指标定理（四） (9-5)

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.

19

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)，. . .

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