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Home , Characteristic class, Chern class, Euler class

CHERN CLASSES Juan Moreno

§1. INTRODUCTION

A associates to a ω (E,π,B) a class α H∗(B), which = ω ∈ respects bundle maps, that is, if f : ω η is a bundle map, then f ∗α α . Chern classes, denoted → η = ω ck(ω) for k N, are characteristic classes associated to complex vector bundles which satisfy the ∈ following properties:

0 2k • c0(ω) 1 H (B), ck(ω) H (B), k N. = ∈ ∈ ∀ ∈ n • Let C denote the trival rank n complex vector bundle. Then ck(E) 0, k 0. = ∀ >

• If rank(ω) n then ck(ω) 0 for all k n. = = > P • Let c(ω) denote the formal sum i ci(ω) called the total . Then we have the Whitney Product formula c(ω η) c(ω)c(η). ⊕ = One can give an axiomatic definition of Chern classes, however, we will instead focus on definitions which are more tractible for computations, and give intuition as to what the Chern classes measure. We will give two different perspectives on Chern classes: the first is uses an auxilliary characteristic class, while the second uses the of a connection. These notes do not present a full intro- duction to Chern classes. In particular, we do not provide any proofs of these important properties described above. The purpose of these notes is simply to give explicit constructions of Chern classes so that one can compute them at least for simple examples. For a detailed account of Chern classes and the theory of characteristic classes in general see [Sta74]. In order to construct characteristic classes we must fix a particular cohomology theory to work with. In this note, we work exclusively with . For an introduction to the theory of differential forms and de Rham cohomology see [Pol10] or [Tu95]

§2. CHERNCLASSESVIATHE EULERCLASS

§2.1. ORIENTATIONSAND POINCARÉDUALITY

Let V be a real vector space of finite dimension n. We begin with a quick review of orientations.

Definition 1. An orientation on V is an equivalence class of basis b : Rn V, where two bases n → b, b0 : R V are equivalent if and only if there is an element A GLn(R) such that b0 b A and → ∈ = ◦ detA 0. > Remark 1. Note that there are precisely two possible orientations on a real vector space V corre- sponding to the two connected components of GLn(R). In fact the equivalence above could be re- stated as follows. Two bases b, b0 are equivalent if and only if there is a path γ : I GLn(R) such → that γ(0) In and b γ(1) b0. Equivalently, if B(V) is the space of all bases of V,then B(V) is = ◦ = GLn(R)-torsor, and we can topologize it as such. Then an orientation is simply a choice of connected component of B(V).

1 Let M be a closed1 smooth n-.

Definition 2. An orientation of M is a choice of orientation [bx] of Tx M for each x M subject to ∈ the following compatibility condition: For each x M there is a neighborhood U of x and a diffeomor- ∈ phism n 1 b : U R π− (U) TM × → ⊂ such that [b(x,_)] [bx]. = Recall that choosing an orientation on a manifold allows one to integrate differential forms over the manifold. Now if M is a closed oriented manifold, Stokes’ Theorem implies that integration over M descends to cohomology. The proof is a simple computation: Z Z Z Z Z Z Z Z n n 1 ω dη ω dη ω η ω η ω, ω Ω (M),η Ω − (M). M + = M + M = M + ∂M = M + = M ∈ ∈ ; There results a bilinear pairing Z k n k : H (M) H − (M) R. M ⊗ → Z ω η ω η ⊗ 7→ M ∧ Theorem 1. (Poincaré duality)[Tu95] This pairing is nondegenerate, implying n k k H − (M) (H (M))∗. =∼ Let i : S , M be an embedded oriented submanifold of M of dimension k such that i(S) is closed → k R (as a subspace) in M. This determines a linear functional H (M) R given by [ω] S i∗ω. By n k → 7→ Poincaré duality, this corresponds to a unique element [ηS] H − (M) which is characterized by ∈ Z Z k i∗ω ω ηS, ω H (M). S = M ∧ ∀ ∈

This element [ηS] is called the Poincaré dual to the submanifold S.

§2.2 THE EULERCLASSOFASMOOTHORIENTEDVECTORBUNDLE

Recall that y N is a regular value of a smooth map f : M N of if d fx has full rank for 1 ∈ → all x f − (y). The regular value theorem tells us that this condition guarantees that the solution set ∈ 1 {x M f (x) y} f − (y) is a submanifold of M. The definition that follows generalizes this notion by ∈ | = = giving us a sufficient condition for which the solution set {x M f (x) Z} for some submanfold Z N ∈ | ∈ ⊂ is a submanfiold of M. Definition 3. Let Z N be a submanifold and f : M N a smooth map. We say that f is transverse 1 ⊂ → to Z if for all x f − (Z), ∈ d fxTx M T f (x)Z T f (x)N. + = 1 Proposition 1. [Pol10] If f : M N is a smooth map transverse to a submanifold Z N, then f − (Z) → 1 ⊂ is a submanifold of M. Furthermore, the of f − (Z) in M is the same as the codimension of Z in N. Definition 4. Let π : E M be a smooth vector bundle of rank r, and Z(E) E the image of the zero → ⊂ . A section s : M E is called generic if it is transverse to Z(E). The of E, denoted r → 1 e(E) H (M) is the Poincare dual to s− (Z(E)), the zero-set of a generic section, s. ∈ Remark 2. There are more general definitions of the Euler class for oriented non-smooth vector bundles, but this one is convenient in our setting. 1compact & boundaryless

2 §2.3 CHERNCLASSES

We now give an inductive definition of Chern classes starting with the Euler class. Let ω (E,π, M) = be a complex vector bundle. Forgetting the complex structure gives an underlying real vector bundle, which we denote by ωR.

Proposition 2. The underlying real vector bundle of a complex vector bundle has a canonical orien- tation.

Proof. First, let V be a complex vector space and VR its underlying real vector space. Any complex basis v1,...,vn gives rise to an ordered real basis v1, iv1,...,vn, ivn which corresponds to the orien- tation on VR. This orientation is independent of the original basis of V since GLn(C) is connected. For a general complex vector bundle we can apply this construction fiberwise and get compatible orientations, again because GLn(C) is connected.

Let rank(ω) k. We define the top Chern class of ω, ck(ω), to be the Euler class of its underlying = 2k real vector bundle, e(ωR) H (M). Now let B0 E \ Z(E), where Z(E) is the zero section of ω. There ∈ = is a natural vector bundle over B0 defined as follows. First, choose a Hermitian metric on ω. This choice of metric is immaterial and simply used for conveniece. Now any point of B0 is a pair (x,v) 1 with x M and v Ex π− (x), so we may take the fiber above (x,v) to be v⊥ Ex, the perpendicular ∈ ∈ = ⊂ subspace of E. This defines a vector bundle ω0 (E0,π0,B0) of rank k 1. As such, it’s top Chern class 2(k 1) = − 2(k 1) ck 1(ω0) H − (B0) is well defined, however, we wish to get a class in H − (B). For this we need − ∈ the following fact [Sta74]: Let ω (E,π,B) be an oriented vector bundle of real dimension n, and B0 E \ Z(E). Then there = = is an

i e i n π∗ i n i 1 H (B) ∪ H + (B) H + (B0) H + (B) ··· → −−→ −→ → → ··· called the Gysin sequence. The map π∗ in the above sequence is simply the pullback via the restriction of π to B0 E. Going ⊂ back to our situation above, since Hi(B) 0 for i 0, we have that = < i 2k i 2k π∗ : H + (B) H + (B0) → 2(k 1) 2(k 2) is an isomorphism for i 1 so that π∗ : H − (B) H − (B0) is an isomorphism. At last, we can < − → define 1 1 ck 1(ω) (π∗)− (ck 1(ω0)) (π∗)− (e(ωR0 )). − = − = We can then repeat this procedure. At the ith step we get a vector bundle ω(i) of rank 2(k i) over (i) (i 1) (i 1) − B E − \ Z(E − ), as well as a sequence of isomorphisms = (i 1) 2(k i) π∗ 2(k i) (π0)∗ 2(k i) (π − )∗ 2(k i) (i) H − (B) H − (B0) H − (B00) H − (B ). −→ −−−→ → ··· −−−−−→ (i) (i) We can then define c2(k i)(ω) to be the inverse image of c2(k i)(ω ) e(ωR ) under these isomor- − − = phisms.

Exercise 1. Suppose ω is a rank n complex vector bundle which admits k linearly-independent sections. Show that cn(ω) cn 1(ω) cn k 1(ω) 0. = − = ··· = − + =

3 §3. CHERNCLASSESVIACONNECTIONS

The perspective on Chern classes of the previous section had the advantage of being manifestly a measure of the non-triviality of a complex vector bundle: If we could find a nowhere vanishing section, then the top Chern class vanishes by definition. Similarly, from Exercise 1 we have that if we have k linearly independent nonvanishing sections then ci 0 for i n, n 1,... n k 1. The = = − − + down-side is that if I want an explicit form representing a Chern class then I have to figure out how to represent a Poincaré dual. In this section, we show how the additional structure of a connection on a vector bundle can be used to find an explicit form representing a Chern class. We begin with a quick review of the geometric notions that we will need. Let ω (E,π, M) be a = smooth complex vector bundle of rank k, and let T∗MC denote the complexified cotangent bundle of M, explicitly T∗MC HomR(T∗M,C). = Definition 5. A connection on ω is a C-linear map

: Γ(ω) Γ(T∗MC ω) ∇ → ⊗ satisfying the Leibniz rule:

(f s) f (s) d f s, f C∞(M,C), s Γ(ω). ∇ · = ∇ + ⊗ ∀ ∈ ∈ Here Γ denotes the space of smooth sections of a given vector bundle.

Note that for any s Γ(ω), the value of (s) at x M only depends on the value s on an arbitrarily ∈ ∇ ∈ small neighborhhood of x. To see this, take f to be a bump around x. Then d fx 0 and = f (x) 1 implying, = (f s)(x) f (x) (s)(x) d fx s(x) (s)(x). ∇ · = ∇ + ⊗ = ∇ Since f can be taken to have support arbitrarily close to x, this shows that (s)(x) only depends on ∇ the restriciton of s to an arbitarily small neighborhood of x, as desired. Because of this, it makes sense to talk about the restriction of a connection to an open neighborhood U of M. Now if U is a k trivial neighborhood of ω so that there exist a basis of sections {si}i 1, then one can use the Leibniz = rule to check that U is completely determined by U (si) for i 1,..., k. Furthermore, ∇| ∇| = X (si) ηi j s j, ∇ = j ⊗ where [ηi j] is a uniquely determined matrix of 1-forms on M. The following result, whose proof can be found in [Sta74], implies that the methods of this section can be applied to any vector bundle.

Proposition 3. Any complex vector bundle admits a connection.

A connection gives us a way of differentiating sections of vector bundles along vector fields. It ∇ turns out that this automatically determines a ’second derivative’:

2 ˆ : Γ(T∗MC ω) Γ(Λ T∗MC ω) ∇ ⊗ → ⊗ given by θ s dθ s θ (s). ⊗ 7→ ⊗ − ∧ ∇ Furthermore, for f C∞(M), ∈ ˆ (f (θ s)) d f (θ s) f ˆ (θ s). ∇ ⊗ = ∧ ⊗ + ∇ ⊗

4 Let K denote the composition ˆ . One can show that, as before, if U is a trivial neighborhood of ω k ∇ ◦ ∇ with basis of sections {si}i 1, then K U is completely determined by K U (si). Furthermore, = | | X K U (si) Ωi j s j, | = j ⊗ where [Ωi j] is now a matrix of 2-forms on M, whose components are given explicitly by X Ωi j dηi j ηil ηl j. = − l ∧

2 Proposition 4. The map K : Γ(ω) Γ(Λ T∗MC ω) is C∞(M)-linear. → ⊗ Proof. This follows from the following computation:

K(f s) ˆ ( (f s)) ˆ (d f s f (s)) d f (s) ˆ (f (s)) d f (s) d f (s) f K(s) f K(s). = ∇ ∇ = ∇ ⊗ + ∇ = − ∧ ∇ + ∇ ∇ = − ∧ ∇ + ∧ ∇ + =

Now if we look at the intersection of two trivializing neighborhoods of ω, U,U0 with bases si and s0i. P We may write s0 ai j s j, for some ai j C∞(M). The preceding proposition then implies that the i = j ∈ components of the matrix [Ωi j] transform simply as the components of a matrix under conjugation 2 by an element of GLn(C). Indeed, one can compute

X X X X XX XXX 1 K(s0i) K( ai j s j) ai jK(s j) ai j Ω jk sk ai jΩ jk sk ai jΩ jk[a− ]kl s0k, = j = j = j k ⊗ = j k ⊗ = j k l ⊗

1 where [a− ]kl are the components of the inverse matrix of [a]. It therefore, makes sense to take the 2 trace of K, TrK Ω (M) C, which locally is just the trace of the matrix of 2-forms Tr[Ωi j]. It’s value ∈ ⊗ at a given point is independent of the choice of coordinates since the trace of a matrix is invariant under conjugation. Similarly, letting K i be the i-fold product of K, which locally is an i-fold product of matrices of 2-forms, we can take the trace TrK i.

Exercise 2. Show that this does not hold for the connection matrix of 1-forms. Specifically, show that under coordinate transformations given by a matrix (ai j) as above,

X¡ 1 X 1 ¢ ηi j η0i j dail[a− ]l j aikηkl[a− ]l j . 7→ = l + k

Definition 6. Let ω (E,π, M) be a smooth complex vector bundle with a connection and = ∇ K . Then the kth Chern class of ω is defined to be ∇

¡ i ¢k£ k ¤ 2i ck(ω) Tr(K ) H (M). = 2π ∇ ∈ Of course, there are many things to check here including independence of the chosen connection and that the resulting class is in the real de-Rham cohomology of M. For this one is refered to [Sta74]. So we have an explicit description of the Chern classes of a complex vector bundle in terms of a connection. The problem is then to define a connection on our vector bundle. It turns out to be easier to define a metric first.

2Here we use the fact that 2 2 Γ(Λ T∗MC C ω) Γ(Λ T∗MC) Γ(ω). ⊗ = ⊗C∞(M)

5 Definition 7. A Hermitian metric on a complex vector bundle ω (E,π, M) is a section , of Hom(ω = 〈 〉 ⊗ ω), C), such that for each x M, , x is a Hermitian inner product on Ex. ∈ 〈 〉 Example 1. The trivial complex n-plane bundle Cn has a canonical metric, which in each fiber is the n n canonical Hermitian metric on C . Namely, for zx,wx C , ∈ x

zx,wx x zx wx. 〈 〉 = · Definition 8. Let ω be a smooth complex vector bundle with a Hermitian metric, , . A connection, 〈 〉 , on ω is compatible with the metric if ∇ d z,w z,w z, w , z,w Γ(ω). 〈 〉 = 〈∇ 〉 + 〈 ∇ 〉 ∀ ∈ Remark 3. One can show that being compatible with a metric is a local property given by the re- quirement that the local connection matrix, [η], be skew-Hermitian, i.e. ηi j η . = − ji Let ω (E,π, M) be a smooth complex vector bundle with Hermitian metric, and let U M be an = ⊂ open set over which M has local coordinates {xk} and ω has an orthonormal frame {si}. We claim that if M has a cover by such open sets, then X Ai j si,∂ks j dxk = k 〈 〉 defines a local connection matrix for a connection on ω. To show this it suffices to check that Ai j ∇ transforms as such a local connection matrix.

Exercise 3. Prove this. (Hint: You will use the fact that the matrix [a] whose components are ai j C∞(M) is unitary. It will also be useful to recall the chain rule.) ∈ Exercise 4. Show that the resulting connection is compatible with the given metric. (Hint: Use remark 3.) P Now suppose rankω 1. Then the local connection matrix has a single component A s,∂ks dxk = = k〈 〉 for a local nonvanishing section s. The local curvature matrix then also consists of the single compo- nent Ω dA A A dA. = + ∧ = The following computation then gives an explicit description for Ω in terms of s:

X X X 2 dA d s,∂ks dxk d(s ∂ks)dxk ds ∂ksdxk s d∂ksdxk ds, ds s, d s ds, ds . = k 〈 〉 = k · = k · + · = 〈 〉 + 〈 〉 = 〈 〉

§4. EXAMPLECOMPUTATIONS

Let σi for i x, y, z, denote the Pauli matrices and write ~σ for the 3-dimensional ‘vector’ whose entries = are these matrices. Explicitly,

µ0 1¶ µ0 i¶ µ1 0 ¶ σx , σy − , σz . = 1 0 = i 0 = 0 1 − Consider the trivial complex bundle C2 over S2. The bundle EndC2 over S2 is then isomorphic to the trivial bundle M2(C). The map 2 H S M2(C), −→

6 ~x I ~x ~σ 7→ ~x + · 2 2 defines a global section of M2(C), and hence a bundle map C C , which we will also denote by H. It → is an exercise in linear algebra to show that H~x is a Hermitian operator for each ~x with eigenvalues 0,1. Let K~x denote the 0-eigenspace of H~x. The union G K K~x =~x S2 ∈ is a vector subbundle of C2. To see that this is so, it suffices to note that the kernel of I ~x ~σ is ~x + · a constant dimension [And89]. Alternatively, we can exhibit nonvanishing sections within K locally around any point of S2. On U S2 \{(0,0,1)} we can take the section =

ξ µ z 1 ¶ ~x − , 7−→ x iy + and on V {~x S2 z 0} we can take the section = ∈ | > Ã z(x iy) ! s − − ~x z 1 . 7−→ +z

So we have a complex , K, over S2. We will compute the first Chern class first as the Euler class as in §2 and then using a connection as in §3. To compute the first Chern class as the Euler class we need a section that is transverse to the zero section. Consider ξ : S2 K given as above and extended to be 0 on {(0,0,1)}. This is evidently a → smooth section which intersects the zero section at a single point, z 1. To check that this intersec- = tion is transverse we use the local trivialization at z 1 given by s: = sˆ 1 C π− (v). −→ (v,~x) vs(~x) 7→ Under this trivialization, we have 1 ³ x iy ´ sˆ− ξ(~x) + ,~x . ◦ = z x iy For f (~x) + , = z ³ 1 i x iy ´ d f + , ~x = z z z2 has full rank at ~x (0,0,1), so ξ is indeed transverse to the zero section. It follows that c1(K) is = 2 Poincaré dual to {(0,0,1)} S , from which we can deduce that c1(K) 0 since ⊂ 6= Z Z Z c1(K) 1 c1(K) 1 1. S2 = S2 · = (0,0,1) =

Now we provide an explicit form representing c1(K). First we must specify a choice of connection. As in Example 1 above, we have a natural Hermitian metric on C2, which descends to a metric on K C2. We may then take the connection defined at the end of §3. Now since ξ gives us a nonzero ⊂ section over U, we can use the normalized unit section ξˆ ξ to obtain a form representing the first = ξ Chern class as || || i dξˆ, dξˆ c1(K) 〈 〉. = 2π

7 Remark 4. Although this form is technically only defined on S2 \{(0,0,1)}, since {(0,0,1)} has measure zero in S2 this won’t affect calculations of integration. It is useful to rewrite ξˆ in spherical coordinates

µ θ ¶ ˆ sin 2 ξ − θ iφ . = cos 2 e Then µ 1 cos θ dθ ¶ dξˆ − 2 2 = 1 sin θ eiφdθ i cos θ eiφdφ − 2 2 + 2 i θ θ i θ θ 1 i dξˆ, dξˆ i£ sin cos dθ dφ cos sin dφ dθ¤ sinθdθ dφ. =⇒ 〈 〉 = − 4 2 2 ∧ − 4 2 2 ∧ = 2 ∧ Thus sinθ c1(K) dθ dφ, = 4π ∧ R 2 and we can then verify that 2 c1(K) 1. Using the metric on C , we can decompose this trivial S = bundle into 2 C K K⊥. = ⊕ Then, by the Whitney product theorem,

2 1 c(C ) c(K)c(K⊥) c1(K⊥) c1(K). = = =⇒ = −

REFERENCES

[And89] Michael Francis Atiyah & D. W. Anderson. K-theory. Pennsylvania State University, 1989. [Pol10] Victor Guillemin & Allan Pollack. Differential . American Mathematical Society, 2010. [Sta74] John W. Milnor & James D. Stasheff. Characteristic Classes. Princeton University Press, 1974. [Tu95] & Loring W. Tu. Differential Forms in . Springer New York, 1995.

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