指标定理（四） (9-3)

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.

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