 Aﬃne and projective variety . . . . . . . . . . . . . . . . . . . 17
2.2 Regular and rational functions . . . . . . . . . . . . . . . . . 18
2.3 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Rational diﬀerential forms and canonical divisors. . . . . . . . 22
2.5 Dualizing sheaf . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 A look to some complex constructions 25
3.1 Almost complex structure . . . . . . . . . . . . . . . . . . . . 25
3.2 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 (Real) Symplectic manifolds . . . . . . . . . . . . . . . 28
3.2.2 (Complex) Symplectic manifolds . . . . . . . . . . . . 29
3.3 Compatible almost complex structures . . . . . . . . . . . . . 29
4 Calaby-Yau manifolds 32
4.1 Symplectic Calabi-Yau manifolds . . . . . . . . . . . . . . . . 32
4.2 Complex Calabi-Yau manifolds . . . . . . . . . . . . . . . . . 33
4.2.1 1-dimensional Calabi-Yau . . . . . . . . . . . . . . . . 33
4.2.2 2-dimensional Calabi-Yau . . . . . . . . . . . . . . . . 43
i

CONTENTS
4.2.3 3-dimensional and higher-dimensional Calabi-Yau man-
ifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Kähler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.1 Kähler and Calabi-Yau manifolds . . . . . . . . . . . . 63
4.4 Toric geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.1 Cones and fans . . . . . . . . . . . . . . . . . . . . . . 66
4.4.2 Toric divisors . . . . . . . . . . . . . . . . . . . . . . . 69
4.4.3 Toric Calabi–Yau threefolds . . . . . . . . . . . . . . . 70
5 Counterexamples 74
5.1 The Kodaira-Thurston example . . . . . . . . . . . . . . . . . 74
5.2 Hopf surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Symplectic but not complex . . . . . . . . . . . . . . . . . . . 79
6 Calabi–Yau Manifolds and String Theory 81
6.1 Compactiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.1.1 Dimensional reduction . . . . . . . . . . . . . . . . . 82
ii

Introduction
Superstring theory is an attempt to explain all of the particles and funda-
mental forces of nature in one theory by modelling them as vibrations of
tiny supersymmetric strings.
Our physical space is observed to have three large spatial dimensions and,
along with time, is a boundless four-dimensional continuum, known as space-
time.
The problem is that if we want to apply superstring theory to our spacetime
universe we need it to be ten-dimensional. The discrepancy between the
critical dimension d = 10, required by string theory, and the number of
observed dimensions d = 4 is resolved by the idea of compactification.
Looking for a ten-dimensional manifold consistent with the requirements
imposed by this theory, the simplest possibility is to have a space time that
takes the product form M4 × N6, where M4 is a four-dimensional Minkowski
space and N6 is some compact six-dimensional manifold. This compact man-
ifold is usually taken to have a sufficiently small size as to be unobservable
with present technology, and thus we would only see the four-dimensional
manifold M4. Requiring that supersymmetry is preserved at the compactifi-
cation scale restricts us to a special class of manifolds known as Calabi-Yau
manifolds.
Eugenio Calabi, an American mathematician of Italian origins, in 1953
proposed that certain specific geometric structures are allowed under some
topological condition.
In particular he surmised that whether a certain kind of complex manifold,
namely the compact (finite in extent) and "Kähler" ones, could satisfy the
general topological conditions of vanishing first Chern class and could also
satisfy the geometrical condition of having a Ricci-flat metric (excluding the
flat torus).
In 1976 the Chinese professor of mathematics at Harvard, Shing-Tung Yau,
proved the existence of the geometric structure as surmised by the Calabi’s
conjecture. He was able to prove, by expressing the conjecture in terms of non
linear partial differential equations, the existence of many multi-dimensional
shapes that are Ricci-flat, i.e., satisfying the Einstein equation in 3 complex
dimensions and empty space.
iii

Chapter 0. Introduction
Such structure is now called Calabi-Yau manifold.
Its special properties are indispensable for compactification in Superstring
Theory.
This thesis provides an introduction to Calabi-Yau manifolds. We will
not go into details of the physical problem of superstring theory but we will
point out the mathematical aspect.
The outline is as follows. In Chapter 1 and 2 we prepare the ground to
understand and built the matter that will be later developed: respectively
we give a short review of complex geometry and we analyse some algebraic
geometry tools. More specifically on the former chapter we describe the
analogue of the differential manifold on the complex world: we give the
holomorphic definition of tangent and cotangent bundle leading us to the
one of canonical bundle; while on the latter, following [Smi], we study the
concept of divisor and sheaf that will be very useful in order to analyse some
examples of Calabi-Yau.
Chapter 3 follows up the topic developed in the first chapter, moreover it
tries to connect the symplectic real world with the complex one, up to the
definition of symplectic complex manifold.
In Chapter 4 we come to the definition of Calaby-Yau manifold and we discuss
several example focusing on some algebraic geometry’s ones. Following
[Shu12] and [Raa10] we begin showing how a complex one dimensional torus
(first example of one-dimensional Calabi-Yau) can be seen as an elliptic curve.
In order to present two-dimensional examples of Calabi-Yau we refer to
[Mor88] and [Kut10]. Concerning the higher dimension we set the problem
on weighted projective space Pn(w) and we give an algeabric condition on
a variety to be Calabi-Yau. Furthermore we give the definition of Kähler
manifold in order to see how they are linked with the Calabi-Yau structures.
Pursuing the aim of understanding the links between the complex world and
the symplectic one, some famous counterexamples are showed in Chapter 5.
Finally the dissertation ends with a view on compactifications: the last
chapter is an adaptation of [FT] and provides a glance on the fascinating
dimension that we could not detect.
iv

Chapter 1
Complex manifolds
This chapter is concerned with the theory of complex manifolds. Apparently,
their definition is identical to the one of smooth manifolds. It is only necessary
to replace open subsets of Rn by open subsets of Cn, and smooth functions
by holomorphic functions. Nevertheless we will illustrate how the complex
and real worlds are fundamentally different.
We try to enter into this world because it provides the setting of our issue.
1.1 Complex coordinates
We will give the definition of complex manifold following the definition of
smooth manifold, with the appropriate changing.
Definition 1.1. M is a complex manifold of complex dimension n if:
• it is a topological space T2 and countable;
• it is endowed with a complex atlas i.e. a collection of coordinate charts
F = {(Ui, ϕi)| i ∈ I},
where I is a set of index such that
– Ui ⊂ M are open sets and {(Ui)}i∈I is a cover of M i.e.
M =
i∈I
Ui;
– ϕi : Ui → Cn is a homeomorphism onto an open set in Cn,
– the change of coordinates
ϕi ◦ ϕ−1
j : ϕj(Ui ∩ Uj) → ϕi(Ui ∩ Uj)
are biholomorphic, ∀ i, j s.t. Ui ∪ Uj = ∅;
1

Chapter 1. Complex manifolds
– the collection F is maximal towards the previous condition. This
means that if (U, ϕ) is a system of charts such that ϕi ◦ ϕ and
ϕ ◦ ϕ−1
i are biholomorphic ∀i ∈ I, then (U, ϕ) ∈ F.
In other words M is a differentiable manifold endowed with a holomorphic
atlas.
Observation 1.2. Charts define coordinates. Suppose ϕ: U → Cn is some chart
centered around a point p ∈ U (meaning p maps to the origin.) Then ϕ can
be written as local coordinates (z1, . . . , zn), where each zi is a holomorphic
function on U. These are complex coordinates on M. Note that if M is a
complex n−dimensional manifold, it can be realized as a real 2n−dimensional
manifold with coordinates (xi, yi) coming from zj = xj + iyj.
Example 1.3 (Complex Torus). Trivially Euclidean space Cn is a comlplex
manifold. We can now consider the action of Zn on Cn by translation:
(m1, n1, . . . , mn, nn) · (z1, . . . , zn) = (z1 + m1 + in1, . . . , zn + mn + inn),
where mi, ni ∈ Z, zi ∈ C, ∀i.
Since this action is properly discontinuous and holomorphic, the quotient
Cn/Zn inherits the structure of a complex manifold from the standard atlas
on Cn. Thus Tn := Cn/Zn is a manifold, called complex torus.
Example 1.4 (Complex projective space). CPn is the set of equivalence classes
{[z] = [z0 : · · · : zn], | zi ∈ C},
where y ∈ Cn and w ∈ Cn, belong to the same class iff there exist λ ∈ C{0}
such that y = λw.
Let
Ui = {[z0 : · · · : zn] | zi = 0} .
An open cover of CPn is given by
{Ui}i=0,...,n .
The maps ϕi : Ui → Cn given by
[z0 : · · · : zn] → (z0/zi, . . . , zi/zi, . . . , zn/zi)
are bijective, so it remains to show that the transition functions are holo-
morphic.
These are given by ϕij := ϕi ◦ ϕ−1
j : ϕj(Ui ∩ Uj) → ϕi(Ui ∩ Uj).
Let wk be the coordinates on Uj and assume i < j. Then ϕij is defined by
(w0, . . , ˆwj, . . , wn)
ϕ−1
j
−−→ [w0 : . . : 1 : . . : wn]
ϕi
−→ (
w0
wi
, . . ,
ˆwi
wi
, . . ,
1
wi
, . . ,
wn
wi
).
Since wi = 0 on ϕj(Ui ∩ Uj), we see that the coordinate functions ϕij are of
the form wk/wi or 1/wi, which are holomorphic on the domain of ϕij. So
CPn is a complex manifold.
2

1.2 Tangent space
Let M be a complex manifold of complex dimension n and x be a point of
M.
Let Cn be identified with R2n via the map (z1, . . . , zn) → (x1, y1 . . . , xn, yn).
Let (z1, . . . , zn) be a local coordinate system near x with zj = xj + iyj,
j = 1, . . . , n.
Then the real tangent space Tx(M) is spanned by
∂
∂x1 x
,
∂
∂y1 x
, . . . ,
∂
∂xn x
,
∂
∂yn x
.
Define an R−linear map J from Tx(M) onto itself by
J
∂
∂xi x
=
∂
∂yi x
, J
∂
∂yi x
= −
∂
∂xi x
for all j = 1, . . . , n.
Obviously, we have J2 = −1, and J is called the complex structure on Tx(M).
We observe that the definition of J is independent of the choice of the local
coordinates (z1, . . . , zn). The complex structure J induces a natural splitting
of the complexified tangent space
CTx(M) = Tx(M) ⊗R C.
First we extend J to the whole complexified tangent space by
J(x ⊗ α) = (Jx) ⊗ α.
It follows that
J : CTx(M) → CTx(M)
is a C−linear map with eigenvalues i and −i.
Denote by T1,0
x (M) and by T0,1
x (M) the eigenspaces of J corresponding to i
and −i respectively. It is easily verified that
• T1,0
x (M) = T0,1
x (M),
• T1,0
x (M) ∩ T0,1
x (M) = {0},
• T1,0
x (M) is spanned by
∂
∂z1 x
, . . . ,
∂
∂zn x
,
where ∂
∂zi x
= 1
2
∂
∂xi
− i ∂
∂yi x
, for 1 ≤ i ≤ n.
Consequently T0,1
x (M) is spanned by
∂
∂ ¯z1 x
, . . . ,
∂
∂ ¯zn x
,
3

Any vector v ∈ T1,0
x (M) is called a vector of type (1, 0), and we call v ∈
T0,1
x (M) a vector of type (1, 0). The space T1,0
x (M) is called the holomorphic
tangent space at x.
Therefore
CTx(M) = Tx(M) ⊗R C = T1,0
x (M) ⊕ T0,1
x (M).
Elements of CTx(M) can be realized as C−linear derivations in the ring
of complex valued C∞ functions on M around x.
Indeed, given the real derivation v ∈ Tx(M), the elementary tensor v ⊗ z ∈
Tx(M) ⊗R C acts on f + ig in the following way
(v ⊗ z)(f + ig) = z · (v(f) + iv(g)).
1.3 Complex forms: the (p,q)-forms
1.3.1 1-forms
Let CT∗
x (M) be the dual space of CTx(M). By duality, J also induces a
splitting on
C∗
Tx(M) = T1,0
x
∗
(M) ⊕ T0,1
x
∗
(M) := Λ1,0
x (M) ⊕ Λ0,1
x (M), (1.1)
where Λ1,0
x (M) and Λ0,1
x (M) are eigenspaces corresponding to the eigenvalues
i and −i respectively. Let U be an open set containing x and let z1, . . . , zn
be local coordinates. It is easy to see that the vectors
(dz1
)x, . . . , (dzn
)x
span Λ1,0
x (U) and that the space Λ0,1
x (U) is spanned by
(d¯z1
)x, . . . , (d¯zn
)x.
This means that any differential 1-form with complex coefficients can be
written uniquely as a sum
n
j=1
fjdzj
+ gjd¯zj
where fj, gj ∈ C∞(U), i.e. fj, gj : U → C are holomorphic.
Because of this splitting the complex 1-forms are also called the (1, 1)-forms.
Example 1.5. Let f : M → C be a smooth function, the exterior derivative
d = ∂ + ¯∂ is defined by
df =
n
j=1
∂f
∂zj
dzj
+
∂f
∂ ¯zj
d¯zj
= ∂f + ¯∂f.
4

1.3.2 Higher degree forms
Let p and q be a pair of non-negative integers ≤ n. One can define the wedge
product of complex differential forms in the same way as with real forms.
The space Λ
(p,q)
x of (p, q)-forms is defined by taking linear combinations of the
wedge products of p elements from Λ
(1,0)
x (M) and q elements from Λ
(0,1)
x (M),
i.e.,
Λ(p,q)
x (M) = Λ(1,0)
x (M) ∧ · · · ∧ Λ(1,0)
x (M)
p−times
∧ Λ(0,1)
x (M) ∧ · · · ∧ Λ(0,1)
x (M)
q−times
.
At this point the isomorphism (1.1) leads to the following splitting
Λk
x(M) := Λk
x(C∗
Tx(M)) = Λk
(T1,0
x
∗
(M) ⊕ T0,1
x
∗
(M)) (1.2)
=
p+q=k
Λp
x(T1,0
x
∗
(M)) ∧ Λq
x(T0,1
x
∗
(M)) (1.3)
=
p+q=k
Λ(p,q)
x (M). (1.4)
This means that given u a (p, q)−form, it can be written in local coordinates
in an open set U as
u(z) =
|I|=p,|J|=q
uIJ (z)dzI
∧ d¯zJ
where uIJ ∈ C∞(U.)
If we consider the disjoint union of the Λk
x(M) over the points of the manifold,
we get the space of all complex differential forms of total degree k, i.e.,
Λk
(M) =
x∈M
Λk
x(M).
Example 1.6. Let u be a (p, q)−form. We can define de following operator
∂ : Λ(p,q)
(M) → Λ(p+1,q)
(M) and ¯∂ : Λ(p,q)
(M) → Λ(p,q+1)
(M) (1.5)
and respectively in an open coordinate set their actions are given by
∂u =
I,J 1≤k≤n
∂uI,J
∂zk
dzk
∧ dzI
dzJ
, ¯∂u =
I,J 1≤k≤n
∂uI,J
∂¯zk
d¯zk
∧ dzI
dzJ
.
According to the previous consideration we can consider the differential
operator
d = ∂ + ¯∂. (1.6)
It satisfy
• d: Λk(M) → Λk+1(M), ∀k
5

• d2 = 0
• if ω ∈ Λr(M) and α ∈ Λs(M) then d(ω ∧ α) = dω ∧ α(−1)rω ∧ dα
• if f ∈ C∞(U) then the operator is the same as in the Example 1.5.
Remark. If M is a complex manifold, as in the case of the real manifolds,
Λk(M) defines a complex vector bundle on M, of "complex" rank n
k . In the
following section we will recall the definition in the real case and it will turn
out that it is a useful construction to study complex manifolds.
1.4 Canonical bundle
Let M be a real smooth manifold and let T∗M be its cotangent bundle.
As we saw in the complex case, the vector bundle of the r−forms on M is
defined by
Λr
(M) =
x∈M
Λr
x(T∗
x M) :=
x∈M
Λr
x(M)
and it has rank n
r as real vector bundle.
Definition 1.7 (Canonical bundle). Let r = n. The bundle of the n−forms
of a smooth manifold of dimension n is called the canonical bundle.
Since n
n = 1 the canonical bundle is a line bundle.
Definition 1.8 (Holomorphic canonical bundle). Let M be a complex man-
ifold. The holomorphic line bundle
Λn
(M) := K
is called the canonical line bundle.
Our aim will be to investigate manifold with trivial canonical bundle. It
is known that in the real case this condition is equivalent to:
Λn
(M) is trivial ⇐⇒ ∃ a nowhere vanishing section
⇐⇒ M is orientable.
The second condition means that there exist a differentiable n form ω nowhere
vanishing, i.e., ω(p) = 0, for every p in M. While the first condition still
applies in the complex case, since it is a topological condition, the second
equivalence it is not still true, as the following theorem proves.
Theorem 1.9. Any complex manifold M of dimension n is orientable.
6

Proof. Let (Ui, ϕi)i∈I be a system of holomorphic coordinates around a point
p, and
ϕi : Ui → ϕ(Ui), p → (z1(p), . . . , zn(p)),
ϕj : Uj → ϕ(Uj), p → (Z1(p), . . . , Zn(p)).
By identifying Cn with R2n in the usual way we can write:
(z1, . . . , zn) = (x1, y1, . . . , xn, yn) and (Z1, . . . , Zn) = (X1, Y1, . . . , Xn, Yn)
and thus we obtain a real manifold structure on M.
We will now calculate the Jacobian matrices of the transition functions for
both structures.
In the complex case we have
Jacϕi(p)(ϕj ◦ ϕ−1
i ) =
∂(ϕj ◦ ϕ−1
i )k
∂zl 1≤k,l≤n
=
∂Zk(z1, . . . , zn)
∂zl 1≤k,l≤n
= (ckl)1≤k,l≤n ∈ GL(n, C).
Instead in the real case we will find
Jacϕi(p)(ϕj ◦ ϕ−1
i ) =


∂Xk(x1,y1...,xn,yn)
∂xl
∂Xk(x1,y1...,xn,yn)
∂yl
∂Yk(x1,y1...,xn,yn)
∂xl
∂Yk(x1,y1...,xn,yn)
∂yl


1≤k,l≤n
.
Using the Cauchy-Riemann conditions, this coincides with
Re(ckl) −Im(ckl)
Im(ckl) Re(ckl)
1≤k,l≤n
∈ GL(2n, R).
We will now calculate the determinant of these matrices and we will see that
it is always positive, which is equivalent to state that M is orientable as a
real manifold.
Consider the following homomorphism
ρ: Mn(C) → M2n(R),
defined by
(ckl)1≤k,l≤n →
Re(ckl) −Im(ckl)
Im(ckl) Re(ckl)
1≤k,l≤n
.
The map ρ is continuous, since it is R−linear and the spaces involved are
finite dimensional. Also, being a Lie algebra homomorphism, we have
det(ρ(P−1
AP)) = det(ρ(P−1
)ρ(A)ρ(P)) = det(ρ(A)).
7

Finally, the diagonalizable matrices are dense in Mn(C), so we can restrict
our calculations to diagonal matrices in Mn(C). Therefore we obtain:
det(ρ(Diag(c1, . . . , cn))) = det Diag
Re(c1) −Im(c1)
Im(c1) Re(c1)
, . . . ,
Re(cn) −Im(cn)
Im(cn) Re(cn)
=
n
i
det
Re(ci) −Im(ci)
Im(ci) Re(ci)
=
n
i
|ci|2
= | det (Diag(c1, . . . , cn)) |2
.
So we can conclude that,
det(ρ(A)) = | det(A)|2
, ∀A ∈ Mn(C).
Finally, we get that the Jacobian matrices of the transition functions for the
chart ϕi for M have positive determinants, thus the real underlying manifold
M is orientable.
Synthetically if we want to verify if a holomorphic canonical bundle is
trivial, the following equivalence applies:
Λn
(M) is trivial ⇐⇒ ∃ a nowhere vanishing section of Λn
(M).
The nowhere vanishing section is also called a holomorphic volume form.
Example 1.10. We are going to examine the triviality of the canonical bundle
of some well-known manifolds.
• The canonical bundle of a complex n−dimensional torus K → Tn is
trivial. In fact we can explicitly show a nowhere vanishing section of
its canonical bundle. With the same notation used in Example 1.3, we
can see that
dz1
∧ · · · ∧ dzn
is a non vanishing complex n−form belonging to Λn(M).
• The canonical bundle of the projective line K → CP1 is not trivial.
One could prove that using the general following property for a closed
oriented Riemann surface M:
Λn
(M) is trivial ⇔ χ(M) = 0. (1.7)
Indeed χ(CP1) = 2, but we will not go into details. Furthermore asking
K to be trivial is equivalent to ask TCP1 to be trivial, and as we can
8

Figure 1.1: Non vanishing section on torus.
Figure 1.2: As we can see from the picture it is not possible to ﬁnd a holomorphic
non vanishing form on the sphere: the poles correspond to the two pots with zero
net ﬂow.
9

see from figure 1.2 it is not possible to find a non vanishing form on
this manifold1. In fact since the following proposition applies we have
that the tangent bundle of CP1 is diffeomorphic to the tangent bundle
of S2 thus it is not trivial.
Proposition 1.11. The complex projective line CP1 is diffeomorphic to
the 2-sphere S2.
Proof. Consider a point [z : w] with w = 0. Then we send this point to
z
w ∈ C. We use then the stereographic projection C → S2N, and we
send the point [1 : 0] to the north pole of S2. It is easy to check that
the composition
CP1
→ C∗
→ S2
is a diffeomorphism.
The inverse of the first map sends a point h ∈ C to
h
1 + |h|2
,
1
1 + |h|2
and infinity to [1, 0], so we can easily obtain the inverse
S2
→ CP1
explicitly.
• The canonical bundle of a Riemann surface Σ, K → Σ is not trivial
if its genus, g(M), is greater than one. One could see this using (1.7)
together with the fact that for a closed Riemann surface its genus is
related to the Euler characteristic by the formula
2 − 2g(M) = χ(M).
1.4.1 Line bundles over M
We should now open a parenthesis on line bundles in order to clarify some
ideas and to understand the potential of these structures.
Let M be a complex manifold and L1, L2, L3 be elements of the set
G = {isomorphism classes of all holomorphic line budle overM} .
They satisfy:
1
This topic is closely related to the problem of combing a hairy sphere, see "hairy ball
theorem" for further details.
10

1. Closure: if L1 and L2 are holomorphic line bundles then the tensor
product L1 ⊗ L2 is a holomorphic line bundle.
In fact by definition the tensor product of two bundles is the bundle
whose fiber is the tensor product of the fiber. Therefore, L1 ⊗ L2 has
fiber (p, C) ⊗ (p, C)), ∀p. That is because
C ⊗ C = C
since
C ⊗ C = C ⊗ C∗
= Hom(C, C).
Moreover if we consider the following isomorphism
Φ: C → Hom(C, C)
a → Φ(a): C → C
b → a · b
we conclude that Hom(C, C) = C.
2. Associativity:
(L1 ⊗ L2) ⊗ L3 = L1 ⊗ (L2 ⊗ L3)
3. Identity element idL1 : it is the trivial bundle C × M since
L1 ⊗ C × M = L1.
4. Inverse element: L∗
1. Since
idL1 ∈ Hom(L1, L1) = idL1 .
idL1 is a nowhere vanishing section of idL1 and this means that
L1 ⊗ L∗
1 = C.
5. Commutativity:
L1 ⊗ L2 = L2 ⊗ L1,
even if they are different bundles, they are isomorphic.
According to the consideration we have made, we can conclude that the space
of line bundles modulo equivalence forms a group under the tensor product.
Often instead of studying fiber bundles it is convenient to analyse their
smooth sections. In the case of the holomorphic line bundles, the holomorphic
sections correspond to the holomorphic functions f : M → C. But this is
quite useless since, as we will prove, holomorphic functions over a compact
connected manifold have to be constant. First we will examine the following
lemma that will let us to prove what we claimed.
11

Lemma 1.12. Let B be an open set of Cn and f : B → C. If |f| has a
maximum in B, then f is constant.
Proof. Let x, y ∈ Rn such that z = x + iy ∈ Cn and f(x, y) = u(x, y) +
iv(x, y) ∈ Cn.
If |f| has a maximum, then |f|2 = u(x, y)2 + v(x, y)2 has a maximum. Since
|f|2 is a real valued function, to find its stationary point, we set the partial
derivative to zero, namely



∂|f|2
∂xi
= 2 ∂u
∂xi
· u + 2 ∂v
∂xi
· v = 0
∂|f|2
∂yi
= 2 ∂u
∂yi
· u + 2 ∂v
∂yi
· v = 0
, ∀i ∈ {1, . . . , n}. (1.8)
By linear algebra, solving the above system is equal to impose:


∂u
∂xi i
∂v
∂xi i
∂u
∂yi i
∂v
∂yi i

 u
v
= 0, ∀i ∈ {1, . . . , n}, (1.9)
and using the Cauhy-Riemann conditions this is equivalent to ask


∂u
∂xi i
− ∂u
∂yi i
∂u
∂yi i
∂u
∂xi i

 u
v
= 0 ∀i ∈ {1, . . . , n}. (1.10)
It is easily seen that the last matrix has rank 1, so the condition (1.10) is
equivalent to impose
∂u
∂xi
2
+
∂u
∂yj
2
= 0, ∀i, j. (1.11)
Hence
∂u
∂xi
=
∂u
∂yj
= 0, ∀i, j. (1.12)
Therefore we are asking f to be constant.
Theorem 1.13. Let M be a complex compact and connected manifold,
f : M → C be a holomorphic function. Then f is constant.
Proof. Since f is holomorphic, |f| is a continuous function defined on a
compact set and for the Weirstrass theorem |f| has a maximum on M.
This means that if (Uα, ϕα) is a finite cover of M around the point p where
f reaches his maximum, the function
|f ◦ ϕ−1
α |: ϕα(Uα) ⊆ Cn
→ C
has a maximum. Using Lemma 1.12 we get that f is constant on Uα. Now
we can conclude considering that every holomorphic function is an analytic
function, more specifically it is a continuous function on Uα. Since f is
constant on Uα ∪Uβ, ∀α, β, because of the uniqueness of the Taylor expansion
f is constant on M.
12

1.5 Cohomologies
Before turning to the Dolbeault cohomology of complex manifolds, we will
give a very brief summary of these concepts in the real situation which, of
course, also applies to complex manifolds if they are viewed as real analytic
manifolds.
Definition 1.14. A sequence C∗ = (Cn, ∂n)n∈Z of modules Cn over a ring
R and homomorphisms ∂n : Cn → Cn−1 is called a chain complex, if for all
n ∈ Z we have that
∂n−1 ◦ ∂n = 0
holds. The ∂n functions are usually called the boundary operators or differ-
entials.
A chain complex is usually visualised in a diagram as such,
· · ·
∂n+1
−→ Cn
∂n
−→ Cn−1
∂n−1
−→ · · ·
Observation 1.15. Since ∂n−1 ◦ ∂n = 0 we immediately get that
Im(∂n) ⊂ ker(∂n−1).
Note that these are both submodules of Cn.
Definition 1.16. The n-cycles of a chain complex C∗ is
Zn(C∗) = ker(∂n).
The n-boundaries of a chain complex C∗ is
Bn(C∗) = Im(∂n+1).
The n-th homology module of a chain complex C∗ is
Hn(C∗) = Zn/Bn.
Similarly one can define a cochain to consist of the modules of R−linear
maps from your modules to R, together with special boundary maps
dn : Cn → Cn+1.
1.5.1 The de Rham cohomology
In order to apply what we have seen to manifolds we will use for R−modules in
this case real vector spaces, namely the vector spaces of differential k−forms
Λk(M), for all k = 1 . . . n, if n is the dimension of the manifold.
13

Definition 1.17. A differential form θ is called closed if dθ = 0. And a
differential k−form α is exact if there exist a differential (k − 1)−form β
such that dβ = α.
Observation 1.18. Note that because d ◦ d = 0 we have that all exact forms
are also closed.
Lemma 1.19. Let M be a smooth manifold, the following diagram is a
cochain
· · ·
d
−→ Λn−1
(M)
d
−→ Λn
(M)
d
−→Λn+1
(M)
d
−→ · · ·
Observation 1.20. The n-cycles are exactly the closed n-forms on M. And
the n-boundaries are the exact n-forms on M.
Observation 1.21. We also use the fact that Λn = 0 for n > dim(M).
Proof. Note that Λn(M) is a real vector space, thus an R−module. And
d is a R−linear map. Furthermore d ◦ d = 0 which is the same as saying
Im(d) ⊂ ker(d). Thus we are dealing with a cochain.
Definition 1.22. The p − th de Rham cohomology group is equal to the
p − th cohomology groups of the cochain in Lemma 1.19. This is usually
denoted
Hp
deRham(M).
We will now analyse a property that can be applied to symplectic mani-
folds.
Proposition 1.23. Let M be a 2n dimensional oriented differentiable man-
ifold which is compact and without boundary.
For all 0 ≤ p ≤ n it exists an isomorphism
Hp
deRham(M) = Hn−p
deRham(M)
∗
.
In particular
bk(M) = bn−k(M).
For a proof of the theorem and for a general clarification the reader is
referred to [Huy06].
To be more precise we should write Hp
deRham(M, R) instead of Hp
deRham(M).
The symbol R is used here to stress that we are considering real valued
p− forms; of course one can introduce a similar group
Hp
deRham(M, C)
for complex valued forms, i.e. forms with values in C ⊗ Λp(M). Then
Hp
deRham(M, C) = C ⊗ Λp
(M) (1.13)
is the complexification of the real De Rham cohomology group.
14

1.5.2 Dolbeault cohomology
Most of the facts about homology and De Rham cohomology on real manifolds
are also valid on complex manifolds if one views them as real analytic
manifolds. However one can use the complex structure to define as we have
seen in (1.5) the ∂-cohomology or Dolbeault cohomology. With the same
notation as those in Chapter 3
¯∂ : Λ(p,q)
(M) → Λ(p,q+1)
(M).
In particular, we get differential cochain complexes
· · ·
¯∂
−→ Λ(p,q−1)
(M)
¯∂
−→ Λ(p,q)
(M)
¯∂
−→Λ(p,q+1)
(M)
¯∂
−→ · · ·
Definition 1.24. We say that a (p, q)−form α is ¯∂-closed if ¯∂α = 0. The
space of ¯∂-closed (p, q)−forms is denoted by
Z
(p,q)
¯∂
(M).
A (p, q)−form β is ¯∂-exact if it is of the form β = ¯∂γ for γ ∈ Λ(p,q−1)(M).
Observation 1.25. Since ¯∂2 = 0, ¯∂(Λ(p,q)(M)) ⊂ Z
(p,q+1)
¯∂
(M).
Definition 1.26. Dolbeault cohomology groups are then defined as
H
(p,q)
¯∂
(M) =
Z
(p,q)
¯∂
(M)
¯∂(Λ(p,q−1)(M))
. (1.14)
Definition 1.27. The dimensions of the (p, q) cohomology groups are called
Hodge numbers
h(p,q)
(M) = dimC H
(p,q)
¯∂
(M).
They are finite for compact complex manifolds. The Hodge numbers of a
compact complex manifold are often arranged in the Hodge diamond:
h0,0
h1,0 h0,1
h2,0 h1,1 h0,2
h3,0 h2,1 h1,2 h0,3
h3,1 h2,2 h1,3
h3,2 h2,3
h3,3
which we have displayed here for a three complex dimensional manifold. The
general diagrams take the following form
15

hn,n
hn,n−1 hn−1,n
hn,n−2 hn−1,n−1 hn−2,n
. . .
... . . .
h2,0 h1,1 h0,2
h1,0 h0,1
h0,0
The decomposition (1.4) does not carry over the cohomology group in fact
we have that
dim(Hk
deRham(M, C)) ≤
p+q=k
h(p,q)(M)
. (1.15)
We will see in Chapter 4 that the equality (1.15) is for a special types of
complex manifolds, namely Calabi-Yau manifolds and Kähler manifolds.
Deﬁnition 1.28. dim(Hk
deRham(M, C)) is called the k − th Betty number.
16

Chapter 2
Some Algebraic Geometry
tools
In this chapter we analyse some tools of algebraic geometry that will be used
in the following discussion in order to understand the algebraic aspect of
some Calaby-Yau manifolds.
2.1 Affine and projective variety
Let k be an algebraically closed field and fix S ⊆ k[x1, . . . , xn]. Let An
k be
the n-dimensional affine space over k.
Definition 2.1. The set
V(S) = {p ∈ An
k | f(p) = 0 ∀f ∈ S}.
is called affine algebraic set.
Let I(S) the ideal generated by S, i.e., the smallest ideal of k[x1, . . . , xn]
which contains S.
Observation 2.2. We have that V(S) = V(I(S)).
Moreover the ring k[x1, . . . , xn] is nöetherian, thus every ideal is finitely
generated. For every ideal I of k[x1, . . . , xn] there exist polynomials f1, . . . , fr ∈
k[x1, . . . , xn] such that
V(I) = V(f1, . . . , fr)
= {(a1, . . . , an) ∈ An
k | fi(a1, . . . , an) = 0, 1 ≤ i ≤ r}.
Definition 2.3. The Zariski topology on An
k is the topology whose closed
set are the algebraic subset of An
k .
If X ⊂ An
k the Zariski topology on X is the one induced by the Zariski
topology on An
k .
17

Chapter 2. Some Algebraic Geometry tools
We can now give the definition of reducible and irreducible set which are
purely topological concepts.
Definition 2.4. Let X be a topological space and Y ⊆ X such that Y = ∅.
Y is irreducible if it is not the union of two closed proper subsets of Y.
A subset of X is reducible if it is not irreducible.
Thus every non empty set can be expressed as union of irreducible subsets
Y = Yi
where Yi are the irreducible components.
Definition 2.5. An algebraic variety over the field k is an irreducible closed
subset of An
k , endowed with the Zariski topology. An open subset of an affine
variety is called quasi-affine variety.
The definition we have previously given can be extended, in a natural
way, to the projective space.
The projective space Pn
k can be considered an extension of the affine space
by the following immersion
An
k → Pn
k
(a1, . . . , an) → (1, a1, . . . , an)
Let k be an algebraically closed field and consider the projective space Pn
k . Let
R = k[x1, . . . , xn, xn+1] be the ring of polynomials in n + 1 indeterminates
and f be an homogeneous polynomial of R, then it makes sense to ask
whether or not f(p) = 0 for a point p ∈ Pn. As in the affine case we have the
following definition.
Definition 2.6. A projective algebraic subset of Pn
k is the common zero set
of a collection of homogeneous polynomials in R.
We can define as well projective varieties and the Zariski topology in Pn
k .
2.2 Regular and rational functions
Fix X ⊂ An
k , algebraic set.
Definition 2.7. A function
X → k
is regular if it agrees with the restriction to X of some polynomial function
on the ambient An
k .
18

The set of all regular functions on X has a natural ring structure (where
addition and multiplication are the functional notions). This is the coordinate
ring of X, denoted k[X].
For all open set U ⊂ X, we will denote OX(U) (or simply O(U)) the set of
regular functions on U. Whereas sum and product of regular function is a
regular function, the set OX(U) is actually a ring, rather a k−algebra. We
call OX(U) the sheaf of regular functions on U.
Fix now an affine algebraic set X and assume that X is irreducible. Even
though we will not prove the following fact we have to remark that
X is irreducible ⇐⇒ k[X] is a domain.
Definition 2.8. The function field of X is the fraction field of k[X], denoted
k(X).
Definition 2.9. A rational function on X is an element ϕ ∈ k(X) i.e., ϕ is
an element of the equivalence class f
g , where f, g ∈ k[X], g = 0. Here
f
g
∼
f
g
⇐⇒ fg = gf
as elements of k[X]. A rational function ϕ ∈ k(X) is regular at p ∈ X if it
admits a representation ϕ = f
g where g(p) = 0.
The domain of definition of ϕ ∈ k(X) is the locus of all points p ∈ X where
ϕ is regular.
The definition or regular and rational function is a little different from
the one of the affine case.
Definition 2.10. Let X ⊂ Pn
k an algebraic set and U an open subset of
X. A function f is regular around a point p ∈ U if there exist an open
neighbourhood V of p in U and two homogeneous polynomials g, h of the
same degree such that for all (a1, . . . , an+1) ∈ V,
h(a1, . . . , an+1) = 0
and
f|V =
g
h
.
The function f is regular in U if it is regular in every points of U.
As in the affine case, for every open subset U of X we denote OX(U) (or
O(U)) the ring of the regular functions in U.
19

2.3 Divisors
Let X be an irreducible variety.
Definition 2.11. A prime divisor or irreducible divisor on X is a codimen-
sion 1 irreducible (closed) subvariety of X. A divisor D on X is a formal
Z−linear combination of prime divisors
D =
t
i=1
kiDi, ki ∈ Z.
In P2, C = V(xy − z2), L1 = V(x) and L2 = V(y) are prime divisors,
while 2C, 2L1 − L2 are divisors which are not prime.
We say a divisor D = t
i=1 kiDi is effective if each ki ≥ 0.
The support of D is the list of prime divisors occurring in D with non-zero
coefficient.
The set of all divisors on X form a group Div(X), the free abelian group on
the set of prime divisors of X. The zero element is the trivial divisor
D = 0Di,
and Supp(0) = ∅.
Example 2.12. Consider
ϕ =
f
g
=
(t − λ1)a1 · · · (t − λn)an
(t − µ1)b1 · · · (t − µm)bm
∈ k(A1
) = k(t)
where f, g ∈ k[t].
The divisor of zeros and poles of ϕ is
a1{λ1} + a2{λ2} + · · · an{λn}
Divisors of zeroes
− b1{µ1} − · · · − bm{µm}
Divisors of poles
.
Example 2.13. Let An = X. A prime divisor is D = V(h), where h ∈
k[x1, . . . , xn] is irreducible. Write
ϕ =
f
g
=
fa1
1 · · · fan
n
gb1
1 · · · gbm
m
∈ k(An
) = k(x1, . . . , xn),
where f, g ∈ k[x1, . . . , xn] and fi, gi irreducible, ai ∈ N.
Denoting the divisor of zeros and poles of ϕ by div(ϕ), we have
div(ϕ) = a1V(f1) + a2V(f2) + · · · + anV(fn) − b1V(g1) − · · · − bmV(gm)
20

On almost any X, we will associate to each ϕ ∈ k(X){0} some divisor,
div(ϕ), the divisor of zeros and poles, in such a way that the map
k(X)∗
= k(X){0} → Div(X)
ϕ → div(ϕ) =
D⊆X
prime
νD(ϕ) · D
preserves the group structure on k(X)∗, i.e.,
(ϕ1 ◦ ϕ2) → div(ϕ1) + div(ϕ2).
The image of this map will be the group of principal divisors:
P(X) ⊆ Div(X).
We will write
div(ϕ) =
D⊆X
prime
νD(ϕ) · D
where νD(ϕ) = ord(ϕ) which corresponds to the order of vanishing of ϕ
along D and it is computed as follows: take u1, . . . , un local coordinates for
a point x ∈ D; write
ϕ = fdu1 ∧ · · · ∧ dun,
where f ∈ k(X). Then νD(ϕ) = νD(f).
We should thus focus on the definition of order of vanishing of ϕ ∈ k(X){0}
along a prime divisor D, denoted νD(ϕ).
Assuming that X is non-singular in codimension 1, we distinguish two cases.
Case 1.Let X be affne, ϕ ∈ k[X], D = V (π) is a hypersurface defined by π ∈
k[X]. We say that ϕ vanishes along D provided that D = V(π) ⊆ V(ϕ).
So by the Nullstellensatz, (ϕ) ⊆ (π).
Definition 2.14. The order of vanishing of ϕ along D, denoted νD(ϕ),
is the unique integer k ≥ 0 such that ϕ ∈ (πk)(πk+1).
Observation 2.15. νD(ϕ) = 0 =⇒ ϕ ∈ (π0)(π1) = k[X](π), i.e., ϕ
does not vanish on all of D.
If ϕ is rational and ϕ = f
g , where f, g ∈ k[X], define
νD(ϕ) = νD(f) − νD(g).
Case 2.General case: ϕ ∈ k(X){0}, D ⊆ X arbitrary prime divisor. Choose
U ⊆ X open affine such that
• U is smooth;
21

• U ∩ D = ∅;
• D is a hypersurface: D = V(π) for some π ∈ k[U] = OX(U)).
We have ϕ ∈ k(X) = k(U). Define νD(ϕ) as in case 1.
Example 2.16. Let ϕ = x
y ∈ k(x, y) = k(A2) we have that
div(ϕ) =
D⊆A2
prime
νD
x
y
D
where νD
x
y is 0 for all divisors D except for L1 = V(x), where the order
of vanishing is 1, and L2 = V(y), where νL2 (ϕ) = −1.
2.4 Rational differential forms and canonical divi-
sors.
A rational differential form on X is intuitively f1dg1 + · · · + frdgr, where fi
and gi are rational functions on X. Formally:
Definition 2.17. A rational differential form on X is an equivalence class
of pairs (U, ϕ) where U ⊆ X is open and ϕ ∈ ΩX(U) and
(U, ϕ) ∼ (U , ϕ ) ⇐⇒ ϕ|U∩U = ϕ U∩U
We can define the divisor of a rational differential form. If ω is a rational
differential form on X, then div(ω) ∈ Div(X) is called a canonical divisor.
The canonical divisors form a linear equivalence class on X, denoted KX.
We are going to present another way to define the canonical divisor of a
compact complex manifold.
Definition 2.18. Let X be a compact complex manifold and Yi ⊂ X
codimension 1 subvarieties then we define the canonical divisor of X
KX = niYi
where ni ∈ Z.
From this definition we can glimpse that there exist a connection between
the canonical divisors and the canonical bundle of a complex manifold but
only later we will clarify this link.
A differential form ψ on X is regular if ∀x ∈ X, there is an open
neighborhood U such that x ∈ U and ψ|U agrees with t
i=1 gidfi, where
fi, gi ∈ OX(U). In other words, viewing ψ as a section of the cotangent
bundle of X, the section map is regular.
22

Example 2.19. The differential form
ψ = 2xd(xy)
= 2x(xdy + ydx)
= 2x2
dy + 2xydy
is a regular differential form in A2.
Definition 2.20. For U ⊂ X open, let ΩX(U) be the set of regular differ-
ential forms on the variety U.
2.5 Dualizing sheaf
The following discussion is far from being a complete presentation about
sheaves. We will not enter in details and we will just give a sketchy presenta-
tion of them.
A sheaf of rings F on a topological space X is a functor from the category
of open subsets of X, where the morphisms are inclusions, to the category of
rings where the objects are rings and the morphisms are ring homomorphism,
satisfying the standard sheaf axioms. In particular, for an open subset U
we have F(U) is a ring and if U, V are both subsets of X such that U → V ,
then the induced morphism
F(U) → F(V )
is a ring homomorphism.
A ringed space is a pair (X, OX) where X is a topological space and OX is
a sheaf of unital rings. The sheaf OX is called the structure sheaf of the
ringed space (X, OX).
Definition 2.21. Let (X, OX) be a ringed space. Let F be a sheaf of
OX− modules. We say F is locally free if for every point x ∈ X there exists
a set I and an open neighbourhood x ∈ U ⊂ X such that F|U is isomorphic
to
i
∈ OX|U
as an OX|U −module.
ΩX(U) is a module over OX(U). In fact, ΩX(U) is a sheaf of OX-modules.
On An, ΩX is the free OX−module generated by dx1, . . . , dxn.
Theorem 2.22. If X is smooth then the sheaf Ω(X) is a locally free
OX−module of rank d = dim X.
23

We will not prove this fact but we have that the set of rational differential
forms forms a vector space over k(X). We recall the definition of canonical
bundle that we have seen in Chapter 1 in order to analyse it in an algebraic
point of view.
For each p ∈ N, look at the sheaf ΛpΩ(X) of p−differentiable forms on
X, which assigns to open U ⊆ X the set of all regular p−forms, ∀x ∈ U
ϕ(x): Λp
TxX → K
Locally these look like
fidgi1 ∧ · · · ∧ dgip
Rational p−forms are defined analogously.
Observation 2.23. The set of rational p−forms on X is a k(X)−vector space
of dimension n
p .
Definition 2.24. Let X be a smooth n−dimensional, the canonical sheaf
(or dualizing sheaf ) of X is
ωX = Λn
ΩX.
Observation 2.25. The canonical sheaf satisfy the following properties:
• ωX is locally free of rank 1.
• The set of rational canonical n−forms is a vector space of dimension 1
over k(X).
Thus we have establish a connection between holomorphic sections and
divisors however we will better develop it in Chapter 4.
24

Chapter 3
A look to some complex
constructions
In this chapter we will equip smooth manifolds with a smooth linear complex
structure on each tangent space. The existence of this structure is a necessary,
but not sufficient, condition for a manifold to be a complex manifold. That
is, every complex manifold is an almost complex manifold, but not vice
versa. Almost complex structures have important applications in symplectic
geometry. Therefore we will also have a look to some aspect of symplectic
geometry in order to connect them to the complex world.
3.1 Almost complex structure
Definition 3.1 (Almost complex structure). An almost complex structure
on a differentiable manifold M is a differentiable endomorphism of the tangent
bundle
J : T(M) → T(M)
such that
J2
x = −id, (3.1)
where Jx : Tx(M) → Tx(M).
A differentiable manifold with some fixed almost complex structure is called
an almost complex manifold.
Almost complex manifolds must be even dimensional. In fact the following
preposition applies.
Proposition 3.2. If M admits an almost complex structure, it must be
even-dimensional.
25

Chapter 3. A look to some complex constructions
Proof. This can be seen as follows. Suppose M is n−dimensional, and let
J : T(M) → T(M) be an almost complex structure. Since the determinant
det: GL(n, R) → R∗
A → det(A)
is a group homomorphism, we have that
det(J2
x) = det(Jx)2
.
Using (3.1) we obtain
det(Jx)2
= det(−id) = (−1)n
.
But if M is a real manifold, then det(J) is a real number, thus n must be
even if M has an almost complex structure.
For instance every a complex manifold has a natural structure of almost
complex manifold, while the vice versa is not always true.
Theorem 3.3. Every complex manifold has a canonical almost complex
structure.
Proof. Let M be a complex manifold of dimension n, and p ∈ M. Let
(U, (z1, . . . , zn)) be a holomorphic chart around p. In this case , if we set
xk := Rezk and yk := Imzk,
then (U, (x1, . . . , xn, y1, . . . , yn)) is a local chart of M, seen as a smooth
manifold.
Firstly set forall q ∈ U,
Jq
∂
∂xi q
=
∂
∂yi q
, Jq
∂
∂yi q
= −
∂
∂xi q
.
In order to conclude we will now show that J is globally well-deﬁned, in other
words we will prove that J does not depend on the choice of coordinates.
Let (U, (zk)k) and (V, (wk)k) holomorphic charts around p; set
zk := xk + iyk, wk = uk + ivk
for all k ∈ {1, . . . , n} and denote by J and J the almost complex struc-
ture on (U, (zk)k) and (V, (wk)k) respectively. If we think that xk =
xk(u1, . . . , un, v1, . . . , vn) and yk = yk(u1, . . . , un, v1, . . . , vn), in U ∩ V we
have, 


∂
∂xk
= n
j=1
∂uj
∂xk
∂
∂uj
+
∂vj
∂xk
∂
∂vj
∂
∂yk
= n
j=1
∂uj
∂yk
∂
∂uj
+
∂vj
∂yk
∂
∂vj
. (3.2)
26

The Cauchy-Riemann conditions are,



∂uj
∂xk
=
∂vj
∂yk
∂uj
∂yk
= −
∂vj
∂xk
(3.3)
Therefore,
J
∂
∂xk
= J


n
j=1
∂uj
∂xk
∂
∂uj
+
∂vj
∂xk
∂
∂vj

 (3.4)
=
n
j=1
∂uj
∂yk
∂
∂uj
+
∂vj
∂yk
∂
∂vj
(3.5)
=
∂
∂yk
(3.6)
= J
∂
∂xk
in U ∩ V. (3.7)
Similarly,
J
∂
∂yk
= J
∂
∂yk
in U ∩ V. (3.8)
Example 3.4. In order to make it clear here some simple examples of almost
complex manifolds.
• Let (x, y) be the standard coordinates on R2. It is east to show that
J : R2 → R2, deﬁned by
J
∂
∂x
=
∂
∂y
, J
∂
∂y
= −
∂
∂x
,
satisﬁes J2 = id. Therefore R2 admits an almost complex structure J.
Identifying R2 with C in the usual way, z = x + iy, we can see J as a
multiplication by i, i.e., a rotation of π
2 in the plane.
• More in general, R2n admits an almost complex structure, for every inte-
ger n ≥ 0. In fact if we consider global coordinate (x1 . . . xn, y1, . . . , yn),
as we saw before,
J
∂
∂xi
=
∂
∂yi
, J
∂
∂yi
= −
∂
∂xi
,
is an almost complex structure.
27

3.2 Symplectic manifolds
3.2.1 (Real) Symplectic manifolds
Definition 3.5. Let M be a smooth 2n−dimensional manifold, a 2-form
ω ∈ Λ2(M) is said symplectic form if it is closed and non degenerate, i.e. if
satisfies the two condition
1. dω = 0;
2. ωp = 0 for every p ∈ M.
If M is a smooth manifold and ω is a symplectic form on M, then the pair
(M, ω) is called symplectic manifold.
In other words a symplectic form is a section of the bundle of the 2-forms
on M such that:
1. dω = 0 (analytic condition);
2. (TpM, ωP ) is a symplectic vector space for all p ∈ M (algebraic condi-
tion).
Example 3.6. Let x1, . . . , x2n be local coordinates around a point p ∈ R2n
and endow R2n with the 2-form
n
i=1
dxi
∧ dxn+i
.
R2n, ω is a symplectic manifold and the matrix of ω in the base
∂
∂x1 p
, . . . ,
∂
∂x2n p
of TpM
is
0 In
−In 0
.
Example 3.7 (Cotangent bundles). Let N be a smooth manifold of dimension
n and let M = T∗N be its cotangent bundle. This has a natural symplectic
structure, which may be defined locally as follows. Choose local coordinates
x1, . . . , xn on N. Then the 1−forms dx1, . . . , dxn provide a local trivialisation
of T∗N, so we obtain local coordinate functions ξ1, . . . , ξn on the fibres of
T∗N. Thus M has local coordinates (x1, . . . , xn, ξ1, . . . , ξn.) We may define
a 1-form
θ =
n
i=1
ξidxi
28

locally on M and it turns out that this local definition in fact defines a global
one-form (the so-called “Liouville form”) on M. The exterior derivative
ω = dθ
is a natural symplectic form on M. Clearly it is closed (since it is exact), and
it is nondegenerate because in local coordinates has the following expression
ω =
n
i=1
dξi
∧ dxi
.
Example 3.8. The 2-sphere S2 endowed with the symplectic form (the volume
form)
ω = sin θdθ ∧ dφ.
In general it is not true that the 2n−sphere S2n, with n > 1 is a symplectic
manifold, for the proof of this fact see Corollary 5.11.
3.2.2 (Complex) Symplectic manifolds
A complex symplectic manifold is a pair (M, ω) consisting of a complex
manifold M and a holomorphic 2-form ω (of type (2, 0)) such that:
1. ω is closed, i.e., dω = 0;
2. ω is non degenerate, i.e., the associated linear map
TpM → T∗
p M
v → (ω)p(v, −)
from the holomorphic tangent space to the holomorphic cotangent
space, is an isomorphism at each point p ∈ M.
These are sometimes also referred to as holomorphic symplectic manifolds.
3.3 Compatible almost complex structures
Definition 3.9. Let (M, ω) be a symplectic manifold and p a point of
M. An almost complex structure J on M is called compatible with ω (or
ω− compatible) if
• ω(Ju, Jv) = ω(u, v) ∀u, v ∈ TpM;
• ω(Ju, u) > 0, ∀u ∈ TpM such that u = 0.
29

Remark. g: TpM × TpM → R defined by
g(u, v) = ω(Ju, v), u, v ∈ TpM,
is a Riemannian metric. This bilinear form is simmetric since
g(w, v) = ω(w, Jv) = ω(Jw, J2
v) = ω(Jw, −v) = ω(v, Jw) = g(v, w),
where v, w ∈ TpM. By definition of ω it is positive definite, i.e. given
u ∈ TpM, we have
g(v, v) = ω(Ju, u) > 0.
Remark. The triple (ω, J, g) is said a compatible triple. Moreover any one of
J, ω, g can be written in terms of the other two by the following formulas.
• g(u, v) = ω(u, Jv);
• ω(u, v) = g(Ju, v);
• J(u) = ˜g−1(˜ω(u)),
where ˜ω: TM → T∗
M, ˜g: TM → T∗
M
u → ω(u, ·) u → g(u, ·).
Theorem 3.10. On any symplectic manifold (M, ω), there exists almost
complex structures J that are compatible with ω.
Proof. First we show this is true for a symplectic vector space V. Let g be a
Riemannian metric on V and define A by
ω(u, v) = g(Au, v).
Since ω is skew-symmetric and g is a metric, thus it is symmetric, we have
ω(u, v) = ω(−v, u) = g(−Av, u) = g(u, −Av). (3.9)
By definition of adjoint matrix and by eqution (3.9) we have
g(u, A∗
v) = g(Au, v) = g(u, −Av),
then
A∗
= −A.
Furthermore AA∗ is
• symmetric, i.e. (AA∗)∗ = AA∗;
• positive definite, i.e. g(AA∗v, v) = g(A∗v, A∗v) > 0, ∀v = 0.
30

Since every symmetric matrix B can be factored into B = Q∆Q−1 where Q
is orthogonal matrix, i.e. Q−1 = QT and ∆ = diag(λ1, . . . , λ2n), AA∗ can be
written as
AA∗
= Q∆Q−1
,
moreover λi > 0, for every i = 1, . . . , 2n because AA∗ is positive deﬁned. Set
J := (
√
AA∗)−1
A. (3.10)
The factorization A =
√
AA∗ is called the polar decomposition of A. Using
the diagonalization we can write J = Q
√
∆Q−1A, this implies that JJ∗ = id
and J∗ = −J, in fact
JJ∗
= Q(
√
∆)−1
Q−1
A · A∗
(Q−1
)∗
((
√
∆)−1
)∗
Q∗
,
since ((
√
∆)−1)∗ = (
√
∆)−1 and Q is orthogonal,
= Q(
√
∆)−1
Q−1
· Q∆Q−1
· Q(
√
∆)−1
Q−1
= Q(
√
∆)−1
∆(
√
∆)−1
Q−1
= id,
where the last equality applies since the diagonal matrices commute. Thus
J2 = −id is an almost complex structure and
• ω(Ju, Jv) = g(AJu, Jv) = g(JAu, Jv) = g(Au, v) = ω(u, v)
where the second equality applies since A commutes with
√
AA∗ and
thus J commutes with
√
AA∗;
• ω(u, Ju) = g(−JAu, u) = g(
√
AA∗u, u) > 0,
so J is compatible.
Finally we can use this construction to each tangent space TpM of every point
of M. It turns out that J is globally well-deﬁned since the polar decomposition
is canonical after a choice of Riemannian metric, i.e. it does not depend on
the choice of Q nor of the ordering of the eigenvalues {λ1, . . . , λ2n}. Hence J
is smooth.
31

Chapter 4
Calaby-Yau manifolds
Currently, research on Calabi-Yau manifolds is a central focus in both
mathematics and mathematical physics. The spread of this study is partially
propelled by the prominent role of the Calabi-Yau in superstring theories.
Many beautiful properties of Calabi-Yau manifolds have been discovered and
nowadays this subject represents an extremely active research field.
In this chapter we will give the definition of Calabi-Yau manifolds and we will
provide many examples, using the tools we have previously studied. Moreover
we will try to place this type of manifold in the complex geometry world and
we will investigate how they interact with others complex manifolds.
4.1 Symplectic Calabi-Yau manifolds
Following [JD08] we firstly analyse Calabi-Yau manifolds from the symplectic
point of view.
Definition 4.1. Let M be a symplectic real manifold of dimension 2n. M
is a symplectic Calabi-Yau if its canonical bundle
K → M
is trivial.
There are many examples of compact symplectic Calabi-Yau manifolds.
Firstly, many nilmanifolds are of this type.
Example 4.2 (Nilmanifolds). A Nilmanifold is a compact quotient space of a
nilpotent Lie group modulo a closed discrete subgroup, or (equivalently) a
homogeneous space with a nilpotent Lie group acting transitively on it.
The most famous example of a nilmanifold is the so-called Kodaira Thurston
surface, which will be discussed in Chapter 5.
32

Chapter 4. Calaby-Yau manifolds
4.2 Complex Calabi-Yau manifolds
We give now the definition of a Calabi-Yau manifold from the complex point
of view.
Definition 4.3. A complex Calabi-Yau is a complex compact manifold such
that its canonical bundle
K → M
is trivial.
Remark. Here the canonical bundle is a holomorphic line bundle while in
the case of the symplectic Calabi-Yau manifold it does not exist a notion of
holomorphic and the triviality is a topological condition.
In the following sections will now give some examples of complex Calabi-
Yau manifolds.
4.2.1 1-dimensional Calabi-Yau
In one complex dimension, the only compact examples are tori, see Example
1.10.
In order to expand the horizon of the 1-dimensional Calabi-Yau manifolds
we will show now how complex tori C/Λ can also be viewed as cubic curves.
These cubic curves are called elliptic despite not being ellipses, due to a
connection between them and the arc length of an actual ellipse.
Elliptic curves
Definition 4.4 (Elliptic curves over a field K). An elliptic curve is an
irreducible cubic in P2(K) which is non singular.
By an appropriate change of variables, a general elliptic curve over a field
K with field characteristic different from 2, 3, can be written in the affine
form
y2
= 4x3
− ax + b,
with a3 − 27b2 = 0 (condition of non singularity).
Let K = C we will now see how the points of an elliptic curve form a torus.
The meromorphic functions on a complex torus relate the manifold to a
cubic curve. Given a lattice Λ, the meromorphic functions f : C/Λ → C on
the torus, where C = C ∪ {∞}, are naturally identified with the Λ-periodic
meromorphic functions f : C → C on the plane. Let Λ be a lattice, a function
f such that
f(z + ω) = f(z),
for any z ∈ C and ω ∈ Λ, is called elliptic function.
It follows that if Λ = ω1Z ⊕ ω2Z,
f(z + ω1) = f(z + ω2) = f(z),
33

and we call ω1, ω2 the periods of f.
A non-trival example of an elliptic function is the Weierstrass ℘-function.
Given a lattice Λ, we define the Weierstrass ℘-function by
℘(z) =
1
z2
+
ω∈Λ{0}
1
(z − ω)2
−
1
ω2
, (4.1)
where z ∈ C, z /∈ Λ.
Proposition 4.5. The ℘(z) function satisfies the following properties:
1. ℘(z) is well defined in the sense that the sum converges absolutely and
uniformly on compact sets Ω such that Ω ∩ Λ = ∅.
2. ℘(−z) = ℘(z) for all z ∈ C.
3. The derivative
℘ (z) = −2
ω∈Λ{0}
1
(z − ω)3
(4.2)
has period Λ.
4. ℘(z) has period Λ.
Proof. We will just sketch the proofs. For 1 it is convenient to state the
following lemma.
Lemma 4.6. If k > 2 then the following series
ω∈Λ{0}
1
|ω|k
(4.3)
converges over the entire lattice Λ.
The convergence of the series is proven using an integral comparison test
and estimates on the diagonal of the fundamental parallelogram for Λ,
Π = {x1ω1 + x2ω2 : x1, x2 ∈ [0, 1]} .
Then the absolute and uniform convergence is a consequence of another
estimate and the exclusion of finitely many terms.
(2) If ω ∈ Λ then it is also true that −ω ∈ Λ, by multiplying by −1. Hence
℘(−z) =
1
(−z)2
+
ω∈Λ{0}
1
(−z − ω)2
−
1
ω2
=
1
(z)2
+
−ω∈Λ{0}
1
(−z + ω)2
−
1
(−ω)2
=
1
(z)2
+
−ω∈Λ{0}
1
(z − ω)2
−
1
ω2
= ℘(z).
34

(3) The sum ℘ (z) converges uniformly and absolutely by the lemma with
simple comparison to 1
|w|3 . Moreover if ω, ρ ∈ Λ then it is also true that
ω − ρ ∈ Λ; thus
℘ (z + ρ) = −2
ω∈Λ{0}
1
(z + ρ − ω)3
= −2
ω−ρ∈Λ{0}
1
(z − (ω − ρ))3
= ℘ (z).
(4) Using 3 we find out that the derivative of ℘(z + ω) − ℘(z) is equal to
zero, if z /∈ Λ. Thus there exists a constant cω such that
℘(z + ω) − ℘(z) = cω,
for all z /∈ Λ. Setting z = −ω
2 we obtain
cω = ℘ −
ω
2
− ℘ −
ω
2
.
By the fact that ℘(z) is even we can conclude that
cω = 0.
Therefore
℘(z + w) = ℘(z) (4.4)
for all w ∈ Λ.
Remark. Let Λ = ω1Z ⊕ ω2Z. Since ℘ satisfies the condition (4.4), we say
that ℘ is a doubly periodic function, as it has two independent periods ω1
and ω2.
We will now see some functions that will appear in the Laurent expansion
of the Weierstrass ℘(z)-function for Λ.
Definition 4.7. Let Λ be a lattice and k be an integer, the Eisenstein series
are functions defined by
Gk(Λ) =
ω∈Λ{0}
1
ωk
, (4.5)
for k > 2, even.
Remark. The Eisenstein series satisfy the homogeneity condition
Gk(mΛ) = m−k
Gk(Λ)
for all m ∈ C{0}.
35

Theorem 4.8. Let ℘ be the Weierstrass function with respect to a lattice Λ.
Then
(i) The Laurent expansion of ℘ is
℘(z) =
1
z2
+
∞
n=1
(2n + 1)G2n+2(Λ)z2n
, (4.6)
for all z such that 0 < |z| < inf{|ω| : ω ∈ Λ{0}}.
(ii) The functions ℘ and ℘ satisfy the relation
(℘ (z))2
= 4(℘(z))3
− g2(Λ)℘(z) − g3(Λ) (4.7)
where g2(Λ) = 60G4(Λ) and g3(Λ) = 140G6(Λ).
(iii) Let Λ = ω1Z ⊕ ω2Z and let ω3 = ω1 + ω2. Then the cubic equation
satisﬁed by ℘ and ℘ , y2 = 4x3 − g2(Λ)x − g3(Λ) is
y2
= 4(x − e1)(x − e2)(x − e3), (4.8)
where ei = ℘ ωi
2 for i = 1, 2, 3.
This equation is non singular, meaning its right side has distinct roots.
Proof. (i) Firstly we observe how the geometric series squares i.e.,
∞
n=0
qn
2
=
∞
n=0
qn
∞
k=0
qk
=
∞
k=0
qk
+ q
∞
k=0
qk
+ · · · + qn
∞
k=0
qk
+ . . .
= 1 + q + q2
+ . . . + q + q2
+ q3
+ . . . + · · · + qn
+ qn+1
+ qn+2
+ . . . + . . .
= 1 + 2q + 3q2
+ 4q3
+ · · · + (n + 1)qn
+ . . .
=
∞
n=0
(n + 1)qn
,
where q ∈ C. As a result, if 0 < |z| < inf{|ω| : ω ∈ Λ{0}}, we obtain that
℘(z) =
1
z2
+
ω∈Λ{0}
1
(z − ω)2
−
1
ω2
=
1
z2
+
ω∈Λ{0}
1
ω2
1
1 − z
ω
2 − 1
=
1
z2
+
ω∈Λ{0}
1
ω2


∞
n=0
z
ω
n 2
− 1


36

and by using the result of the geometric square series we get
=
1
z2
+
ω∈Λ{0}
1
ω2
∞
n=0
(n + 1)
z
ω
n
− 1
=
1
z2
+
ω∈Λ{0}
1
ω2
1 +
∞
n=1
(n + 1)
z
ω
n
− 1
=
1
z2
+
ω∈Λ{0}
1
ω2
∞
n=1
(n + 1)
z
ω
n
=
1
z2
+
ω∈Λ{0}
∞
n=1
(n + 1)
zn
ω2+n
.
Convergence results allow the resulting double sum to be rearranged, and
then the inner sum cancels when n is odd. In fact the last expression is equal
to
1
z2
+
∞
n=1
(n + 1)
ω2
1
zn
ωn
1
+
∞
n=1
(n + 1)
(−ω1)2
zn
(−ω1)n + . . .
+
∞
n=1
(n + 1)
ω2
i
zn
ωn
i
+
∞
n=1
(n + 1)
(−ωi)2
zn
(−ωi)n + . . .
with ωi ∈ Λ{0}. Therefore we can conclude that
℘(z) =
1
z2
+
∞
n=1 ω∈Λ{0}
(2n + 1)
ω2n+2
z2n
and by defenition of Einstein series
=
1
z2
+
∞
n=1
(2n + 1)G2n+2(Λ)z2n
.
(ii) The series expansions of ℘ and ℘ from the previous proposition leads to
℘(z) =
1
z2
+ 3G4z2
+ 5G6z4
+ O(z5
)
℘ (z) = −
2
z3
+ 6G4z + 20G6z3
+ O(z4
)
and taking the cube and the square we have
℘(z)3
=
1
z6
+
9G4
z2
+ 15G6 + O(z2
)
℘ (z)2
=
4
z6
−
24G4
z2
− 80G6 + O(z2
).
37

Deﬁne a function f as follows,
f(z) = (℘ (z))2
− 4(℘(z))3
+ g2(Λ)℘(z) + g3(Λ).
Since f is an elliptic function, as a polynomial of ℘ and ℘ , with no poles,
for the First Liouville Theorem1 it is constant. If we substitute the value we
found above, we get
f(z) =
4
z6
−
24G4
z2
− 80G6 − 4
1
z6
+
9G4
z2
+ 15G6 + g2(Λ)
1
z2
+ 3G4z2
+ 5G6z4
+ g3(Λ) + O(z2
)
=
4
z6
−
24G4
z2
− 80G6 − 4
1
z6
+
9G4
z2
+ 15G6 + 60G4(Λ)(
1
z2
+
+ 3G4z2
+ 5G6z4
) + 140G6(Λ) + O(z2
)
= 180 G2
4z2
+ 300 G4G6z4
+ O(z2
)).
Noticing that the last right-hand side grows at the order of z2, and considering
that lim
z→0
= O(z2), we conclude that f(z) ≡ 0.
(iii) We start observing that ℘(z) and ℘ (z) have a double pole at each
ω ∈ Λ. Moreover since ℘ (z) is doubly periodic, i.e. ℘ (z + ω) = ℘ (z), letting
Λ = ω1Z ⊕ ω2Z, we see that if z = −ωi
2 , and ω = ωi
2 ,
℘ (
ωi
2
) = ℘ (−
ωi
2
) (4.9)
for i = 1, 2, 3. Secondly, since ℘ is odd,
℘ (−
ωi
2
) = −℘ (
ωi
2
). (4.10)
Equation (4.9) together with (4.10) allow to conclude that zi = ωi
2 are points
of order 2 with ℘ (zi) = 0, for i = 1, 2, 3. The relation between ℘(z) and
℘ (z) from (ii) shows that the corresponding values xi = ℘(zi) for i = 1, 2, 3
are roots of the cubic polynomial
pΛ(x) = 4x3
− g2(Λ)x − g3(Λ),
so it factors as claimed. Each xi is a double value of ℘ since, as we have seen,
℘ (zi) = 0, furthermore ℘ has degree 2, meaning it takes each value twice
counting multiplicity, this makes the three xi distinct. That is, the cubic
polynomial pΛ has distinct roots.
1
Theorem 4.9 (First Liouville Theorem). If an elliptic function has no poles, then it is
constant.
38

The isomorphism between complex tori and complex elliptic curves
Part (ii) of Theorem 4.8 shows that the map
C/Λ → C2
z → ℘(z), ℘ (z)
takes nonlattice points of C to points (x, y) ∈ C2 satisfying the nonsingular
cubic equation of part (iii), y2 = 4x3 − g2(Λ)x − g3(Λ). Moreover this
application extends to all z ∈ C by mapping lattice points to a suitably
defined point at infinity, but we will see the details in the following theorem.
Theorem 4.10. Let Λ be a lattice on C and E be the elliptic curve
E = {(x, y) ∈ C2
| y2
= 4x3
− g2(Λ)x − g3(Λ)}.
Then the map φ defined by
C/Λ → E
z → ℘(z), ℘ (z) if z = 0
0 → {∞},
is a group isomorphism.
Before going into the proof of the theorem we need to state the following
result.
Theorem 4.11. Let f be an elliptic function for the lattice Λ and let Π be
a fundamental parallelogram for Λ. If f is not constant then
f : C → C ∪ {∞}
is surjective. If n is the sum of the orders of the poles2 of f in Π and z0 ∈ C,
then f(z) = z0 has n solutions (counting multiplicities).
We can now see the proof of Theorem 4.10, which allows us to understand
that complex tori (Riemann surfaces, complex analytic objects) are equivalent
to elliptic curves (solution sets of cubic polynomials, algebraic objects).
Proof. We need to show that φ is:
1. injective. Suppose
℘(z1), ℘ (z1) = ℘(z2), ℘ (z2) , (4.11)
where z1, z2 ∈ C/Λ are such that z1 = z2 mod Λ. The only poles for
℘(z) belong to Λ, thus we can distinguish to cases.
2
If f is an elliptic function, we can write it as a Laurent series expansion around ω ∈ Λ
as
f(z) = ar(z − ω)r
+ ar+1(z − ω)r
+ 1 + . . .
with ar = 0. We define the order of f at ω as ordωf = r.
39

• If z1 is a pole of ℘ then z1 ∈ Λ. Since (4.11) applies, z2 is a pole
of ℘, i.e., z2 ∈ Λ. This implies z1 = z2 mod Λ.
• Now suppose z1 is not a pole of ℘, i.e., z1 /∈ Λ. Then h(z) =
℘(z) − ℘(z1) has a double pole at z = 0 and no other poles in
Π = {x1ω1 + x2ω2 : x1, x2 ∈ [0, 1]} . By Theorem 4.11, h(z) has
exactly two zeros (counting multiplicities).
– Suppose z1 = ωi
2 for some i. From the proof of (iii) of Theorem
4.8 we know that ℘ (ωi
2 ) = 0, so z1 is a double root of h(z)
hence the only root. Thus z2 = z1.
– Suppose z1 is not of the form ωi
2 . Since h(−z1) = h(z1) = 0
(by evenness of ℘) and since z1 = z2 mod Λ, two zeros of h
are z1 and z2 = −z1 mod Λ. But
y = ℘ (z2) = ℘ (−z1) = −℘ (z1) = −y.
Hence ℘ (z1) = y = 0. But ℘ (z) has only a triple pole, thus
has only three zeros in Π. But from the proof of (iii) of
Theorem 4.8, we know that these zeros occur at ωi
2 , hence
a contradiction since z = ωi
2 . Thus z1 = z2 mod Λ and φ is
injective.
2. surjective. Let (x, y) ∈ E; we need to prove that there exists z ∈ C
such that φ(z) = (x, y), i.e., that x = ℘(z) and y = ℘ (z).
Since ℘(z) − x has a double pole, Theorem 4.11 implies it has zeros,
hence there exists z ∈ C such that ℘(z) = x. The elliptic equation in
the (ii) part of Theorem 4.8 implies that ℘ (z)2 = y2, so ℘ (z) = ±y.
• If ℘ (z) = y we are done.
• If ℘ (z) = −y, then by the evenness of the ℘ function, ℘ (−z) = y
and ℘(−z) = x, so −z → (x, y).
Hence φ is onto.
3. a group homomorphism. We need to show that
φ(z1 + z2) = φ(z1) + φ(z2),
where z1, z2 ∈ C. The map φ transfers the group law from the complex
torus to the elliptic curve. The proof is not trivial and it is well devel-
oped in [Kna92], thus we will not focus on the demonstration and we
will try to understand the group law on the curve.
40

Let z1 + Λ and z2 + Λ be nonzero points of the torus. The image points
(℘(z1), ℘ (z1)) and (℘(z2), ℘ (z2)) on the curve determine a secant or tangent
line of the curve in C2, ax + by + c = 0. Consider the function
f(z) = a℘(z) + b℘ (z) + c.
This is meromorphic on C/Λ. When b = 0 it becomes
f(z) = a
1
z2
+
ω∈Λ{0}
1
(z − ω)2
−
1
ω2
+ b − 2
ω∈Λ{0}
1
(z − ω)3
+ c. (4.12)
As we can see it has a triple pole at 0 + Λ and zeros at z1 + Λ and z2 + Λ.
One could also prove that its third zero is at the point z3 + Λ such that
z1 + z2 + z3 + Λ = 0 + Λ
in C/Λ. When b = 0, f has a double pole at 0 + Λ and zeros at z1 + Λ and
z2 + Λ, furthermore
z1 + z2 + Λ = 0 + Λ
in C/Λ. In this case let z3 = 0+Λ so that again z1 +z2 +z3 +Λ = 0+Λ, and
since the line is vertical view it as containing the infinite point (℘(0), ℘ (0))
whose second coordinate arises from a pole of higher order than the first.
Therefore for any value of b the elliptic curve points on the line ax+by+c = 0
are the points
(xi, yi) = ℘(zi), ℘ (zi))
for i = 1, 2, 3.
Since z1 + z2 + z3 + Λ = 0 + Λ on the torus in all cases, the resulting group
law on the curve is:
• The identity element of the curve is the infinite point;
• collinear triples on the curve sum to zero.
Our aim now is to find an addition law and a duplication law for the
function ℘.
Consider the affine equation of the elliptic curve and the affine equation of a
line:
E : y2
= 4x3
− g2x − g3, L: y = mx + b.
If a point (x, y) ∈ C2 lies on E ∪ L then its x-coordinate satisfies the cubic
polynomial obtained by substituting mx + b for y in the equation of E,
4x3
− m2
x2
+ −b2
− 2mbx − g2x − g3 = 0.
Thus, given three points collinear points (x1, y1), (x2, y2), and (x3, y3) on
the curve, necessarily
x1 + x2 + x3 =
m2
4
,
41

where m is the slope of their line,
m =



y1−y2
x1−x2
x1 = x2
12x2
1−g2
2y1
x1 = x2
(4.13)
A slight restatement is that
x3 =
m2
4
− x1 − x2, m as above.
(And also y3 = m(x3 − x1) + y1.)
These results translate back to the desired addition law and duplication
law for the Weierstrass ℘-function. Since the three points on the curve are
collinear, we have for some z1 + Λ, z2 + Λ ∈ C/Λ,
(x1, y1) = ℘(z1), ℘ (z1) ,
(x2, y2) = ℘(z2), ℘ (z2) ,
(x3, y3) = ℘(−z1 − z2), ℘ (−z1 − z2) .
But ℘ is even and ℘ is odd, so that in fact
(x3, y3) = ℘(z1 + z2), −℘ (z1 + z2) .
That is,
℘(z1 + z2) =
1
4
℘ (z1) − ℘ (z2)
℘(z1) − ℘(z2)
2
− ℘(z1) − ℘(z2) ifz1 + Λ = ±z2 + Λ,
℘(2z) =
1
4
12℘(z)2 − g2
2℘ (z)
2
− 2℘(z) ifz /∈
1
2
Λ + Λ.
In order to conclude this paragraph we outline that every elliptic curve
over C comes from a torus. That is, given an elliptic curve E, then we can
produce a lattice Λ unique up to some homothetic equivalence. We point
out that not only does every complex torus C/Λ lead via the Weierstrass
℘-function to an elliptic curve
y2
= 4x3
− a2x − a3, a3
2 − 27a2
3 = 0
with a2 = g2(Λ) and a3 = g3(Λ), but the converse holds as well.
Deﬁnition 4.12 (Homothetic Lattices). Let Λ = Zω1 + Zω2 be a lattice in
C. We deﬁne
τ :=
ω1
ω2
.
Since ω1 and ω2 are linearly independent over R, τ cannot be real. Hence,
by switching ω1 and ω2 if necessary, we can assume the imaginary part
42

Im(τ) > 0, i.e., τ lies in the upper half plain H = {x + iy ∈ C | y > 0}. Now
if we let Λτ = Zτ + Z, then Λ is homothetic to Λτ , that is
Λ = λΛτ
for some λ ∈ C. In this case λ = ω2.
Finally we can conclude thanks to the following result.
Theorem 4.13. Let y2 = 4x3 − Ax + b define an elliptic curve E over C.
Then there exists a lattice Λ such that g2(Λ) = A and g3(Λ) = B and there
is an isomorphism of groups C/Λ = E.
Observation 4.14. The existence of such a lattice is a homothetic equivalence,
that is, if we find Λ that works, then any Λ = λΛ for λ ∈ C will suffice.
There are several approaches to proving the statement but we will not
go any further. The reader is referred to [Raa10] for more details.
We have decided to point out the connection between complex tori and
complex elliptic curves in order to give different example of 1-dimensional
Calabi-Yau manifolds; moreover this turned out to be a good tools to analyse
the same object from the algebraic point of view.
4.2.2 2-dimensional Calabi-Yau
Calabi–Yau manifolds of dimension two are two-dimensional complex tori
and complex K3 surfaces, most of the latter are not algebraic. This means
that they cannot be embedded in any projective space as a surface defined
by polynomial equations.
K3 surfaces
Definition 4.15. A K3 surface is a (smooth) surface X which is simply
connected and has trivial canonical bundle.
We will now present a definition of K3 surfaces which is slightly different
from the standard one. Before going into the matter we need to understand
the following definition.
Definition 4.16 (Rational double points). A complex surface X has rational
double points if the dualizing sheaf ωX is locally free, and if there is a
resolution of singularities π: X → X such that
π∗
ωX = ωX
= OX
KX
.
This means that for every P ∈ X there exists a neighborhood U of P
and a holomorphic 2-form
α = α(z1, z2)dz1 ∧ dz2
43

defined on U − {P} such that π∗(α) extends to a nowhere-vanishing holo-
morphic form on π−1(U).
The structure of rational double points (sometimes called simple singularities)
is well-known and each such point must be analytically isomorphic to one of
the following:
An(n ≥ 1) x2 + y2 + zn+1 = 0 Fig4.1
Dn(n ≥ 4) x2 + yz2 + zn−1 = 0 Fig4.2
E6 x2 + y3 + z4 = 0 Fig4.3 a)
E7 x2 + y3 + yz3 = 0 Fig4.3 b)
E8 x2 + y3 + z5 = 0 Fig4.3 c)
and the resolution X → X replaces such a point with a collection of rational
curves of self-intersection -2 in the following configuration: An, Dn, E6, E7, E8.
Figure 4.1: From left to right in the figure are represented A2, A4, A6, A8. Note that
every A2n+1 for 1 ≤ n can not be plotted since all the points satisfying the related
equation are complex. Furthermore we can observe that as n increases, the top of
the surfaces becomes more pointed.
44

Figure 4.2: From left to right the image shows D4, D5, D6, D7. In the foreground
stand out the even Di 4 ≤ i ≤ 7 while in the background stand out the odd ones.
As we can see the conﬁguration of the surfaces and the look of the singular points
change letting i even or odd.
45

(a) E6. (b) E7.
(c) E8.
Figure 4.3
Deﬁnition 4.17 (K3 surfaces). A K3 surface is a compact complex analytic
surface X with only rational double points such that
h(0,1)
= dim(H1
(OX)) = 0
and ωX = OX.
If X is smooth, the dualizing sheaf ωX is the line bundle associated to
the canonical divisor KX, so this last condition implies that the canonical
divisor is trivial since KX= OX.
If X is a K3 surface and π: X → X is the minimal resolution of singularities
(i. e. the one which appeared in the deﬁnition of rational double point) then
it turns out that the pull-back π∗ establishes an isomorphism
H1
(OX) = H1
(OX
),
and also we have ωX
= π∗OX = OX
. Thus, the smooth surface X is also a
K3 surface.
We collect a number of construction methods for K3 surfaces. One should,
however, keep in mind that most K3 surfaces, especially of high degree,
do not admit explicit descriptions. Their existence is solely predicted by
deformation theory.
46

Proposition 4.18. The following properties apply for a K3 surface.
(i) All K3 surfaces are simply connected.
(ii) Every K3 surface over C is diﬀeomorphic to the Fermat quartic (see
Example 4.19).
(iii) The Hodge diamond
h0,0
h1,0 h0,1
h2,0 h1,1 h0,2
h2,1 h1,2
h2,2
=
1
0 0
1 22 1
0 0
1
is completely determined (even in positive characteristic).
Example 4.19. A non-singular degree 4 surface, such as the Fermat quartic,
[w, x, y, z] ∈ CP3
| w4
+ z4
+ y4
+ x4
= 0
is a K3 surface.
Example 4.20. The intersection of a quadric and a cubic in CP4
gives K3
surfaces.
Example 4.21. The intersection of three quadrics in CP5
gives K3 surfaces.
Example 4.22. A Kummer surface is a special type of quartic surface. As a
projective variety, a Kummer surface may be described as the vanishing set
of an ideal of polynomials. However, these surfaces may also be viewed more
abstractly, in terms of Jacobian varieties but we will not enter in details.
However we will devote the next section to the main feature of these surfaces.
See [End03] for more details.
Kummer surfaces
Deﬁnition 4.23. The Kummer surface with parameter µ ∈ R is the projec-
tive variety given by
Kµ : x2
y2
+ z2
− µ2
w2
− λpqrs = 0 ⊂ RP3
,
where µ2 = 1
3, 1, 3,
λ =
3µ2 − 1
3 − µ2
and p, q, r, s are the “tetrahedral coordinates,” given by
p = w − z −
√
2x,
q = w − z +
√
2x,
r = w + z + 2y,
s = w + z − 2y.
47

(a) µ2
= 1
3
(double sphere). (b) µ2
= 1 (Roman surface).
(c) µ2
= 3 (4 planes).
Figure 4.4: Real plots of the three exceptional cases. Pictures taken from [End03].
We exclude the values µ2 = 1
3, 1, 3 because these are exceptional cases,
for which most of the statements we will make about Kummer surfaces Kµ
will not hold. These three cases correspond, repectively, to the double sphere,
the Roman surface, and 4 planes, and are shown in Figure 4.4 (in the plots
in the ﬁgures the parameter w is set to w=1). As a side note, recall the
Veronese surface, which is given by the embedding RP2
→ RP5
by
[x : y : z] → [x2
: y2
: z2
: xy : xz : yz].
The Roman surface referred to above is the projection of this surface into
RP3
.
Example 4.24. We will now give a non-algebraic example of Kummer surface.
Let
T2
= C2
/Γ
48

be a complex torus of complex dimension 2. (Thus, Γ ⊂ C2 is an additive
subgroup such that there is an isomorphism of R−vector spaces Γ ⊗ R = C2.
Let (z, w) be coordinates on C2, and define
i(z, w) = (−z, −w).
Since Γ is a subgroup under addition,
i(Γ) = Γ.
Thus, i descends to an automorphism ˜i: T2 → T2. If we want to find the
fixed points of i, we need to study the solutions to
i(z, w) ≡ (z, w) mod Γ.
These solutions are
(z, w)|(2z, 2w) ∈ Γ
and so ˜i has as fixed points 1
2Γ/Γ. (There are 16 of these.)
If we define
X := T2
/˜i,
it turns out that X is a Kummer surface. This surface in fact has 16 singular
points at the images of the fixed points of ˜i.
To see the structure of these singular points, consider the action of i on a small
neighborhood U of (0, 0) in C2. Then U/i is isomorphic to a neighborhood of
a singular point of X. To describe U/i, we note that the invariant functions
on U are generated by
z2
, zw, w2
.
Thus, if we let r = z2, s = zw and t = w2 we can write
U/i = {(r, s, t) near (0, 0, 0)|rt = s2
}.
This is a rational double point of type A1.
dz ∧ dw is a global holomorphic 2-form on C2, invariant under the action of
Γ, and so descends to a form on T2. Since
d(−z) ∧ d(−w) = dz ∧ dw,
the form is also invariant under the action of i, so we get a form dz ∧ dw on
X{singular points}. In local coordinates, dr = 2zdz, dt = 2wdw so that
dz ∧ dw =
dr ∧ dt
4zw
=
dr ∧ dt
4s
.
It is easy to check that this form induces a global nowhere vanishing holo-
morphic 2-form on the minimal resolution X of X. To finish checking that
X is a K3 surface, we use the fact that
H1
(OX
) = H1
(OX) = { elements of H1(OT2 ) invariant under ˜i }.
49

Now H1(OT2 ) = H(0,1)(T2), the space of global differential forms of type
(0, 1). Since
˜i∗
(d¯z) = −d¯z and ˜i∗
(d ¯w) = −d ¯w,
this space is generated by d¯z and d ¯w. It follows that
H1
(OX) = H1
(OX
) = (0),
and that X and X are both K3 surfaces.
Proposition 4.25. We have the following facts about the Kummer surface
Kµ.
• Kµ is irreducible.
• As suggested by the appearance of the tetrahedral coordinates in the
defining equation, Kµ has tetrahedral symmetry.
• Kµ has 16 singularities, each of which is an ordinary double point, a
three-dimensional analogue of a node singularity. It is interesting to
note that a complex surface may have at most finitely many ordinary
double points. For quartic surfaces, the maximum number of ordinary
double points is 16, which is achieved by Kµ.
• Resolving the 16 singularities3 of Kµ, we obtain a K3 surface. This
K3 surface, which is sometimes given as the definition of a Kummer
surface, contains 16 disjoint rational curves.
Some Kummer surfaces are shown in Figure 4.5. Note that since these
plots are restricted to the real numbers, some of the ordinary double points
may not be visible. In fact, in the case 0 ≤ µ2 ≤ 1
3 (not plotted), Kµ only
contains 4 real points (each of which turns out to be an ordinary double
point). In the following table are collected all the real and complex nodes of
the surfaces obtained by letting µ vary.
3
In algebraic geometry, the problem of resolution of singularities asks whether every
algebraic variety V has a resolution, a non-singular variety W with a proper birational map
W → V. For varieties over fields of characteristic 0 this was proved in Hironaka (1964),
while for varieties over fields of characteristic p it is an open problem in dimensions at
least 4.
50

(a) 1
3
≤ µ2
≤ 1. (b) 1 ≤ µ2
≤ 3.
(c) 3 ≤ µ2
. (d) µ2
= ∞.
Figure 4.5: Some Kummer Surfaces. Pictures taken from [End03].
µ2 Real nodes Complex nodes Picture
0 ≤ µ2 < 1
3 4 12 4 real points
µ2 = 1
3 / / Fig.4.4 a)
1
3 < µ2 < 1 4 12 Fig.4.5 a)
µ2 = 1 / / Fig.4.4 b)
1 < µ2 < 3 16 0 Fig.4.5 b)
µ2 = 3 / / Fig.4.4 c)
µ2 ≥ 3 16 0 Fig.4.5 c)
µ2 = ∞ 16 0 Fig.4.5 d)
51

4.2.3 3-dimensional and higher-dimensional Calabi-Yau man-
ifolds
While the number of Calabi-Yau manifolds with one or two complex dimen-
sions is known, the situation in three complex dimension is much different.
Several thousand Calabi-Yau three-folds have been discovered; with one
exception T3, their metrics are not explicitly known, and it is not even
known (although it is strongly suspected) that the number of topologically
distinct Calabi-Yau three-folds is finite. Therefore we will focus on complex
projective spaces and we will analyse some interesting result.
Before entering into details we need to recall some rudiments about intersec-
tion theory.
Let f1, . . . , fk be complex, homogeneous polynomials of degree d1, . . . , dk in
n + k + 1 complex variables z = (z1, . . . , zn+k+1), define
X(f1, . . . , fk) := {[z] ∈ CPn+k
| fi(z) = 0 for i = 1, . . . , k}.
The set X = X(f1, . . . , fk) is an algebraic variety of dimension n.
I(X) = {Fi(z) ∈ C[z] | Fi(a) = 0, ∀a ∈ X}.
Definition 4.26 (Complete intersection). Let X = X(f1, . . . , fk) be an n
dimensional algebraic variety in CPn+k. X is a complete intersection if there
exists k homogeneous polynomials
Fi(z1, . . . , zn+k+1), for i = 1, . . . , k,
which generate all other homogeneous polynomials that vanish on X, i.e.,
I(X) =< F1, . . . , Fk > .
Proposition 4.27. The n dimensional complete intersection X of k smooth
hypersurfaces of degree d1, . . . , dk in CPn+k
{is a variety with trivial canonical bundle} ⇐⇒ d1+· · ·+dk = n+k+1.
When n = 3, for example, we have the following solutions of 4 + k =
d1 + · · · + dk with di > 1 :
5 = 5, 6 = 4 + 2 or 6 = 3 + 3, and 7 = 3 + 2 + 2.
In looking for further three dimensional examples, we consider complete
intersections in weighted projective spaces.
52

Weighted Projective Spaces. A weighted projective space CPn
(w) is
a generalization of CPn
. Both are quotients of Cn+1{0} by an action of
C∗ = C{0}. The weights w are sequences of natural numbers
w = (w0, . . . , wn) ∈ Nn+1
and the action of λ ∈ C∗ on (z0, . . . , zn) ∈ Cn+1{0} is given by the formula
λ · (z0, . . . , zn) = (λw0
z0, . . . , λwn
zn).
We assume that the greatest common divisor of the wi is 1.
Remark. From now on we will work on C thus we will denote CPn
(w) simply
with Pn(w).
For each subset S =⊂ {w0, . . . , wn} we denote by q(S) the greatest
common divisor of the wi with i ∈ S. Let H(S) denote the subset of all
(zj) ∈ Pn(w) with zi = 0 for i /∈ S. The points in H(S) are cyclic quotient
singularities for the group Zq(S)Z.
Furthermore we need the polynomials to be quasihomogeneous due to the
nature of the weighted projective space. A quasihomogeneous polynomial is
defined as:
Definition 4.28. A polynomial f is called quasihomogeneous of degree d if
the following relation holds:
f(λw0
z0, . . . , λwn
zn) = λd
f(z0, . . . , zn). (4.14)
We will define hypersurfaces in Pn(w) by using transverse polynomials.
We define a transverse polynomial as follows:
Definition 4.29. A polynomial f is transverse if f = 0 only at the origin.
This means that for a given set of weights not any polynomial is quasiho-
mogeneous, as can be seen in the following example:
Example 4.30. We consider the space P1(2, 3) and the polynomial:
f = x2
y + y2
= (λ2
x)2
(λ3
y) + (λ3
y)2
= λ2·2+3·1
x2
y + λ3·2
y2
= λd
f.
Clearly there is no degree d to satisfy the relations and hence this polynomial
is not quasihomogeneous.
53

Complete Intersections in Weighted Projective Spaces. The equa-
tions of hypersurfaces in the weighted projective space Pn(w) of degree d are
given by transverse quasihomogeneous polynomial equations f(z0, . . . , zn) = 0.
Now in order to find Calabi Yau manifold in Pn(w) the following proposition
can help.
Proposition 4.31. The complete intersections of multiple degree (d1, . . . , dk)
in the weighted projective space Pm+k(w) with trivial canonical bundle are
those satisfying the following condition
w0 + · · · + wm+k = d1 + · · · + dk.
For instance elliptic curves arise either as quartic curves in P2(1, 1, 2) and
sextic curves in P2(1, 2, 3). We can take for example
w4
+ x4
+ y2
= 0 and w6
+ x3
+ y2
= 0
respectively.
Example 4.32. Degree 8 hypersurface in P4(1, 1, 2, 2, 2) with equation
X : y8
0 + y8
1 + y4
2 + y4
3 + y4
4 = 0.
The degree 8 = 1 + 1 + 2 + 2 + 2 so the hypersurface in this degree is a
Calabi–Yau manifold.
y12
0 + y12
1 + y6
2 + y6
3 + y2
4 = 0.
The degree 12 = 1 + 1 + 2 + 2 + 6 so the hypersurface in this degree is a
y18
0 + y18
1 + y18
2 + y3
3 + y2
4 = 0.
The degree 18 = 1 + 1 + 1 + 6 + 9 so that the hypersurface in this degree is a
There is a complete classification of Calabi–Yau varieties arising from
transverse hypersurfaces in P4(w) but we will not go further.
4.3 Kähler manifolds
Definition 4.35. Let (M, ω) be a symplectic manifold. If there exist a
genuine complex structure J compatible with ω, then (M, ω, J) is a Kähler
manifold. The symplectic form ω is then called a Kähler form.
54

Remark. Here genuine means coming from holomorphic coordinates making
M a complex manifold.
Hence a Kähler manifold is a symplectic manifold endowed with a com-
patible complex structure, and as we saw in the previous chapters, g defined
by
g(u, v) = ω(u, Jv),
is a Riemannian metric called Kähler metric and ω is called Kähler form.
Locally on the almost complex manifold M, the holomorphic vector fields
T1,0
x (M), as we have already seen, have a basis ∂
∂z1
, . . . , ∂
∂zn
and the holo-
morphic 1-forms Λ
(1,0)
x have the related dual basis denoted dz1, . . . , dzn. The
coefficients of the Kähler form in these local coordinates are
gj,k(z, ¯z) = g
∂
∂zj
,
∂
∂¯zk
,
and the differential form
ω = −2i
n
j,k=1
gj,k(z, ¯z)dzj
∧ ¯zk
.
Therefore a Kähler manifold is a smooth manifold equipped with
• complex structure;
• Riemannian structure;
• symplectic structure,
which are compatible (as seen in Section 3.3).
The existence of a Kähler metric on a compact manifold constraints the
topology. In particular the following properties apply.
Proposition 4.36. If (M, J, ω) is a compact Kähler manifold of real dimen-
sion 2n, then
i. the complex structure on M leads to the Hodge decomposition (which
is equivalent to having equality in (1.15))
Hk
deRham(M) =
p+q=k
Hp,q
deRham(M) =
p+q=k
Hp,q
¯∂
(M);
ii. h(p,q)(M) = h(q,p)(M)
iii. the odd Betty numbers
b2r+1(M) := dim H2r+1
deRham(M) = dim
ker d
Im d
,
where d: Λ2r+1(M) → Λ2r+2(M), are even ∀i = 1, . . . , n;
55

iv. b2r(M) > 0.
Before entering into the details of the proof we will make a brief digression
concerning some operators and results of complex differential geometry.
Each tangent space V = TxM has a positive inner product (·, ·), part of
the Riemannian metric in a compatible triple.
Let e1, . . . , en be a positively oriented orthonormal basis of V. The star
operator is a linear operator ∗: Λ(V ) → Λ(V ) defined by
∗(1) = e1 ∧ · · · ∧ en
(e1 ∧ · · · ∧ en) = 1
(e1 ∧ · · · ∧ ek) = ∗(ek+1 ∧ · · · ∧ en).
We see that ∗: Λk(V ) → Λn−k and satisfies ∗∗ = (−1)k(n−k). The ∗ operator
will be hereafter used to define an inner product on forms.
Let ¯∂ be defined as in Definition 1.5 and d as in Definition 1.6. We now
define the adjoint operator of ¯∂
¯∂∗
: Λ(p,q)
→ Λ(p,q−1)
(M) (4.15)
by requiring that
¯∂∗
ψ, η = ψ, ¯∂η
for all η ∈ Λ(p,q−1)(M).
The Dolbeault cohomology group Hp,q
¯∂
(M) = Zp,q
¯∂
(M)/¯∂Λ(p,q−1)(M) is rep-
resented by the solutions of the two first-order equations
¯∂ψ = 0, ¯∂∗
ψ = 0. (4.16)
These two may be replaced by the single second-order equation
∆¯∂ψ = ¯∂ ¯∂∗
+ ¯∂∗ ¯∂ ψ = 0.
For a complete proof see [GH14].
The operator
∆¯∂ : Λ(p,q)
(M) → Λ(p,q)
(M)
is called the ¯∂−Laplacian or simply the Laplacian (written ∆) if no ambiguity
is likely. Differential forms satisfying the Laplace equation
∆ψ = 0
are called harmonic forms; the space of harmonic forms of type (p, q) is
denoted Hp,q(M) and called the harmonic space.
The isomorphism
Hp,q
(M) = Hp,q
¯∂
(M)
56

can be proved (see for instance [GH14]).
Before stating the Hodge theorem, whose isomorphism (4.3) is a part, we
begin by giving an explicit formula for the adjoint operator ¯∂∗. First we
define the star operator,
∗: Λ(p,q)
(M) → Λ(p−1,q−1)
(M)
by requiring
(ψ, η) =
M
ψ ∧ ∗η
for all ψ ∈ Λ(p,q)(M). If we suppose that M is compact this define an inner
product on forms. Therefore if we write
η =
I,J
ηIJ ϕI ∧ ¯ϕJ
then
∗η = 2p+q−n
I,J
εIJ ¯ηIJ ϕI0 ∧ ¯ϕJ0 ,
where I0 = {1, . . . , n} − I and we write εIJ for the sign of the permutation
(1, . . . , n, 1 , . . . , n ) → (i1, . . . , ip, j1, . . . , jq, i0
1, . . . , i0
n−p, j0
1, . . . , j0
n−q).
The signs work out so that
∗ ∗ η = (−1)p+q
η.
In terms of star the adjoint operator is
¯∂∗
= − ∗ ¯∂ ∗ .
Observation 4.37. Note that ¯∂2 = 0 ⇒ ¯∂∗2 = 0.
We are now ready to state the following theorem.
Theorem 4.38 (Hoge). Let M be a compact complex manifold, then
1. dim Hp,q(M) < ∞ and
2. the orthogonal projection
H: Λ(p,q)
(M) → Hp,q
(M) (4.17)
is well defined and there exists a unique operator, the Green’s operator,
G: Λ(p,q)
(M) → Λ(p,q)
(M),
with G(H(p,q)(M)) = 0, ¯∂G = G¯∂, ¯∂∗G = G¯∂∗ and
I = H + ∆G (4.18)
on Λ(p,q)(M).
57

The content of (4.18) is sometimes expressed by saying that, given η, the
equation
∆ψ = η
has a solution ψ if and only if H(η) = 0, and then
ψ = G(η)
is the unique solution satisfying H(ψ) = 0. Thus we should try to solve the
Laplace equation on a compact manifold. The idea is to find a ψ such that
(ψ, ∆ϕ) = (η, ψ)
for all ϕ ∈ Λ(p,q)(M) and to prove that this ψ is in fact C∞.
Remark. We remark that we may define the adjoint d∗ of d, form the
Laplacian ∆d = dd∗ + d∗d, and arrive at the exact formalism as for ¯∂ on
complex manifolds. Moreover the Hodge theorem is also true.
Let M be a compact complex manifold with Hermitian metric ds2, and
suppose that in some open set U ⊂ M, ds2 is Euclidean; that is there exists
local holomorphic coordinates z = (z1, . . . , zn) such that
ds2
= dzi ⊗ d¯zi.
Theorem 4.39. With the same hypothesis as above, for a differential form
ϕ = ϕIJ dzI ∧ d¯zJ
compactly supported in U,
∆d = 2∆¯∂. (4.19)
Proof. Let zi = xi + iyi, then
∆¯∂(ϕ) = −2
I,J,i
∂2
∂zi∂¯zi
ϕIJ dzI ∧ d¯zJ
= −
1
2 I,J,i
∂2
∂x2
i
+
∂2
∂y2
i
ϕIJ dzI ∧ d¯zJ
=
1
2
∆d(ϕ),
i.e., the ¯∂−Laplacian is equal to the ordinary d−Laplacian in U, up to a
constant. Although very few compact complex manifolds have everywhere
Euclidean metrics, but as it turns out in order to insure Equation (4.19) on a
complex manifold it is sufficient that the metric approximate the Euclidean
metric to the second order at each point.
We can now start the demonstration of Theorem 4.36.
58

Proof. (i) and (ii). Set
Hp,q
d (M) = {η ∈ Λp,q
(M): ∆dη = 0},
Hr
d(M) = {η ∈ Λr
(M): ∆dη = 0}.
Note that the two groups depend on the particular metric whilst the group
Hp,q
deRham(M) is intrinsically defined by the complex structure. By the com-
mutativity of ∆d and Πp,q : Hr
deRham(M) → Hp,q
deRham(M) and the fact that
∆d is real, the harmonic forms satisfy
Hr(M) = p+q=r Hp,q(M)
Hp,q(M) = Hp,q
d (M)
(4.20)
On the other hand, for η a closed form of pure type (p, q),
η = η + dd∗
G(η),
where the harmonic part η also has a pure type (p, q). Thus every de Rham
cohomology class on a compact oriented riemannian manifold M possesses a
unique harmonic representative, i.e.,
Hk
= Hk
DeRahm(M).
We also have the following orthogonal decomposition with respect to (·, ·) :
Λk
= Hk
⊕ ∆(Λk
(M))
= Hk
⊕ d(Λk−1
) ⊕ d∗
(Λk+1
).
The proof involves functional analysis, elliptic differential operators, pseu-
dodifferential operators and Fourier analysis; see [GH14].
When M is Kähler, the Laplacian satisfies ∆d = 2∆¯∂, hence harmonic forms
are also bigraded
Hk
=
p+q=k
Hp,q
.
Combining this with 4.20 and Theorem 4.38 we obtain the Hodge decompo-
sition for the Laplacian ∆d. For a compact Kähler manifold M, the complex
cohomology satisfies
Hr
DeRahm(M, C) = p+q=r Hp,q
DeRahm(M)
Hp,q
DeRahm(M) = Hp,q
DeRahm(M).
(4.21)
Hence, we have the following isomorphisms:
Hk
DeRahm(M) = Hk
=
p+q=r
Hp,q
(M) =
p+q=r
Hp,q
¯∂
(M).
59

(iii) For the point (ii) of the proposition it follows that
b2r+1(M) =
p+q=2r+1
h(p,q)
(M)
and considering that h(p,q)(M) = h(q,p)(M), we obtain
b2r+1(M) = 2
j
h(j,2r+1−j)
(M).
Therefore b2r+1(M) ≡ 0 mod 2.
(iv) This fact directly follows from the fact that every Kähler manifold is
also a symplectic manifold. In fact let ω be the symplectic form ensured by
the Kähler structure. If dα = ωr with 0 ≤ r ≤ n by Stokes’ Theorem we
have that
M
ωn
=
M
d(α ∧ ωn−r
) = 0.
Remark. Note that (i) is true because
dimC (Hr
deRham(M, C)) = dimR (Hr
deRham(M, R)) ,
since we previously deﬁned the Betti numbers for real de Rham cohomology
groups.
Actually Kähler manifolds satisfy several other topological property but
here we have mentioned only the ones that will be used later. For more
details see for instance [DS01].
Deﬁnition 4.40 (Kähler potential). Let (X, J, ω) be a Kähler manifold then
around every point x ∈ X there exists a neighbourhood U and a function
f ∈ C∞(U, R) for which the Kähler form ω can be written as
ω|U = i∂ ¯∂f.
Here, the operators
∂ =
k
∂
∂zk
dzk
and
¯∂ =
k
∂
∂¯zk
d¯zk
are called the Dolbeault operators.
60

For instance, in Cn, the function f = |z|2
2 is a Kähler potential for the
Kähler form, because
i∂ ¯∂
1
2
|z|2
=
1
2
i∂ ¯∂
k
zk ¯zk
=
1
2
i∂
k
zkd¯zk
=
1
2
i
k
dzk ∧ d¯zk
= ω.
We have that the converse holds true in fact we can apply the following
property. Before going into the proposition we will see a necessary definition.
Actually the metric we will talk about is the same as discussed in Section
4.3.
Definition 4.41 (Compatible Riemannian Metric.). Let M be a complex
manifold with corresponding complex structure J. We say that a Riemannian
metric g is compatible with J if
g(JX, JY ) = g(X, Y ) (4.22)
for all vector fields X, Y on M. A complex manifold together with a compat-
ible Riemannian metric is called a Hermitian manifold.
Proposition 4.42. Let M be a complex manifold with a compatible Rieman-
nian metric g, as in (4.22). Then the following assertions are equivalent:
1. g is a Kähler metric.
2. For each point x ∈ M, there is a smooth real function f in a neigh-
bourhood U of x such that ω|U = i∂ ¯∂f.
3. dω = 0.
Proof. By definition, 1 and 3 are equivalent.
3 ⇒ 2 Since ω is real and dω = 0, we have ω = dα locally, where α is a real
1-form. Then α = β + ¯β, where β is a form of type (1, 0). Since ω is of type
(1, 1), we have
∂β = 0, ¯∂ ¯β = 0 and ω = ¯∂β + ∂ ¯β
Hence β = ∂φ locally, where φ is a smooth complex function. Then ¯β = ¯∂ ¯φ
and hence
ω = ¯∂∂φ + ∂ ¯∂ ¯φ
= ∂ ¯∂(¯φ − φ)
= i∂ ¯∂f,
61

with f = i(φ − ¯φ).
2 ⇒ 3 Let d = ∂ + ¯∂, then we simply observe that
∂ω = i∂2 ¯∂f
= 0,
since ∂2 = 0 and that
¯∂ω = i∂ ¯∂2
f
= 0,
since ¯∂2 = 0.
Thanks to the equivalence of the definition of Kähler metric we will show
in the following example how the complex projective space can be seen as a
Kähler manifold.
Example 4.43 (Complex Projective space). Using the same notation as in
example 1.4, let CPn be the complex projective space. To see that CPn has
a natural Kähler manifold structure, we introduce the following functions
which turn out to be Kähler potentials. Let f be defined in an open set Uj
by
fj = log

1 +
k=j
zk
zj
2


= log
n
k=0
|zk|2
− log |zj|2
.
On the intersection of two coordinate charts Uj ∪ Uk, the difference
fj − fk = log |zk|2
− log |zj|2
= log
zk
zj
− log
¯zk
¯zj
.
satisfies the equation
∂ ¯∂ (fj − fk) = 0,
and hence there exists a global form ω on CPn with ω|Uj
= i∂ ¯∂fj.
To see that this form ω is the form associated with a Kähler structure, we
consider its coefficients gk,l in local coordinates where
ω|Uj
= i∂ ¯∂fj =
k,l
gk,ldwk ∧ d ¯wl
62

and wk = zk
zj
.
In order to see that the coefficient matrix (hk,l) is positive definite and
Hermitian symmetric we calculate the differentials using
fj = log

1 +
k=j
zk
zj
2

 = log 1 +
n
k=1
|wk|2
.
Thus
¯∂fj = 1 +
n
k=1
|wk|2
−1
·
n
k=1
wkd ¯wk
and
∂ ¯∂fj = 1 +
n
k=1
|wk|2
−1
·
n
k=1
dwk ∧ d ¯wk − 1 +
n
k=1
|wk|2
−2
·
n
k,l=1
¯wkdwk ∧ wld ¯wl
= 1 +
n
k=1
|wk|2
−2
·
n
k,l=1
δk,l 1 +
n
k=1
|wk|2
− ¯wkwl dwk ∧ d ¯wl.
For a complex vector ξ = (ξ1, . . . , ξn) ∈ Cn, we study the positivity properties
of the following expression using the Hermitian inner product
< w, ξ >=
n
k=1
wk
¯ξk
to obtain the inequality
n
k,l=1
δk,l 1 +
n
k=1
|wk|2
− ¯wkwl ξk
¯ξl
=< ξ, ξ >2
1+ < w, w >2
− | < w, ξ > |2
for ξ = 0, and the Schwarz inequality | < w, ξ > |2 ≤ < ξ, ξ >2< w, w >2
leads to
≥ < ξ, ξ >2
1+ < w, w >2
− < ξ, ξ >2
< w, w >2
=< ξ, ξ >2
≥ 0.
4.3.1 Kähler and Calabi-Yau manifolds
We have decided to introduce Kähler manifold in our dissertation in order
to see their connection with Calabi-Yau’s. We want to clarify that Calabi-
Yau manifolds are a particular kind of Kähler manifolds. Many different
definitions of Calabi–Yau manifolds exist in the literature; we list here some
63

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