指标定理（四） (9-4)

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

18

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].

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