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# Marcaccio_Tesi

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### Marcaccio_Tesi

1. 1. 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
2. 2. 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
3. 3. 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
4. 4. 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
5. 5. 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
6. 6. 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
7. 7. 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
8. 8. Chapter 1. Complex manifolds 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
9. 9. Chapter 1. Complex manifolds 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
10. 10. Chapter 1. Complex manifolds 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
11. 11. Chapter 1. Complex manifolds • 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
12. 12. Chapter 1. Complex manifolds 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
13. 13. Chapter 1. Complex manifolds 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
14. 14. Chapter 1. Complex manifolds 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
15. 15. Chapter 1. Complex manifolds 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
16. 16. Chapter 1. Complex manifolds 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
17. 17. Chapter 1. Complex manifolds 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
18. 18. Chapter 1. Complex manifolds 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
19. 19. Chapter 1. Complex manifolds 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
20. 20. Chapter 1. Complex manifolds 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
21. 21. Chapter 1. Complex manifolds 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
22. 22. 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
23. 23. 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
24. 24. Chapter 2. Some Algebraic Geometry tools 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
25. 25. Chapter 2. Some Algebraic Geometry tools 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
26. 26. Chapter 2. Some Algebraic Geometry tools 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
27. 27. Chapter 2. Some Algebraic Geometry tools • 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
28. 28. Chapter 2. Some Algebraic Geometry tools 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
29. 29. Chapter 2. Some Algebraic Geometry tools 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
30. 30. 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
31. 31. 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
32. 32. Chapter 3. A look to some complex constructions 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
33. 33. Chapter 3. A look to some complex constructions 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
34. 34. Chapter 3. A look to some complex constructions 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
35. 35. Chapter 3. A look to some complex constructions 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
36. 36. Chapter 3. A look to some complex constructions 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
37. 37. 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
38. 38. 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
39. 39. Chapter 4. Calaby-Yau manifolds 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
40. 40. Chapter 4. Calaby-Yau manifolds (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
41. 41. Chapter 4. Calaby-Yau manifolds 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
42. 42. Chapter 4. Calaby-Yau manifolds 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
43. 43. Chapter 4. Calaby-Yau manifolds 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
44. 44. Chapter 4. Calaby-Yau manifolds 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
45. 45. Chapter 4. Calaby-Yau manifolds • 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
46. 46. Chapter 4. Calaby-Yau manifolds 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
47. 47. Chapter 4. Calaby-Yau manifolds 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
48. 48. Chapter 4. Calaby-Yau manifolds 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
49. 49. Chapter 4. Calaby-Yau manifolds 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
50. 50. Chapter 4. Calaby-Yau manifolds 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
51. 51. Chapter 4. Calaby-Yau manifolds (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
52. 52. Chapter 4. Calaby-Yau manifolds 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
53. 53. Chapter 4. Calaby-Yau manifolds (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
54. 54. Chapter 4. Calaby-Yau manifolds 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
55. 55. Chapter 4. Calaby-Yau manifolds 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
56. 56. Chapter 4. Calaby-Yau manifolds (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
57. 57. Chapter 4. Calaby-Yau manifolds 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
58. 58. Chapter 4. Calaby-Yau manifolds 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
59. 59. Chapter 4. Calaby-Yau manifolds 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. Example 4.33. Degree 12 hypersurface in P4(1, 1, 2, 2, 6) with equation 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 Calabi–Yau manifold. Example 4.34. Degree 18 hypersurface in P4(1, 1, 1, 6, 9) with equation 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 Calabi–Yau manifold. 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
60. 60. Chapter 4. Calaby-Yau manifolds 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
61. 61. Chapter 4. Calaby-Yau manifolds 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
62. 62. Chapter 4. Calaby-Yau manifolds 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
63. 63. Chapter 4. Calaby-Yau manifolds 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
64. 64. Chapter 4. Calaby-Yau manifolds 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
65. 65. Chapter 4. Calaby-Yau manifolds (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
66. 66. Chapter 4. Calaby-Yau manifolds 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
67. 67. Chapter 4. Calaby-Yau manifolds 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
68. 68. Chapter 4. Calaby-Yau manifolds 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