指标定理（四）

注意：指标定理（4/5）

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Now we compute its topological index. We need to compute the Chern character of an induced bundle，and we consider this in a general setting.

Let ρ：∪ₙ → ∪ₘ be a homomorphism，and E be a complex vector bundle of rank n. Denote the induced bundle by E × ᵨ ℂᵐ. We want to find how the Chern classes of the two bundles are related.

Without loss of generality，we assume that E＝l₁⨁ · · · ⨁lₙ，that is to say，the structure group has been reduced to the torus Tⁿ. ρ defines a representation of the torus and the representation decomposes into one-dimensional ones，since Tⁿ is compact and abelian.

We first consider α：Tⁿ＝ S¹ × · · · × S¹ → S¹. Suppose it is of the form (z₁，. . .，zₙ) ↦

∏zᵢᵏⁱ，kᵢ ∈ ℤ. The S¹-bundle ET × α S¹ → BT has first chern class c(α) ∈ H²(BT；ℤ)，which is the image of the generator in H²(BS¹；ℤ). H*(BT；ℤ)＝ℤ[t₁，. . .，tₙ] is the tensor product of copies of H*(BS¹；ℤ).

Let ф：Tⁿ → Uₙ be the embedding. The bundle ET ×α ℂ → BT has transition functions ∏ fᵢᵏⁱ when ET × ф ℂⁿ＝l₁⨁ · · · ⨁lₙ → BT has transition functions (f₁，. . .，fₙ)，according to the definition of α. Thus ET × α ℂ → BT is isomorphic to l₁ᵏⁱ ⨂ · · · ⨂lₙᵏⁿ .Thus c(α)＝∑kᵢtᵢ.

In general，if α＝(α¹，. . .，αᵐ)：Tⁿ → Tᵐ，

ch(ET × α ℂᵐ)＝∑eᶜ⁽αʲ⁾＝∑eΣᵢᵏʲᵢᵗᵢ .

ⱼ ⱼ

The total Chern class is

c(ET × α ℂᵐ)＝∏(1＋c(αʲ))＝∏(1＋∑kʲᵢtᵢ).

ʲ ⱼ

By splitting principle，this is true for general rank n complex vector bundles. Here are some examples：i) ρ：∪ₙ → ∪ₙ is conjugation (of complex numbers). E × ᵨ

ℂⁿ＝E*＝ˉE. kⁱⱼ＝–δⁱⱼ.

ₙ ₙ

c(E*)＝∏(1 – xᵢ)＝∑(–1)ⁱcᵢ(E).

ᵢ₌₁ ᵢ₌₀

ii) For ∧ᵏ E，u ∈ Tⁿ acts as eᵢ₁ ∧ · · · ∧ eᵢₖ ↦ ue，∧ · · · ∧ ueᵢₖ.

c(αᵢ₁，. . .，ᵢₖ)＝xᵢ₁＋· · ·＋xᵢₖ and c(∧ᵏE)

＝∏ (1＋(xᵢ₁＋· · ·＋xᵢₖ)).

1≤i₁＜· · ·＜iₖ≤n

iii) For ∧ᵏE*，

c(∧ᵏE*)＝∏ (1–(xᵢ₁＋· · ·＋xᵢₖ)).

1≤i₁＜· · ·＜iₖ≤n

ch(∧ᵏE*)＝∑eˉ(xᵢ₁＋· · ·＋xᵢₖ)).

1≤i₁＜· · ·＜iₖ≤n

iv)

ⁿ ⁿ

∑ ch(∧ᵏE*) · tᵏ＝∏(1＋te⁻ˣⁱ).

ₖ₌₀ ᵢ₌₁

Now let’s consider an oriented real bundle E of rank 2n. By splitting principle， we may assume that it splits into a direct sum of oriented plane bundles (or complex line

17

bundles) and talk about its Chern roots. We want to calculate ∑ⁿₖ₌₀ ch(∧ᵏE* ⨂ ℂ) · tᵏ Let ρ：Tⁿ ⊂ SO₂ₙ → ∪₂ₙ be the inclusion，where Tⁿ denotes the standard maximal torus. This Tⁿ ⊂ ∪₂ₙ is a conjugate subgroup of a torus in the standard maximal torus in ∪₂ₙ， and the conjugation restricts to

(cos(2πtᵣ) –sin(2πtᵣ)

) ↦(ᵉⁱ²πᵗʳ 0)

(sin(2πtᵣ) cos(2πtᵣ) ( 0 ₑ⁻ⁱ²πᵗʳ)

on each S¹. Thus the weights of ρ：Tⁿ ⊂ SO₂ₙ → ∪₂ₙ are (x₁，–x₁，. . .，xₙ，–xₙ). where the xᵢ’s are the Chern roots. By the same argument as above，

ⁿ ₙ

∑ch(∧ᵏE* ⨂ ℂ) · tᵏ＝∏(1＋te⁻ˣⁱ)(1＋teˣⁱ). ᵢ₌₁

ₖ₌₀

Settingt＝–1，we have

ₙ ₙ

∑ch(∧ᵏE* ⨂ ℂ) · (–1)ᵏ＝∏(1 – e⁻ˣⁱ)(1 – eˣⁱ). ᵢ₌₁

ₖ₌₀

Thus for the de Rham complex defined for an oriented compact 2n-dimensional manifold，

ₘ

indₜ(d)＝〈((∑(–1)ⁱ · ch(∧ⁱT* ⨂ ℂ) )

ᵢ₌₀

ₙ xⱼ 1

∏(─── · ───).[X]〉

ⱼ₌₁ 1 – e⁻ˣʲ 1– eˣʲ

ₙ

＝〈(∏(1 – e⁻ˣⁱ)(1 – eˣⁱ)

ᵢ₌₁

ₙ xⱼ 1

∏ (─── · ───))，[X]〉

ⱼ₌₁ 1 – e⁻ˣʲ 1 – eˣʲ

＝e(X)

＝indα(d).

3.2 Dolbeault complex

Let X be a complex n-dimensional manifold，Tℝ its tangent bundle as a real manifold， and T＝T¹，⁰ the holomorphic tangent bundle. Recall that

＿＿

T*ℝ ⨂ ℂ ≅ T* ⨁ T*，

＿＿

∧ⁱ(T*ℝ ⨂ ℂ) ≅ ∧ⁱ(T* ⨁ T*) ≅ ⨁ₚ₊q₌ᵢ (∧ᴾT*

＿＿

⨂ ∧qT*).

＿＿

Let ∧ᴾ，q＝∧ᴾT* ⨂ ∧qT*，Aᴾ，q：＝Γ

＿＿

(∧ᴾT* ⨂ AqT*).The exterior derivative d： Aᴾ，q → Aᴾ⁺¹，q ⨁ Aᴾ，q⁺¹ splits into d＝∂＋ˉ∂，with ∂：Aᴾ，q → Aᴾ⁺¹，q and ˉ∂： Aᴾ，q → Aᴾ，q⁺¹.

One easily check that as before，ˉ∂ defines an elliptic complex of differential operators for each fixed p. This is called the Dolbeault complex. Let Hᴾ，q be the q-th cohomology group of this complex，and hᴾ，q be its dimension.

Definition 5 For fired p，χᴾ；＝∑ⁿq₌₀(–1)q h ᴾ，q is defined to be the αnαlyticαl index of ˉ∂. χ⁰ is αlso cαlled the αrithmetic genus.

Now we would like to find what the topological index is by the index theorem. First set p＝0.

＿＿

∧qT*＝AqT. By the calculations in the former subsection we have

ₙ ₙ

∑ch(∧ᵏT) · tᵏ＝∏(1＋teˣⁱ).

ₖ₌₀ ᵢ₌₁

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Thus

ₘ

χ⁰＝〈((∑(–1)ⁱ · ch(∧ᵏT)

ᵢ₌₀

ₙ xⱼ 1

∏(─── · ───))，[X]〉

ⱼ₌₁ 1 – e⁻ˣʲ 1 – e⁻ˣʲ

ₙ xⱼ

＝〈∏(───))，[X]〉

ⱼ₌₁ 1 – e⁻ˣʲ

=Td(T)[X]＝Td(X).

For general p，define χy＝∑ⁿₚ₌₀ χᴾ·yᴾ to be a formal linear combination of χᴾ. Formally χy is the analytical index of the elliptic complex (Cq，ˉ∂)，whose direct summands consist of yᴾ copies of the p-th complex for each p， i.e. Cq＝⨁ₚyᴾ · ∧ᴾ，q.

Thus

∑(–1)q ch(Cq)＝∑(–1)q yᴾ ch(∧ᴾT*)ch(∧qT)

p，q

= (∑(–1)q ch(∧q T))(∑ yᴾch(∧ᴾT*))

q p

＝∏(1 – eˣʲ)(1＋ye⁻ˣʲ)，

ⱼ

and consequently

ₙ xⱼ

χy＝∏((1＋y · e⁻ˣʲ) ───) [X].

ⱼ₌₁ 1 – e⁻ˣʲ

That is，

ₙ xⱼ

χy＝∫ₓ∏((1＋y · e⁻ˣʲ) ───).

ⱼ₌₁ 1 – e⁻ˣʲ

This finishes the calculation.

For y＝0。the above formula tells us that

χ₀＝χ(X，𝓞 )＝∫ₓ Td(X).

Indeed this is a special case of Hirzebruch-Riemann-Roch theorem，as we will see later.

For y＝–1 and X an n-dimensional compact Kähler manifold，this yields the Gauss-Bonnet formula

ₙ

χ–1＝∑(–1)ᴾ⁺q hᴾ，q＝e(X)＝∫ₓ∏ xⱼ＝∫ₓ cₙ(x). ⱼ₌₁

For y＝1 and X a 2n-dimensional compact Kähler manifold，the χy-genus is

ₙ xⱼ

χ₁＝χ(X，⨁Ωᴾₓ)＝∫ₓ∏((1＋eˉˣʲ)───)

ⱼ₌₁ 1 – e⁻ˣʲ

＝∫ₓL(X)，

where L(X)＝L(p(X)) is the L-class explained later. This is the Hirzebruch signature theorem.

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4 Hirzebruch Signature Theorem

4.1 Multiplicative sequence

Let A be a fixed commutative ring with unit， and A* a graded A-algebra. Write AΠ for the ring of formal power series α₀＋α₁＋. . .， where αᵢ ∈ Aⁱ，and AΠ₀ for its subset containing elements with leading term 1. AΠ₀ form a group under multiplication.

Now consider a sequence of polynomials

K₁(x₁)，K₂(x₁，x₂)，K₃(x₁，x₂，x₃)，. . .

with coefficients in ∧ such that，if the variable xᵢ is assigned degree i，then each Kₙ (x₁，. . .，xₙ)is homogeneous of degree n. Given an element α ∈ AΠ with leading term 1， define a new element K(α) ∈ AΠ by the formula

K(α)＝1＋K₁(α₁)＋K₂(α₁，α₂)＋ . . .

Definition 6 These Kₙ form α multiplicαtiυe sequence if K(αb)＝K(α)K(b) for αll grαded ∧-αlgebrαs A* αnd αll α，b ∈ AΠ₀(i.e.K：AΠ₀ → AΠ₀ is α group homomorphism).

Theorem 11 Giυen α formαl pοωer series f(t)＝1＋λ₁t＋ . . . ωith coefficients in ∧，there ezists α unigue multiplicαtiυe sequence such thαt K(1＋t)＝f(t) for αll 1＋t ∈ AΠ₀.

We omit the proof. Note that if α＝(1＋t₁) . . . (1＋tₙ)，then K(α)＝f(t₁) . . . f(tₙ). tᵢ

Consequently，the Todd class ∏ᵢ ───

1–e⁻ᵗⁱ

∈ HΠ(B∪ₙ；ℚ) is indeed given by the mul-tiplicative sequence associated to the series

x

td(x)＝───，

1–e⁻ˣ

where we take ∧＝ℚ，AΠ＝HΠ(B∪ₙ；ℚ). We may write

Td(c)＝∏ td(tᵢ) ∈ HΠ(B∪ₙ；ℚ).

where c is the total Chern class and tᵢ’ s are the Chern roots.

Here you may notice somet hing strange.tᵢ is indeed in H²(B∪ₙ；ℚ) but not H¹(B∪ₙ；ℚ). To avoid this，we may replace H*(B∪ₙ；ℚ) by H²*(B∪ₙ；ℚ)：＝ ⨁ₖ H²ᵏ(B∪ₙ；ℚ) with grad-ing given by k. We will take this for granted when considering Chern classes，and use H⁴* when considering Pontrjagin classes.

4.2 Hirzebruch signature theorem

We define the L-genus of a 4n-dimensional smooth compact oriented manifold to be

L[M]＝〈L(p(M))，[M]〉＝〈Lₙ(p₁(M)， . . . ，pₙ(M))，[M]〉

where L is the multiplicative sequence associated to

√x 2²ᵏB₂ₖ x x²

──── ＝∑ ────xᵏ＝1＋─ – ─

tanh(√x) k≥0 (2k)！ 3 45

＋ . . . .

For a 4n-dimensional smooth compact oriented manifold，we define its signature σ(M) to be the signature of the symmetric bilinear form defined on H²ⁿ(M⁴ⁿ；ℚ)，

(α，b) ↦〈α∪b，[M]〉.

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Theorem 12 For α 4n-dimensionαl smooth compαct oriented mαnifold.

L[M]＝σ(M).

Both sides are algebra homomorphisms from Ω*.⨂ to ℚ (where Ω* is the oriented cobordism ring)，so we only need to check the result on the generators ℂP²ᵏ of Ω* ⨂ ℚ.

H²ⁿ(ℂP²ⁿ；ℚ) is generated by a single element and we easily see that its signature is one.

For a complex vector bundle ξ viewed as a real bundle，

1 – p₁＋p₂ – · · · ＝(1 – c₁＋c₂ – . . . )(1＋C，c₁＋c₂＋ . . . ).

In particular，for the tangent bundle of ℂPⁿ，

1 – p₁＋p₂ – · · · ＝(1 – α)ⁿ⁺¹(1＋α)ⁿ⁺¹＝(1 – α²)ⁿ⁺¹，

1＋p₁＋p₂＋ · · · ＝(1＋α²)ⁿ⁺¹. Consequently for ℂP²ᵏ，L(p)＝

α

(──)²ᵏ⁺¹，

tαnh(α)

where α is a generator of the cohomology ring.

To calculate〈L(p(ℂP²ᵏ))，[ℂP²ᵏ]〉，it suffices to calculate the coefficient of α²ᵏ

α

in(───)²ᵏ⁺¹.

tαnh(α)

The result follows from direct calculation using contour integration in complex analysis, and this completes the proof. ▢

In the proof above we are using the fact that the total Chern class of ℂPⁿ is of the form (1＋α)ⁿ⁺¹ where α is a generator. We sketch a proof.Let L be the tautological line bundle(which is the dual of the hyperplane bundle H).Let ε be the trivial line bundle，and ω the complementary rank n bundle of L ⊂ εⁿ⁺¹. Thus TℂPⁿ＝Hom(L，ω) and

TℂPⁿ ⨁ ε＝Hom(L，ω ⨁ L)＝Hom(L，εⁿ⁺¹)＝(n＋1)H.

Taking total Chern class of both sides we have c(TℂPⁿ)＝(1＋c₁(H))ⁿ⁺¹，as desired.

Now we show that this is indeed a special case of the Atiyah-Singer index theorem. Recall Aⁱ：＝Γ(∧ⁱ(T* ⨂ ℂ))，〈α，b〉＝ ∫ₓ α∧ *ˉb.

Define τ：＝(–1)ⁱ⁽ⁱ⁻¹⁾/²⁺ᵏ* and d*：＝ – * d*＝–τdτ，where dim(X)＝4k，and * is the Hodge-* operator. Since τ² is the identity， ∧(T* ⨂ ℂ)＝E₊ ⨁ E₋ splits into eigenbundles with eigenvalues ＋1 and –1 with respect to τ. Since

T(d＋d*)＝τd – ττdτ＝τd – dτ＝ –(dτ – τd)

＝ – (dτ＋d*τ)＝ – (d＋d*)τ，

one can consider d＋d*：ΓE₊ → ΓE₋.

Recall H²ᵏ(X；ℂ) is isomorphic to the vector space of harmonic forms 𝓡 ²ᵏ(X)＝ker(d＋d*) ⊂ Γ(∧²ᵏ(T* ⨂ ℂ)). One verifies that 𝓡 ²ᵏ(X) splits into eigenspaces 𝓡 ₊²ᵏ(X)⨁

𝓡 ₋²ᵏ(X) with respect to τ：Γ(∧²ᵏ(T* ⨂ ℂ)) → Γ(∧²ᵏ(T* ⨂ ℂ)). Restricting to the space of real harmonic forms H²ᵏ(X；ℝ) ⊂ H²ᵏ(X；ℂ)， this gives exactly the decomposition into positive and negative parts with respect to the intersection form. To see this，for example， for a real harmonic 2k-form α in 𝓡 ₊²ᵏ(X)，we have

0＜〈α，α〉＝∫ₓ α ∧ *α＝∫ₓ α ∧ τα＝∫ₓ α ∧ α.

Hence σ(X)＝dimℂ(𝓡 ₊²ᵏ)–dimℂ(𝓡 ₋²ᵏ). We have an elliptic operator d+d*：ΓE₊ → ΓE₋ since its square is the Laplace operator Δ：Γ(∧(T* ⨂ ℂ)) → Γ(∧(T* ⨂ ℂ)) which

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