Complex line bundles. Chapter Connections of line bundle. Consider a complex line bundle L M. For any integer k N, let

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1 Chapter 1 Complex line bundles 1.1 Connections of line bundle Consider a complex line bundle L M. For any integer k N, let be the space of k-forms with values in L. Ω k (M, L) = C (M, L k (T M)) Definition A connection of L is a linear map : Ω 0 (M, L) Ω 1 (M, L) satisfying (fs) = df s + f s. for any section s C (M, L) and function f C (M). For any section s, s is called the covariant derivative of s. Assume L is the trivial bundle M C, so that Ω k (M, L) identifies naturally with Ω k (M). Then, for any β Ω 1 (M), we can define a connection by f = df + fβ, f C (M). Conversely, all the connections of the trivial bundle have this form (set β = 1 and compute (f.1) = df + f 1.). We will see that any connection is locally of this form. First let us prove that a connection is a local operator. Lemme Let be a connection of L M and U be an open set of M. Then for any s C (M, L), if s vanishes on U, the same holds for s. Proof. For any x U, there exists a smooth function f on M with support contained in U and identically equal to 1 on a neighborhood of x. So fs = 0. This implies that 0 = (fs) = df s + f s. Since f(x) = 1 and d x f = 0, this shows that s vanishes at x. As a consequence, given a connection of L, for any open set U of M, we can define a connection U on the restriction of L to U such that ( s) U = U (s U ) for any section s of L. Assume that there exists a local frame s C (U, L), that is a section which does not vanish at any point of U. Then any section t of L restricts to fs over U with f C (U). Furthermore U (fs) = df s + fβ s with β = ( U s)/s Ω 1 (U). In the sequel we will denote U by. 3

2 4 CHAPTER 1. COMPLEX LINE BUNDLES Proposition Let L M be a complex line bundle. The space of connections of L is non empty and given by { 0 + β/β Ω 1 (M)}, where 0 is any connection of L. So it is a complex affine space with associated vector space Ω 1 (M). Proof. To construct a connection, consider an open cover (U i ) i I of M with a family of local frames (s i C (U i, L), i I). Let (ϕ i, i I) be a partition of unity subordinate to this cover. Then the following formula defines a connection s := ϕ i df i s i, s C (M, L) (1.1) where for any i I, f i C (U i ) is given by f i = s/s i on U i. Next, if we have two connections and of L, then ( )(fs) = f( )(s) for any f C (M) and section s of L. This has the consequence that there exists β Ω 1 (M), such that is the multiplication by β. Curvature Given a connection and a vector field X of M, we define the covariant derivative with respect to X by X : C (M, L) C (M, L), X s = s, X. If s is a local frame and β = ( s)/s, then X (fs) = (X.f + β(x))s. Proposition Let be a connection of a complex line bundle L M. Then there exists ω Ω 2 (M) such that for any vector fields X, Y we have X Y Y X [X,Y ] = ω(x, Y ). The form ω is determined by this equation, it is closed and is called the curvature of. Proof. It is sufficient to check the result in a local trivialisation, so that = d + β. Then with ω = dβ. [X + β(x), Y + β(y )] =[X, Y ] + X.β(Y ) Y.β(X) =[X, Y ] + β([x, Y ]) + ω(x, Y ) Observe that if = +β, then the curvatures of and satisfy ω = ω +dβ. So the cohomology class of ω does not depends on the choice of the connection. As the proof of Proposition shows, for any local frame s, β = ( s)/s is a local primitive of ω. The following is a partial converse. Lemme Let β Ω 1 (M) such that dβ = ω. Then for any x M, there is a local frame s defined on a neighborhood U of x, such that s = β s. Proof. Consider any local frame s 0 at x so that s 0 = β 0 and dβ 0 = ω. Since β β 0 is closed, restricting U if necessary, there exists f C (U) such that df = β β 0. Set s = exp(f)s 0 so that s = (β 0 + df) s = β s.

3 1.1. CONNECTIONS OF LINE BUNDLE 5 Flat connection A connection is said to be flat if its curvature vanishes. A flat complex line bundle is a complex line bundle endowed with a flat connection. The sections of a flat line bundle which have a vanishing covariant derivative, are called the flat sections. Let (L, ) be a flat line bundle. Then by Lemma 1.1.5, there exists a flat local frame defined on the neighborhood of each point. Furthermore, if s and s are two flat local frame defined on U, then s = fs where f is locally constant. Conversely, one can define a flat structure on a complex line bundle L as follows. Consider an open cover (U i ) of M and a familly (s i C (U i, L)) of local frames. Assume that for any i and j, s i = f ij s j on U i U j, where f ij is locally constant. Then L has a natural flat connection determined by the condition s i = 0, for any i. Hermitian line bundle Assume now L is a Hermitian line bundle, that is a complex line bundle with a Hermitian metric. Then we say that a connection of L is compatible with the metric if we have d(s, t) = ( s, t) + (s, t), s, t C (M, L). Here we denote by (s, t) C (M), the pointwise scalar product of s and t. Our convention is that the scalar product is linear in the first argument and antilinear in the second. In particular, for any one form α, we have (α s, t) = α(s, t) and (s, α t) = ᾱ(s, t). The curvature of a connection compatible with some metric is imaginary, that is of the form iω with ω Ω 2 (M, R). Furthermore if s C (U, L) is a unitary frame then s = iβ with β Ω 1 (U, R). We can easily generalize Proposition First any Hermitian line bundle L has a compatible connection 0 (Equation (1.1) defines one if the s i s are unitary). Second for any connection of L, is compatible if and only if = 0 + iβ where β is a real form of M. So the space of compatible connections is a real affine space with vector space Ω 1 (M, R). Tensor product, dual, pull-back Assume L and L are two complex line bundle with the same base and endowed with connections,, then the dual L 1 and the tensor product L L have natural connections determined by the following equations: (s 1 ) = s, (s s ) = s s + s s for any local frames s, s of L and L respectively. To check that these connections are well-defined, one has to consider change of local frame, this is left to the reader. If the curvature of and are ω and ω, then the curvatures of the connections of L 1 and L L are respectively ω and ω + ω. Similarly if f is a smooth map from M to N and L N is a line bundle with a connection, then f L has a natural connection determined by (f s) = f ( s) for any local frame of L. If the curvature of is ω, the curvature of f is f ω. In all these construction, if the initial connections are flat, we obtain flat connections. If the initial connections are compatible with Hermitian metrics, the connections obtained are compatible with the induced metric.

4 6 CHAPTER 1. COMPLEX LINE BUNDLES 1.2 Connections of T-principal bundles Let T = R/Z. Recall that a T-principal bundle consists of a fiber bundle π : P M with typical fiber T and for any x M, an action of T on P x which is free and transitive. Furthermore, these data must satisfy the following condition: for any x 0 M, there exists a neighborhood U of x 0 and a diffeomorphism ϕ : π 1 (U) U T, such that ϕ(u) = (π(u), τ(u)) where τ : π 1 (U) T satisfy τ(θ.x) = θ + τ(x). Such a map ϕ is called a local trivialisation. Let π : P M be a T-principal bundle. For any θ T, we denote by L θ : P P the action of θ. Introduce the vector field θ of M given by θ y = d L t (y), y P. dt t=0 The flow at time t of θ is L t. Observe also that for any y P, we have an exact sequence 0 θ y R T y P Tyπ T π(y) M 0. (1.2) The vectors in the kernel of T π are called the vertical tangent vectors. Definition A connection of a T-principal bundle π : P M is a 1-form α Ω 1 (P, R) which is T-invariant, i.e. L θα = α for any θ T, and satisfies ι θ α = 1. A connection α determines a splitting of the sequence (1.2) T y P = ker α y ker T y π. The subspace ker α y of T y P is called the horizontal subspace. Since α is T-invariant, the distribution ker α is also T-invariant. Conversely, for any T-invariant subbundle E of the tangent bundle of P such that T P = E ker T π, there is a unique connection α satisfying E = ker α. For any vector field X of M, there exists a unique vector field X hor of P which is horizontal, i.e. α(x hor ) = 0, and such that T π(x hor ) = X. X hor is called the horizontal lift of X. Observe also that any T invariant vector field Y has the form Y = X hor + α(y ) θ, where X is a vector field of M such that T π(y ) = X. The differential forms of P which descend to M, in the sense that they are the pull-back by π of a form of M, are called the basic forms. We have the following characterization. Lemme Let γ Ω k (P ). Then there exists γ Ω k (M) such that π γ = γ iff L θγ = γ, θ T and ι θ γ = 0. Furthermore, if this condition is satisfied, γ is unique. Since the result is important and we need in the sequel a generalisation for any Lie group, we give a detailed proof.

5 1.2. CONNECTIONS OF T-PRINCIPAL BUNDLES 7 Proof. If k = 0, so that γ and γ are functions, the results follows from the fact that the orbits of the T-action are the fibers of the projection π. Consider now γ Ω k (P ) with k N, we look for a form γ such that γ π(y) (T y π(y 1 ),..., T y π(y k )) = γ y (Y 1,..., Y k ) for any y P and (Y 1,..., Y k ) T y π. Since π is a surjective submersion, γ is completely determined by this equation. To show the existence, we have to prove γ y (Y 1,..., Y k ) = γ y (Y 1,..., Y k) if π(y ) = π(y) and T y π(y i ) = T yπ(y i ) for any i. These conditions imply first that y = θ.y for some θ and then using the exact sequence (1.2) and π L θ = π, Finally L θ γ = γ and ι θ γ = 0 imply T y L θ (Y k ) = Y k mod θ y. γ y (Y 1,..., Y k ) =γ y (T y L θ (Y 1 ),..., T y L θ (Y k )) =γ y (Y 1,..., Y k). Unfortunately, this proof doesn t show the form γ is smooth. To check that, it is convenient to work in a local trivialisation U T with coordinates x 1,..., x n on U. Write γ = dθ α + β with α = α I dx I, β = β I dx I. I =k 1 I =k Here I runs over the subsets of {1,..., n}, dx I = dx i1... dx il if I = {i 1 <... < i l }, α I and β I are functions on U T. Observe that ι θ γ = 0 if and only if α = 0. Furthermore, γ is T-invariant if and only if the coefficients α I and β I are T-invariant. If these conditions hold, we have γ = β, which is smooth. The connections of the trivial T-principal bundle M T are the 1-forms of the form β +dθ, where β Ω 1 (M, R) and dθ is the 1-form of T = R/Λ which lifts to the differential of the identity map of R. More generally, for any T-principal bundle, we have the following Proposition Let π : P M be a T-principal bundle. Then the space of connections of P is not empty and given by {α 0 + π β/ β Ω 1 (M, R)}, where α 0 is any connection of P. So it it a real affine space with associated vector space Ω 1 (M, R). Proof. To show the existence of a connection, consider an open cover (U i ) i I of M with a familly of local trivialisations ϕ i : π 1 (U i ) U i T. Let (f i ) be a partition of unity subordinate to the cover (U i ). Then α 0 := i I (π f i )ϕ i dθ is a connection of P. Given any two connections α and α of P, their difference is basic by Lemma 1.2.2, so α α = π β for some β. Proposition Let α be a connection of P. Then there exists a unique 2-form ω Ω 2 (M, R) such that π ω + dα = 0. For any vector fields X and Y of M, one has [ X hor, Y hor] = [X, Y ] hor + π (ω(x, Y )) θ

6 8 CHAPTER 1. COMPLEX LINE BUNDLES ω is called the curvature of the connection. Proof. To prove the existence of ω we have to show that dα is basic. Since α is T-invariant, the same holds for dα. Furthermore the Lie derivative of α with respect to θ vanishes, so that by Cartan s formula ι θ dα + dι θ α = 0 which implies that ι θ dα = 0. Hence by Lemma 1.2.2, dα is basic. Since X hor and Y hor are T-invariant, the same hold for [ X hor, Y hor]. Furthermore T π([x hor, Y hor ]) = [X, Y ], so that [ X hor, Y hor] = [X, Y ] hor + α([x hor, Y hor ]) θ. Since α(y hor ) = 0, we have 0 =L X horα(y hor ) =(L X horα)(y hor ) + α([x hor, Y hor ]) =dα(x hor, Y hor ) + α([x hor, Y hor ]) where in the last line, we have used Cartan formula and α(x hor ) = 0. We conclude by using dα + π ω = 0 so that dα(x hor, Y hor ) = π ω(x, Y ). 1.3 Line bundles and T-principal bundles As a preliminary, let us compare the Hermitian lines and the T-principal homogeneous spaces. A Hermitian line is a 1-dimensional vector space with an Hermitian scalar product. A T-principal homogenous space is a space acted on by T in a free and transitive way. Given a Hermitian line L, B(L) = {u L/ u = 1} is a T-principal homogeneuous space, the action being given by θ.u = e 2iπθ u. Conversely, given a T- principal homogeneous space B, the quotient L(B) = B C/T, where T acts by θ.(u, z) = (θ.u, e 2iπθ z), is a Hermitian line. These constructions are functorial. In particular, if we have an isomorphism ϕ : L L of Hermitian lines, it restricts to an isomorphism B(ϕ) : B(L) B(L ) of T-principal homogeneous spaces. Similarly, for any isomorphism ϕ : B B, we get an isomorphism L(ϕ) : L(B) L(B ) sending [u, z] into [ϕ(u), z]. Furthermore we have natural isomorphisms L(B(L)) L, [u, z] zu B B(L(B)), u [u, 1]. We can apply these constructions to the fibers of the Hermitian line bundles or T-principal bundle. In this way, given a Hermitian line bundle L, we obtain a T-principal bundle B(L) = x B(L x ) = {u L/ u = 1} with the same base. And given a T-principal bundle P, we obtain a Hermitian line bundle L = B C/T with the same base. As a consequence of the previous considerations, the map which sends L into B(L) induces a bijection between the isomorphism classes of Hermitian line bundles and the isomorphism classes of T- principal bundles. The inverse bijection is induced by the map sending P into L(P ). Here all the isomorphism are assumed to lift the identity of the base.

7 1.4. PROJECTIVE SPACE 9 Let us explain the correspondance between the connections. Consider a Hermitian line bundle L with base M and its associated T-principal bundle P = U(L). Observe that there is a natural isomorphism of C (M)-module E : C (M, L) { g C (P )/g(θ.u) = e 2iπθ g(u) } sending the section s into the function g such that g(u)u = s(x) for any x M and u B x. To any connection α of P, we associate the connection of L such that the covariant derivative with respect to a vector field X corresponds to the Lie derivative with respect to the horizontal lift of X: E( X s) = L X hore(s), s C (M, L). We claim that the map sending α into is a bijection from the set of connections of P to the set of connections of L. To check this, it is convenient to work locally with trivialisations. Let s C (U, L) be a unitary frame and ϕ : P U U T be the corresponding trivialisation of P given by ϕ(θ.s(x)) = (x, θ). Then there exists β Ω(U, R) such that α = ϕ ( β + dθ), (fs) = (df + 2π ) i β s. (1.3) So this is the same 1-form which determines locally α and. Observe also that if ω is the curvature of α, the curvature of is 2π i ω. Proof of Equation (1.3). Identify C (U, L) with C (U) by sending fs into s, and C (P U ) with C (U T) using ϕ. Then E(f) = g with g(x, θ) = f(x)e 2iπθ Furthermore since α = β + dθ, X hor = X + β(x) θ so that which prove Equation (1.3). 1.4 Projective space X hor.g = (X.f 2iπβ(X)f)e iθ. As a first non trivial example, consider for any positive n, the tautological bundle T of the projective space CP n, T = {([y], u), u Cy} CP n C n+1. The associated T-principal bundle P is naturally isomorphic with the sphere S 2n+1 of C 2n+1 through the map sending ([y], u) into u. The projection π : S 2n+1 CP n is the Hopf fibration. Lemme The T-principal bundle π : S 2n+1 CP n admits as a connexion α = i 4π j=0,...,n (y j dȳ j ȳ j dy j ). Proof. Since the action is given by θ.(y 0,..., y n ) = (e 2iπθ y 0,..., e 2iπθ y n ), α is clearly T-invariant. Furthermore θ is the restriction to S 2n+1 of 2iπ ( ) y j ȳ j y j ȳ j so that α( θ ) = y j ȳ j = 1.

8 10 CHAPTER 1. COMPLEX LINE BUNDLES The curvature ω of α in the k-th chart of CP n {[y 0 :... : y n ]/ y k 0} C n (z 0,..., ẑ k,..., z n ), z j = y j y k if j k is given by ω = 1 (1 + z 2 ) dz j d z j ( z j dz j ) ( ) z j d z j 2iπ (1 + z 2 ) 2 (1.4) where z 2 = z j 2 and all the sums are over j {0,..., ˆk,... n}. Proof of Equation (1.4). Introduce the local section s = ( 1 + z 2) 1/2 (z0,..., z k 1, 1, z k+1,..., z n ). Then s y k is real so that s (y k dȳ k ȳ k dy k ) = 0. For j k, let us compute s (y j dȳ j ) z jd z j + a real-valued form 1 + z 2 so that and s (y j dȳ j ȳ j dy j ) = z jd z j z j dz j 1 + z 2 ds (y j dȳ j ȳ j dy j ) = dz ( j d z j d z j dz j zj d z j z j dz j ) d z z 2 + (1 + z 2 ) 2. By summing over j k and mulitplying by i/4π, we obtain the formula (1.4). Lemme Let j be the embedding of CP 1 into CP n sending [y 0 : y 1 ] into [y 0 : y 1 : 0 :... : 0]. Then CP 1 j ω = 1. Here the orientation of CP 1 is the one determined by the complex structure, that is for any holomorphic cordinate v, idv d v is positive. Proof. We have in the first chart of CP 1 that so j ω = 1 dz d z 2iπ (1 + z 2 ) 2 j ω = CP 1 [0, ) 2rdr (1 + r 2 ) 2 = 1.