数学联邦政治世界观
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指标定理(四) (9-8)

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

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