Principal Bundles - Day 4: Chern Classeszykoskib/day4.pdf · 2020. 6. 18. · Principal Bundles Day...

Principal BundlesDay 4: Chern Classes

Bradley Zykoski

June 18, 2020

Bradley Zykoski Chern Classes June 18, 2020 1 / 17

Summary of Day 3

Given a principal G -bundle π : P B with a connection ω ∈ Ω1(P, g),the horizontal projection h : TP HP = kerω gives a covariant exteriorderivative dωη = (dη) h⊗k+1 for η ∈ Ωk(P, g).

We thereby defined the curvature of the connection to be

Ω = dωω.

We have seen that the curvature:

satisfies Ω = dω + 12 [ω, ω]∧ (Cartan),

satisfies dωΩ = 0 (Bianchi),

vanishes if and only if HP is integrable,

obstructs(Ω∗hor(P, g)G , dω

)from being a cochain complex.

models the electromagnetic field when B = R4, G = U(1).

Summary of Day 3

Ω = dωω.

Summary of Day 3

Ω = dωω.

Summary of Day 3

Ω = dωω.

Summary of Day 3

Ω = dωω.

Summary of Day 3

Ω = dωω.

Summary of Day 3

Ω = dωω.

The first Chern class

Recall that a connection splits the exact sequence

0 VP TP π∗TB 0,dπ

and so we have an isomorphism dπ : kerω = HP∼−→ π∗TB. Let b ∈ B.

Then for p ∈ π−1(b) and Xb ∈ TbB, there is a unique Xp ∈ HpP with

dπ(Xp) = Xb.

Definition

Let π : P → B be a principal U(1)-bundle with connection ω. Then wedefine Fω ∈ Ω2(B, u(1)) via

Fω(Xb,Yb) = Ω(Xp, Yp)

for any b ∈ B and an arbitrary choice of p ∈ π−1(b).

Observe that this gives Ω = π∗Fω.

dπ(Xp) = Xb.

Definition

Observe that this gives Ω = π∗Fω.

dπ(Xp) = Xb.

Definition

Observe that this gives Ω = π∗Fω.Bradley Zykoski Chern Classes June 18, 2020 3 / 17

F (Xb,Yb) = Ω(Xp, Yp) ∀b ∈ B, p ∈ π−1(b)

Recall that since our connection is principal, we have Hp.gP = dRg (HpP),

and hence Xp.g = dRg (Xp) for every g ∈ U(1).

Therefore

Ω(Xp.g , Yp.g ) = Ω(dRg (Xp), dRg (Yp)) = Adg−1 Ω(Xp, Yp) = Ω(Xp, Yp),

where the final equality follows since U(1) is abelian, and henceAdg−1 = Idu(1) for every g ∈ U(1). We conclude that Fω is independent ofthe choice of p ∈ π−1(b).

Now observe that

dFω(X ,Y ) = dΩ(X , Y ) = dΩ(h(X ), h(Y )) = dωΩ(X , Y ) = 0,

where the final equality follows from the Bianchi identity.

and hence Xp.g = dRg (Xp) for every g ∈ U(1). Therefore

Now observe that

and hence Xp.g = dRg (Xp) for every g ∈ U(1). Therefore

Now observe that

We have now shown that Fω(X ,Y ) = Ω(X , Y ) is a well-defined closedu(1)-valued 2-form on B. But u(1) = R, and so this is just an ordinaryclosed 2-form. We conclude that we may give the following definition.

Definition (First Chern class)

Let π : P → B be a principal U(1)-bundle with connection ω. We definethe first Chern class of P by

c1(P) = [Fω] ∈ H2(B;R).

A homotopy argument shows that c1(P) does not depend on the choice ofω, hence our notation.

c1(P) = [Fω] ∈ H2(B;R).

Theorem

Let π : P → B be a principal U(1)-bundle with connections ω0, ω1. Thenwe have [Fω0 ] = [Fω1 ] ∈ H2(B;R).

Proof.

Consider the principal U(1)-bundle (π × IdR) : P × R→ B × R whereP × R x G via (p, t).g = (p.g , t). We endow this with the connection

ω = (1− t)(projP)∗ω0 + t(projP)∗ω1.

For j = 0, 1, consider the inclusion ιP, j : P → P × R, p 7→ (p, j) andsimilarly ιB, j . Note ι∗P, jω = ωj . By Cartan’s structure equation, we have

Ωj = dωj +12 [ωj , ωj ]∧ = dι∗P, jω+ 1

2 [ι∗P, jω, ι∗P, jω]∧ = ι∗P, j

(dω + 1

2 [ω, ω]∧),

and hence Ωj = ι∗P, jΩ. (cont’d)

Theorem

Proof.

Consider the principal U(1)-bundle (π × IdR) : P × R→ B × R whereP × R x G via (p, t).g = (p.g , t).

We endow this with the connection

Ωj = dωj +12 [ωj , ωj ]∧ = dι∗P, jω+ 1

2 [ι∗P, jω, ι∗P, jω]∧ = ι∗P, j

(dω + 1

2 [ω, ω]∧),

Theorem

Proof.

Ωj = dωj +12 [ωj , ωj ]∧ = dι∗P, jω+ 1

2 [ι∗P, jω, ι∗P, jω]∧ = ι∗P, j

(dω + 1

2 [ω, ω]∧),

Theorem

Proof.

For j = 0, 1, consider the inclusion ιP, j : P → P × R, p 7→ (p, j) andsimilarly ιB, j . Note ι∗P, jω = ωj .

By Cartan’s structure equation, we have

Ωj = dωj +12 [ωj , ωj ]∧ = dι∗P, jω+ 1

2 [ι∗P, jω, ι∗P, jω]∧ = ι∗P, j

(dω + 1

2 [ω, ω]∧),

Theorem

Proof.

Ωj = dωj +12 [ωj , ωj ]∧

= dι∗P, jω+ 12 [ι∗P, jω, ι

∗P, jω]∧ = ι∗P, j

(dω + 1

2 [ω, ω]∧),

Theorem

Proof.

Ωj = dωj +12 [ωj , ωj ]∧ = dι∗P, jω+ 1

2 [ι∗P, jω, ι∗P, jω]∧

= ι∗P, j(dω + 1

2 [ω, ω]∧),

Theorem

Proof.

Ωj = dωj +12 [ωj , ωj ]∧ = dι∗P, jω+ 1

2 [ι∗P, jω, ι∗P, jω]∧ = ι∗P, j

(dω + 1

2 [ω, ω]∧)

Theorem

Proof.

Ωj = dωj +12 [ωj , ωj ]∧ = dι∗P, jω+ 1

2 [ι∗P, jω, ι∗P, jω]∧ = ι∗P, j

(dω + 1

2 [ω, ω]∧),

Theorem

Proof (cont’d).

Recall that X denotes the lift to HP of a vector X ∈ TB. It is an exerciseto see that dιP, j(X ) = ˜dιB, j(X ) for every X ∈ TB.

We now have

ι∗B, jFω(X ,Y ) = Fω(dιB, j(X ), dιB, j(Y )) = Ω( ˜dιB, j(X ), ˜dιB, j(X ))

= Ω(dιP, j(X ), dιP, j(Y )) = ι∗P, jΩ(X , Y )

= Ωj(X , Y ) = Fωj (X ,Y )

Thus Fωj = ι∗B, jFω. Since ιB, 0 ' ιB, 1, we conclude that [Fω0 ] = [Fω1 ].

Theorem

Proof (cont’d).

Recall that X denotes the lift to HP of a vector X ∈ TB. It is an exerciseto see that dιP, j(X ) = ˜dιB, j(X ) for every X ∈ TB. We now have

ι∗B, jFω(X ,Y ) = Fω(dιB, j(X ), dιB, j(Y ))

= Ω( ˜dιB, j(X ), ˜dιB, j(X ))

= Ωj(X , Y ) = Fωj (X ,Y )

Theorem

Proof (cont’d).

= Ωj(X , Y ) = Fωj (X ,Y )

Theorem

Proof (cont’d).

= Ω(dιP, j(X ), dιP, j(Y ))

= ι∗P, jΩ(X , Y )

= Ωj(X , Y ) = Fωj (X ,Y )

Theorem

Proof (cont’d).

= Ωj(X , Y ) = Fωj (X ,Y )

Theorem

Proof (cont’d).

= Ωj(X , Y )

= Fωj (X ,Y )

Theorem

Proof (cont’d).

= Ωj(X , Y ) = Fωj (X ,Y )

Theorem

Proof (cont’d).

= Ωj(X , Y ) = Fωj (X ,Y )

In complex geometry, one often sees the first Chern class defined via thelong exact sequence in cohomology that is induced by the exponentialexact sequence.

We will not discuss this definition, as it involves a background with Cechcohomology that we do not assume, but we remark that it differs from ourdefinition by a constant multiple. See page 141 of Griffiths-Harris for theproof of this.

Symplectic manifolds

Definition

Let M be a smooth manifold, and let α ∈∧2 T ∗M be a closed 2-form.

Then we say α is symplectic if the bilinear pairing α(·, ·) is nondegenerateon every tangent space. We call the pair (M, α) a symplectic manifold.

It is a consequence of the definition that the dimension of M is an evennumber 2n, and that α∧n ∈

∧2n T ∗M is a volume form.

Since α is nondegenerate, it induces a duality T ∗M ↔ TM. In particular,for every smooth function H : M → R, there exists a vector field XH on Mcalled the Hamiltonian vector field for H given by α(XH , ·) = dH(·).

Cf. when (M, g) is a Riemannian manifold, and we haveg(grad(H), ·) = dH(·).

Definition

Since α is nondegenerate, it induces a duality T ∗M ↔ TM. In particular,for every smooth function H : M → R, there exists a vector field XH on Mcalled the Hamiltonian vector field for H given by α(XH , ·) = dH(·).Cf. when (M, g) is a Riemannian manifold, and we haveg(grad(H), ·) = dH(·).

Examples of symplectic manifolds

If M is a 2-dimensional manifold, then every volume form on M issymplectic.

If M is any n-manifold, then T ∗M admits a symplectic form asfollows. Let U be a chart on M over which T ∗M is trivial, and letq1, . . . , qn be coordinates on U. Then we have coordinate functionspj : T ∗U → R that give the dqj -coefficient of a cotangent vector.Together q1, . . . , qn, p1, . . . , pn form a coordinate system on T ∗U.We thereby define a 1-form θ ∈ T ∗(T ∗M) via the local expressions

θ|T∗U =n∑

pjdqj .

Then α = −dθ is closed because it is exact, and the coordinateexpression α|T∗U =

∑j dqj ∧ dpj shows that α is nondegenerate.

Let h be a hermitian metric on a complex manifold M. Then h isKahler if and only if α = −Imh ∈

∧2 T ∗M is symplectic.

If M is any n-manifold, then T ∗M admits a symplectic form asfollows. Let U be a chart on M over which T ∗M is trivial, and letq1, . . . , qn be coordinates on U.

Then we have coordinate functionspj : T ∗U → R that give the dqj -coefficient of a cotangent vector.Together q1, . . . , qn, p1, . . . , pn form a coordinate system on T ∗U.We thereby define a 1-form θ ∈ T ∗(T ∗M) via the local expressions

θ|T∗U =n∑

pjdqj .

If M is any n-manifold, then T ∗M admits a symplectic form asfollows. Let U be a chart on M over which T ∗M is trivial, and letq1, . . . , qn be coordinates on U. Then we have coordinate functionspj : T ∗U → R that give the dqj -coefficient of a cotangent vector.Together q1, . . . , qn, p1, . . . , pn form a coordinate system on T ∗U.

We thereby define a 1-form θ ∈ T ∗(T ∗M) via the local expressions

θ|T∗U =n∑

pjdqj .

θ|T∗U =n∑

pjdqj .

θ|T∗U =n∑

pjdqj .

Hamiltonian actions

Let (M, α) be a symplectic manifold, and let M x G viasymplectomorphisms. That is, R∗gα = α for every g ∈ G . As we’ve seen

before, each A ∈ g defines a vector field A on M via A = ∂∂tRexp(tA).

say that the action is Hamiltonian if each A is the Hamiltonian vector fieldof a function HA : M → R. That is,

dHA(·) = α(A, ·).

We then define the moment map µ : M → g∗ via

µ(x)(A) = HA(x) ∀x ∈ M, A ∈ g.

We will be concerned with analyzing the level sets µ−1(a) for HamiltonianU(1)-actions.

Hamiltonian actions

before, each A ∈ g defines a vector field A on M via A = ∂∂tRexp(tA). We

dHA(·) = α(A, ·).

µ(x)(A) = HA(x) ∀x ∈ M, A ∈ g.

Hamiltonian actions

dHA(·) = α(A, ·).

µ(x)(A) = HA(x) ∀x ∈ M, A ∈ g.

Hamiltonian actions

dHA(·) = α(A, ·).

µ(x)(A) = HA(x) ∀x ∈ M, A ∈ g.

The reduced space

Let (M, α) be a symplectic manifold, and let M x U(1) be a Hamiltonianaction. We see that µ is U(1)-invariant, i.e. µ is constant along theflowline of any B ∈ u(1) = R, as follows.

Let A ∈ u(1). Then

∂tµ(x . exp(tB))(A) =

∂tHA(x . exp(tB)) = dHA(B) = α(A,B) = 0,

since A,B ∈ u(1) = R are multiples of each other.

One may rephrase this by saying that each level set µ−1(a) is preserved bythe U(1)-action. Suppose µ is proper and the actions µ−1(a) x U(1) arefree. Hence for each a ∈ u(1)∗, we have a principal U(1)-bundle

πa : µ−1(a)→ µ−1(a)/U(1) = Ma.

The reduced space

Let (M, α) be a symplectic manifold, and let M x U(1) be a Hamiltonianaction. We see that µ is U(1)-invariant, i.e. µ is constant along theflowline of any B ∈ u(1) = R, as follows. Let A ∈ u(1). Then

∂tµ(x . exp(tB))(A)

πa : µ−1(a)→ µ−1(a)/U(1) = Ma.

The reduced space

∂tHA(x . exp(tB))

= dHA(B) = α(A,B) = 0,

πa : µ−1(a)→ µ−1(a)/U(1) = Ma.

The reduced space

∂tHA(x . exp(tB)) = dHA(B)

= α(A,B) = 0,

πa : µ−1(a)→ µ−1(a)/U(1) = Ma.

The reduced space

πa : µ−1(a)→ µ−1(a)/U(1) = Ma.

The reduced space

πa : µ−1(a)→ µ−1(a)/U(1) = Ma.

A product model

One may verify that the reduced space Ma = µ−1(a)/U(1) admits a uniquesymplectic form νa satisfying π∗aνa = α|µ−1(a). We seek a convenientmodel for these Ma so that we may compute how νa depends on a.

Let us consider the product space µ−1(0)× (−ε, ε), and letω ∈ Ω1(µ−1(0)) be a principal connection for the bundleπ0 : µ−1(0)→ µ−1(0)/U(1). Let t denote the (−ε, ε)-coordinate.

Theorem

The form α = α|µ−1(0) + d(tω) ∈ Ω2(µ−1(0)× (−ε, ε)) is symplectic if εis small enough.

Proof.

By definition, α|µ−1(0) is closed and d(tω) is exact; hence α is closed.

Theexpression α|t=0 = α|µ−1(0) + dt ∧ ω shows that αt=0 is nondegenerate,and thus the general expression α = α|µ−1(0) + dt ∧ ω + tΩ shows that αis nondegenerate for t 1.

A product model

Theorem

Proof.

A product model

Theorem

Proof.

A product model

Theorem

Proof.

A product model

Theorem

Proof.

By definition, α|µ−1(0) is closed and d(tω) is exact; hence α is closed. Theexpression α|t=0 = α|µ−1(0) + dt ∧ ω shows that αt=0 is nondegenerate,

and thus the general expression α = α|µ−1(0) + dt ∧ ω + tΩ shows that αis nondegenerate for t 1.

A product model

Theorem

Proof.

By definition, α|µ−1(0) is closed and d(tω) is exact; hence α is closed. Theexpression α|t=0 = α|µ−1(0) + dt ∧ ω shows that αt=0 is nondegenerate,and thus the general expression α = α|µ−1(0) + dt ∧ ω + tΩ shows that αis nondegenerate for t 1.

A product model

Let us consider the action µ−1(0)× (−ε, ε) x U(1) given by(x , t).g = (x .g , t). It is an exercise to verify that this action is Hamiltonianwith moment map J : µ−1(0)× (−ε, ε)→ u(1)∗ = R given by J(x , t) = t.

It is a consequence of the coisotropic embedding theorem (see Guillemin,pages 25-26), that there is a neighborhood M ⊃ U ⊃ µ−1(0) and aU(1)-equivariant symplectomorphism

(U, α)∼−→ (µ−1(0)× (−ε, ε), α)

for small enough ε, so that µ−1(a)∼−→ µ−1(0)× a.

Our reduced space Ma = µ−1(a)/U(1) with symplectic form νa given byπ∗νa = α|µ−1(a) is therefore symplectomorphic to M0 × a withsymplectic form νa given by

π∗νa = α|t=a

= α|µ−1(0) + dt|t=a ∧ ω + aΩ

= α|µ−1(0) + aΩ.

A product model

(U, α)∼−→ (µ−1(0)× (−ε, ε), α)

π∗νa = α|t=a

= α|µ−1(0) + dt|t=a ∧ ω + aΩ

= α|µ−1(0) + aΩ.

A product model

(U, α)∼−→ (µ−1(0)× (−ε, ε), α)

π∗νa = α|t=a

= α|µ−1(0) + dt|t=a ∧ ω + aΩ

= α|µ−1(0) + aΩ.

A product model

(U, α)∼−→ (µ−1(0)× (−ε, ε), α)

π∗νa = α|t=a = α|µ−1(0) + dt|t=a ∧ ω + aΩ

= α|µ−1(0) + aΩ.

A product model

(U, α)∼−→ (µ−1(0)× (−ε, ε), α)

π∗νa = α|t=a = α|µ−1(0) + dt|t=a ∧ ω + aΩ

= α|µ−1(0) + aΩ.

Variation of reduced spaces

For small enough a, we have now shown that

(Ma, νa) ∼= (M0, νa)

π∗νa = α|µ−1(0) + aΩ

= π∗ν0 + aπ∗Fω

ν0 = νa + aFω.

In particular, when we identify (Ma, νa) = (M0, νa) and take cohomologyclasses, we find that

[νa] = [ν0] + ac1(µ−1(0)).

(Ma, νa) ∼= (M0, νa)

π∗νa = α|µ−1(0) + aΩ

ν0 = νa + aFω.

[νa] = [ν0] + ac1(µ−1(0)).

(Ma, νa) ∼= (M0, νa)

π∗νa = α|µ−1(0) + aΩ

ν0 = νa + aFω.

[νa] = [ν0] + ac1(µ−1(0)).

(Ma, νa) ∼= (M0, νa)

π∗νa = α|µ−1(0) + aΩ

ν0 = νa + aFω.

[νa] = [ν0] + ac1(µ−1(0)).

Variation of volume is a polynomial

Let n = dimMa, and recall that ν∧na is a volume form. The expression[νa] = [ν0] + ac1(µ−1(0)) then gives

vol(Ma) =n∑

)ak∫M0

[ν0]n−k ^ c1(µ−1(0))k .

References

P. Griffiths and J. Harris, Principles of algebraic geometry

V. Guillemin, Moment maps and combinatorial invariants ofHamiltonian T n-spaces

P. Michor, Topics in Differential Geometry

M. Mirzakhani, Weil-Petersson volume and intersection theory on themoduli space of curves

T. Walpuski, Notes on the geometry of manifolds, https://math.mit.edu/~walpuski/18.965/GeometryOfManifolds.pdf

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