Metrics and Holonomy

Metrics and Holonomy Jonathan Herman The goal of this paper is to understand the following definitions of Kähler and Calabi-Yau manifolds: Definition. A Riemannian manifold is Kähler if and only if it admits a metric g with Hol(g) O(n) Definition. A Riemannian manifold is Calabi-Yau if and only if it admits a Kähler metric g with Hol(g) SU(n). These definitions provide a modern Riemannian geometric interpretation of Kähler and Calabi- Yau manifolds. As we will see, the holonomy of a connection is defined in terms of parallel transport, and furthermore the Ambrose-Singer theorem demonstrates the relationship between the curvature and holonomy of a connection. If not stated otherwise, we always assume that (M, g) is a connected Riemannian manifold, γ I M is a curve from an interval I R to M and π E M is a vector bundle of rank p. The holonomy of a connection is defined in terms of the parallel transport so we first recall what this is. 1 Parallel Transport Parallel transport gives a way to differentiate vector fields on a manifold, which agrees with the usual notion of vector field differentiation in R n. We first explain how this is done along a curve γ I M. Given such a curve recall that we can define the pullback bundle γ (T M) = T γ(t) M t I a vector bundle over I. A section V Γ(γ (T M)) is called a vector field along γ. It is just a smooth map V I γ (T M) such that V t T γ (t)m for each t I. Proposition. Let be any connection on M. Then there exists a unique connection, denoted D t (D t = γ ), satisfying D t (av + bw ) = ad t V + bd t W for a, b R and V, W Γ(γ (T M)) (i.e. linear) D t (fv ) = df dt V + fd tv for V Γ(γ (T M)), f C (Y ) (i.e. satisfies Leibniz rule) If V is the restriction to γ of a vector field Ṽ Γ(T M) then (D t V )(t 0 ) = ( γ (t 0 )Ṽ ). 1

Proof. Let x 1,..., x n denote local coordinates. Then V Γ(γ (T M)) looks like V (t) = V k (t) Also, γ (t) = dγi (t) dt x i γ(t). If such a D t exists then the defining properties give that D t V (t 0 ) = (D t (V k = dv k dt (t 0) = dv k dt (t 0) x k ))(t 0 ) γ(t 0 ) x k γ(t 0 ) + (V k (D t x k ))(t 0) x k + V k (t 0 ) dγ i γ(t 0 ) = x k + V k (t 0 ) dγi γ(t 0 ) dt (t 0) dt (t 0) x i γ(t 0 ) x i γ(t 0 ) = x k + V k (t 0 ) dγi γ(t 0 ) dt (t 0)Γ m ik (γ(t 0)) x k = ( dv k dt (t 0) + Γ k ij(γ(t 0 )) dγi dt (t 0)V j (t 0 )) x k x m γ(t 0 ) x k γ(t 0 ) x k γ(t). Hence if such a D t exists it must be this one, which shows uniqueness. For existence, just define D t V = ( dv k dt + (Γ k ij γ) dγi dt V j ) x k in a coordinate chart. γ. Given a curve γ and a vector field V Γ(T M) we call D t V the covariant derivative of V along Definition. A vector field V Γ(γ (T M)) is said to be parallel along γ if D t V 0. A vector field X Γ(T M) is called parallel if for any curve γ I M we have D t X 0. Proposition. Fix p M and let γ ( ε, ε) M be a curve with γ(0) = p. For any V p T p M, there exists a unique vector field Ṽ Γ(γ (T M)) such that Ṽ (0) = V p and D t Ṽ 0. Proof. The previous proposition shows that Ṽ is found by solving the equation dṽ k dt (t) = Γk ij(γ(t)) dγi dt (t)ṽ j (t). This is a first-order systemm of ODE s and is linear. Putting the initial condition Ṽ (0) = V p on this equation guarantees a unique solution by the Picard-Lindelof theorem. The vector field Ṽ in this proposition is called the parallel translate of V p. The equations in this proof are called the parallel transport equations. Definition. Let p, q M be two distinc points. Let γ [0, 1] M be a curve with γ(0) = p and γ(1) = q. Given arbitrary V p T p M let Ṽ denote the parallel translate along γ. The map is called the parallel transport of V p along γ. P γ T p M T q M V p Ṽ (1) 2

It s not hard to check that the parallel transport is a linear isomorphism; we have that Pγ 1 is equal to P γ 1 where γ 1 is the curve γ 1 (t) = γ(1 t). Parallel transport can naturally be extended to tensors of any type. Given p, q and γ as above, let α p Λ n (Tp M). We denote this extended parallel transport map by Pγ. It is defined by (Pγ α p )(X 1,..., X n ) = α p (Pγ 1 (X 1 ),..., Pγ 1 (X n )), where X 1,..., X n T q M. We would then define, for σ p and (k, l)-tensor at p, (Pγ σ p )(X 1,..., X n, β 1,..., β n ) = σ p (Pγ 1 (X 1 ),..., Pγ 1 (X k ), (Pγ ) 1 (β 1 ),..., (Pγ ) 1 (β l )). 3

2 Holonomy To motivate the definition of holonomy, consider R n with the Euclidean connection. Then Γ k ij = 0 in the trivial coordinate chart. Hence the parallel transport equations become dṽ k dt = 0. That is given V p T p R n the parallel translate along any curve is Ṽ (t) = V p. In particular, if γ is a loop based at p, (i.e. a curve with initial and end points equal to p), then parallel transport of V p along the loop is just the identity. That is, after translating we end back up with V p. However, this is not true on the sphere. We explain this in an example below. The difference here is that (S 2, g std ) has curvature where-as (R n, g std ) is flat. Definition. Let p M and a connection on M. The Holonomy of is defined to be Hol p ( ) = {P γ γ is a loop based at p.}. g. If ever we write Hol p (g) we mean Hol p ( ) where is the Levi-Civita connection associated to Proposition. Let p, q M with p /= q. Then Hol p ( ) is a group, and Hol p ( ) is conjugate to Hol q ( ). Proof. Let γ and τ be loops based at p. Then we know γ 1 (t) = γ(1 t) is also a loop based at p. That is, Pγ 1 Hol p ( ). Also, the uniqueness of parallel transport shows that P γ P τ = P γ τ. Since γ τ is a loop based at p we have P γ P τ Hol p ( ). Now let η be a curve from p to q. Then it s easy to see that the map Hol p ( ) Hol q ( ) P γ P η P γ P 1 η is an isomorphism. Identifying T p M R n we see that Hol p ( ) is a subgroup of GL(n, R),. We also see there is only one holonomy group up to conjugation, and so sometimes we will write Hol( ) without referring to a specific point. Example. (Holonomy of S n ) Recall that S n SO(n)/SO(n 1). To see this, recall that SO(n) acts transitively on S n. With the action of SO(n) on S n the isotropy group (or stabilizer) of e n, the last standard basis vector, is readily computed to be SO(n 1). Then given p M and v T p M, we have that p-corresponds to an element in the orbit of SO(n). Saying that P γ (v) = v just means that P γ is in the isotropy group of v. But by transitivity this is the isotropy group of e n. Hence we see that Hol p ( ) SO(n 1). Theorem. (Holonomy Principle) Let M be a connected manifold. Let Tl k be the vector bundle of (k, l)-tensors, where k and l are arbitrary. Let be a connection on Tl k. The following are equivalent: 1. There exists σ Tl k that is parallel (i.e. P γ σ p = σ q for any p, q M and γ a curve connecting p to q). 2. There exists σ T k l that is constant (i.e. σ 0) 4

3. There exists p M and σ p in the fibre of Tl k at p that is fixed under the action of Hol p ( ) (i.e P γ σ p = σ p for P g Hol p ( )). Proof. (1) (3) : If P γ σ p = σ q for any p, q M and any γ, then it holds for p = q and γ a loop. (3) (1) : Suppose that there exists some p and σ p that is invariant under Hol p ( ). Define σ point wise by σ q = P τ σ p, where τ is a curve from p to q. This is well defined by our assumption. Indeed if τ and η are two different curves connecting p and q that τ η 1 is a loop at p and so P τ η 1σ p = σ p, so that P τ σ p = P η σ p (1) (2) Let p M be arbitrary let X p T p M. Let γ be a curve starting at p with initial velocity X p. Let X = γ and choose parallel vector fields and 1-forms X 1,..., X n and β 1,..., β n along γ. Then, by the definition of the total covariant derivative of a (k, l)-tensor, at p we have σ(x 1,..., X k, β 1,..., β l, X) = X (σ(x 1,..., X k, β 1,..., β l )) k σ(x 1,..., X X i,..., X k, β 1,..., β l ) σ(x 1,..., X k, β 1,..., X β i,..., β l ). i=1 But X 1,..., X n, β 1,..., β n are parallel and so this equation reduces to σ(x 1,..., X k, β 1,..., β l, X) = γ (σ(x 1,..., X k, β 1,..., β l )). k i=1 Hence σ 0 if and only if σ(x 1,..., X k, β 1,..., β l ) is constant along curves. But X 1,..., X k, β 1,..., β l are parallel and so σ 0 if and only if σ is invariant under parallel transport. Example. (Riemannian Manifolds) Let (M, g) be a connected Riemannian manifold and let denote the Levi-Civita connection. Then g 0. Thus for any p M, X, Y T p M and γ a closed loop we have g p (P γ X, P γ Y ) = (P γ 1 g p )(X, Y ) = g p (X, Y ), where the last equality is by the holonomy principle. Hence each P γ is an isometry. Identifying T p M R n we see that on a Riemannian manifold Hol( ) O(n). Example. (Oriented Riemannian Manifolds) Let (M, g) be a connected Riemannian manifold with Levi-Civita connection. Suppose that M is oriented, and let µ denote the Riemannian volume form. The volume form is constant, i.e. µ 0. Let E 1,..., E n be any orthonormal frame. The characterizing property of the Riemannian volume form is that µ(e 1,..., E n ) = 1. For a closed loop γ at p M, the holonomy principle shows that 1 = µ p (E 1,..., E n ) = (P γ µ p )(E 1,..., E n ) = µ p (P 1 γ E 1,..., Pγ 1 E n ) = det Pγ 1. This shows that Hol p ( ) SO(n). Conversely, suppose that Hol p ( ) SO(n). Let α p Λ n (T p M ), be a non-zero n-covector. By hypothesis, det P γ = 1 and so P γ α p = α p. But then by the holonomy principle we may extend, in a well-defined way, α p to a nowhere vanishing top degree form on M. This shows that M is orientable. Hence (M, g) is oriented if and only if Hol( ) SO(n). 5

3 The Holonomy of Kähler Manifolds Let (M 2n, J) be an almost complex manifold. That is, J is a (1, 1)-tensor satisfying J 2 = I. Let h be a Hermitian metric. That is, h(jx, JY ) = h(x, Y ). From the Hermitian metric we get the fundamental form ω, which is defined by ω(x, Y ) = h(jx, Y ). The metric h is called Kähler if J is integrable and ω is closed. Recall that J is integrable means that it comes from a complex structure. That is, there exists a holomorphic atlas {(U α, φ α )} α I such that in any such chart, J = ϕ 1 J 0 ϕ where J 0 is the canonical complex structure on R 2n, J 0 = [ 0 I I 0 ]. Before considering the holonomy of such manifolds, we first determine equivalent conditions for the manifold to be Kähler. The condition we want is that h is Kähler J 0 where is the Levi-Civita connection of h. In order to get there we first need the Newlander-Nirenberg theorem. Recall that a distribution D is called involute if for any p m there exists a local frame E 1,..., E k for D such that [E k, E l ] is in the span of E!,..., E k. Theorem. (Newlander-Nirenberg) Let (M 2n, J) be an almost complex manifold. Then J is integrable if and only if T 0,1 M is involutive. Proof. The proof of the ( ) direction is beyond the scope of this paper. It can be found in [1] for example. ( ) Let e 1,..., e n, e n+1,..., e 2n denote the standard basis for R 2n. Supposing J is integral, in a given coordinate chart (U, ϕ) = (U, x 1,..., x n, y 1,..., y n ) we have by definition x j = ϕ 1 ( e j ) and y j = ϕ 1 ( ). e n+j Since J = ϕ 1 and so defining J 0 ϕ it follows that J ( x j ) = y j and J ( y j ) = x j z α 1 2 ( x α i y α ) and z α = 1 2 ( x α + i y α ) we see give local bases for T 1,0 M and T 0,1 M respectively. Given Z = Z α z α and W = W α have that [Z, W ] = W β Z α z α W Z β α z α z β, which is a local section of T 0,1 M. n β,α=1 z α we We can use this theorem to give an alternative characterization of the integrability of J. This involves the Nijenhuis tenser, which is the (2, 1)-tensor associated to J defined by N J (X, Y ) = [X, Y ] + J[JX, Y ] + J[X, JY ] [JX, JY ]. Proposition. Let (M 2n, J) be an almost complex manifold. Then J is integrable if and only if N J 0. 6

Proof. We show that N J 0 T 0,1 M is involutive. Let X, Y Γ(T M) be arbitrary. Consider Z = [X + ijx, Y + ijy ]. Direct computation gives Z ijz = N J (X, Y ) ijn J (X, Y ). This proves the claim, as T (0,1 M are precisely the i-eigenvalues of J. We can now prove what we wanted to. Theorem. Let (M, J, h) be a Hermitian manifold. Let denote the Levi-Civita connection with respect to h. Then h is Kähler if and only if J 0 Proof. Let p M and X p, Y p, Z p T p M be arbitrary. Fix a curve γ passing through p and let X, Y, Z be vector fields on M which extend X p and Y p and are parallel at p. Since is torsion free we have [X, Y ] = X Y Y X. Using this together with the fact that JX JY = J( JX Y ) + ( JX J)(Y ) and that X and Y are parallel, we obtain N J (X, Y ) = ( JX J)Y ( JY J)X + J( Y J)X J( X J)Y. Also, by the coordinate free definition of the exterior derivative we have dω(x, Y, Z) = X(ω(Y, Z)) Y (ω(z, X))+Z(ω(X, Y )) ω([x, Y ], Z)+ω([Y, Z], X) ω([z, X], Y ). Since X, Y, Z are parallel, by definition of ω, this reduces to Direct computation then gives that dω(x, Y, Z) = h( X JY, Z) h( Y JZ, X) h( Z JX, Y ). h(n J (X, Y ), Z) = dω(jx, Y, Z) dω(x, JY, Z) + 2h(( Z J)X, JY ). So if h is Kähler, i.e. N J 0 and dω = 0, then by the non-degeneracy of h it follows J 0. Conversely if J 0 then the second last equation shows dω is 0 and then from the last equation it follows N J 0, We have now characterized Kähler manifolds by the property J 0, suggesting that the holonomy principle will be applied. We claim that a Riemannian manifold (M, g) admits a Kähler structure if and only if Hol(g) U(n). However, we have g is a real inner product, and so what does U(n) even mean? First consider R 2n. Let g denote any metric on R 2n. Then we can always find a complex structure J R 2n R 2n such that g(jx, JY ) = g(x, Y ). Indeed, let e 1,..., e 2n denote any g- orthonormal basis and define Je k = e n+k and Je n+k = e k. As usual, J turns R 2n into a complex vector space (i.e. (a+ib)v = av +bjv). The first question to address is, what are the complex linear transformations? We denote this space GL(n; C). We have the correspondence GL(n; C) {S GL(m; R); SJ = JS} A + ib ( A B B A ) This correspondence follows as any matrix that commutes with J must be of the above form, since in the given basis J = ( 0 I ). It is also easy to show that the matrix representation, in this basis, I 0 of any C-linear transformation is of this form. 7

From g and J we can define the form ω, where ω(x, Y ) = g(jx, Y ). From g and ω we can then define the metric h = g + iω. This is a complex inner product which satisfies h(jx, JY ) = h(x, Y ). It thus makes sense to define the unitary group of (R 2n, g) as the subgroup of GL(2m; R) which preserves h. But a linear transformation preserves h if and only if it preserves its real and imaginary parts, namely g and ω. To preserve g means the linear transformation is orthogonal. If this is the case then direct computation gives that the linear transformation preserves ω only if it commutes with J. Conversely, if the linear transformation commutes with J and preserves g then it preserves h. Hence U(n) = GL(n; C) O(2n). We are now ready to prove what we set out too. We state the result as a proposition. Proposition. Let (M 2n, g) be a Riemannian manifold and let be the Levi-Civita connection. Then (M 2n, g) admits a Kähler structure if and only if Hol( ) U(n). Proof. ( ) Fix p M. Suppose that h is a Kähler metric and J is the associated complex structure. Then J 0. Hence by the honomy principle for any P γ Hol p ( ) we have that Pγ J p = J p. We know that J p is a (1, 1)-tensor J p T p M T p M. But it can also be thought of as a map J p T p M Tp M R under the identification By assumption then J p (X p, α p ) = α p (J p X p ). α p (J p X p ) = J p (X p, α p ) = (P γ J p )(X p, α p ) = J p (Pγ 1 X p, Pγ 1 α p ) = Pγ 1 α p (J p Pγ 1 X p ) = α p (P γ J p Pγ 1 X p ) Since this holds for all X p T p M it follows that P γ J p Pγ 1 = J p which shows P γ GL(n; C). But since is the Levi-Civita connection of h we have that h 0 and so P γ h = h. Hence P γ g = g showing P γ O(2n). Thus Hol p ( ) U(n). Conversely, suppose that Hol( ) U(n). This means we ve fixed some p M and J p T p M T p M such that g(j p X, J p Y ) = g(x, Y ). We form the Hermitian metric h p = g p +iω p. By hypothesis, we have that each P γ Hol p ( ) fixes h. That is Pγ J p = J p and Pγ g p = g p. Tholonomy principlethe holonomy principle, we may extend J p and g p to parallel tensors on M. Since we can extend J p and g p we can also extend ω p and h p to parallel tensors ω and h. It follows by construction that h is Hermitian, and since h 0, we have that is the Levi-Civita connection for h. Since J 0 it follows from the above theorem that (M 2n, h, J) is a Kähler manifold. 8

4 The Holonomy of Calabi-Yau Manifolds We now discuss the holonomy of Calabi-Yau manifolds. Recall that a Calabi-Yau manifold is a Kähler manifold (M, g, J) with c 1 (M) = 0. Roughly speaking, Calabi-Yau manifolds are the orientable Kahler manifolds, in the sense that their holonomy group is contained SU(n) as compared to U(n), just like oriented Riemannian manifolds have holonomy contained in SO(n) as compared to O(n). We characterized the holonomy of oriented Riemannian manifolds by extending a nonzero top degree co-vector to a top degree form. We will give an analogous argument here, by constructing a nowhere vanishing holomorphic (n, 0)-form. The characterization of the holonomy of Calabi-Yau manifolds will follow by showing that the condition c 1 (M) = 0 is equivalent to c 1 (Λ n (T (n,0) M))) = 0. To see this, we need some general definitions of characteristic classes. If not stated otherwise, we let π E M be a vector bundle of rank p. Let V be a vector space over K, where K = R or C. Let P V V K be a symmetric map and define P V K by P (v) = P (v,..., v). Then P (λv) = λ k P (v) for any λ K. The map P is said to be homogeneous of degree k. Given any such P, we can find a symmetric map P such that P (v) = P (v,..., v) by using the formula k P (v 1,..., v k ) = 1 P (t1 v 1 + + t k v k ). k! t 1 t k This result follows by direct computation. Now let V = gl(n; K) = M n (K). A homogeneous map of degree k, P, is called invariant if for all Q GL(n, C) and B gl(n; C) we have P (QBQ 1 ) = P (B). Proposition. Let P be a n-linear symmetric map. If P is invariant then n i=1 for all B 1,..., B k, B GL(n; K). P (B 1,..., B i 1, [B, B i ], B i+1,..., B k ) = 0 Proof. Take Q = e tb GL(n; C). Then Q 1 = e tb. By hypothesis, Differentiating with respect to t then gives P (QB 1 Q 1...., QB k Q 1 ) = P (B 1,..., B k ). 0 = d dt P (QB k Q 1 ) = P d dt (etb B k e tb ) = P (Be tb B k e tb e tb B k e tb ). Thus this function is constant and setting t = 0 gives the desired result. Using this proposition we get an induced mapping into the 2k-forms of M. This is described by the following proposition. Proposition. Let P be an invariant n-linear symmetric map on GL(n; K). For any vector bundle π E M of rank p and any partition i 1 + + i p = n, there exists a naturally induced map defined by P Ω i 1 (End(E)) Ω i k (End(E)) Ω 2n K (M) P (ω 1 T 1,..., ω k T p ) = ω 1 P (T 1,..., T p ) for all ω j Ω i j and T j Γ(End(E)). Here Ω i k (End(E)) is denoting the space of functions that take in i k vector fields and spit out a section in Γ(End(E)). 9

Proof. This follows from the above since in any chart (or trivialization) each fibre E x is isomorphic to K p and the fibre End(E) x is isomorphic to GL(n; K). Since P is invariant by assumption, this map is well defined. We will give an example of how this induced map works below, which will also give the first Chern class. Recall that given a connection on E we can form the curvature tensor, F (X, Y )s = X Y s Y X s [X,Y ] s which is in Γ(E), for any section s Γ(E) and vector fields X, Y Γ(T M). That is, F Ω 2 (End(E)). Theorem. (Chern-Weil) Let π E M be a vector bundle. Let denote any connection on E. Then for any k-linear symmetric invariant map P gl(n; K) gl(n; K) K the K-valued 2-form P (F ) is closed, i.e. defines a cohomology class. Furthermore, this cohomology class is independent of the connection. That is, if 1 and 2 are two different connections, then [ P (F 1 )] = [ P (F 2 )]. Proof. A proof can be found in [2]. A particular instance comes from the trace operator. The trace is an invariant homogeneous map of degree 1. That is, tr(qbq 1 ) = tr(b) and tr(λb) = λtr(b). We get the induced mapping tr Ω 1 (End(E)) Ω 2 (M). In particular, tr(f ) is defined by tr p (Fp (X, Y )), which makes sense since F (X, Y ) is just an n n matrix in a local trivialization. The Chern-Weil theory guarantees that tr(f ) is a closed 2-form, whose cohomology class is independent of the chosen connection. We define the first chern class by c 1 (E) = [tr( i 2π F )]. This agrees with the definition given in class, as the Ricci curvature is the trace of the curvature tensor multiplied by. Using this definition, a straightforward computation gives that given two vector bundles π 1 E 1 M and π 2 E 2 M we get c 1 (E 1 E 2 ) = c 1 (E 1 ) + c(e 2 ). Computing the holonomy of Calabi-Yau manifold will rely on the following propositions. Recall that a vector bundle π E M of rank p is called trivial if E M K p. Recall that a line bundle is a vector bundle of rank 1. Proposition. A line bundle L is trivial if and only if it admits a nowhere vanishing section. Proof. Let s Γ(L) be nowhere vanishing. Then for each p M, we have s p is a basis for L p. The map L K L (p, λ) λs p is readily seen to be an isomorphism. Conversely, if L = M K then the map p (p, 1) is a nowhere vanishing section. Proposition. Let π E M be a line bundle and let E denote the dual bundle. Then E E is trivial. Proof. Since E E Hom(E, E), we have that s M E E given by p id p is a nowhere vanishing section. Then the map (p, λ) λs p is an isomorphism of M K onto Hom(E, E). Corollary. With the notation in the previous proposition, c 1 (E) = c 1 (E ). i 2π 10

Proof. This follows since 0 = c 1 (E E ) = c 1 (E) + c 1 (E ). Proposition. Let E = M K p denote the trivial bundle of rank p. A trivial vector bundle has c 1 (M) = 0. Proof. By the Chern-Weil theory, we can choose any connection we like as the cohomology classes are independent of the chosen connection. Let 0 denote the trivial connection on this trivial bundle, which is defined as follows. Since we are working with a trivial bundle, the sections are just p-tuples of functions in C (M). The trivial connection is defined by 0 X s 1 Xs 1 = s p Xs p where s 1,..., s p C (M). It follows immediately from the definition of the curvature tensor that F 0. Hence c 1 (M) = 0. We now are ready to discuss the holonomy of Calabi-Yau manifolds Proposition. Let (M 2n, g) be a Riemannian manifold and let denote the Levi-Civita connection. The M admits a Calabi-Yau structure if and only if Hol( ) SU(n). Proof. Using the same argument as in the Kähler case, we have that SU(n) = SL(n; C) O(2n) where SL(n; C) are the elements of GL(2m; R) which commute with J and have determinate equal to one. Suppose first that Hol( ) SU(n). Since SU(n) U(n), we know that there exists a Kähler structure (M, h, J). Let α p be a nonzero section of the holomorphic n-covectors at p, i.e. α Λ n ((T ) (n,0) p M). Since det P γ = 1, by the exact same argument as in the Kähler case, we may extend α p to a parallel holoomorphic nowhere-vanishing n-form α Λ n ((T ) (n,0) M). But Λ n ((T ) (n,0) M) is a complex line-bundle. Thus by the previous propositions it is trivial. Thus But also the above propositions show that Hence c 1 (Λ n ((T ) (n,0) M)) = 0. 0 = c 1 (Λ n ((T ) (n,0) M) Λ n (T (n,0) M)) = c 1 (Λ n ((T ) (n,0) M)) + c 1 (Λ n (T (n,0) M)). But T M T (n,0) M and so Λ n (T (n,0) M) = 0. c 1 (M) = 0, showing that manifold is Calabi-Yau. Conversely if the manifold is Calabi-Yau then we have a Kähler structure (M, h, J) such that c 1 (M) = 0. The Kähler structure gives that Hol( ) U(n). Applying the argument above backwards, we see that c 1 (Λ n (T (n,0) M)) = 0 11

and so c 1 (Λ n ((T ) (n,0) M)) = 0. But Λ n ((T ) (n,0) M) is a line bundle, and so the above propositions show that we have a nowhere vanishing parallel section α Λ n ((T ) (n,0) M). But then α p is a non-zero element of Λ n ((T ) (n,0) p M). Since α p can be extended only uniquely to a parallel section (i.e. by the assignment α q = P γ α p, where γ is any curve from p to q) it must be that P γ α p = α p for all closed loops γ centred at p. But since α p (Pγ 1 Z 1,..., Pγ 1 Z n ) = (det Pγ 1 )α p (Z 1,..., Z n ) for any P γ Hol p ( ), it must be that det P γ = 1. Hence Hol( ) SU(n). 12

5 References 1. Lectures on Kähler geometry, Andrei Moroianu. http://moroianu.perso.math.cnrs.fr/tex/kg.pdf. 2. Riemannian Holonomy Groups and Calibrated Geometry, Dominic D. Joyce, Oxford Graduate Texts in Mathematics 3. Riemannian Manifolds: An Introduction to Curvature, John M. Lee. Springer Graduate Texts in Mathematics. 13