指标定理（五） (9-7)

There’s another fact that ∫ᴄc₁(L)＝deg(L). This is because the first Chern class of a line bundle Lᴅ associated to a divisor D is the Poincaré dual of the divisor，i.e.

i

∫᙮ ─ Ω∇∧α＝∫ᴅ

2π

α for all closed real form α，

which holds for divisors on general complex manifolds X.

We always assume Riemann surfaces to be connected. Let C be a compact Riemann surface，and L ∈ Pic(C). Then the formula tells us that χ(C，L)＝∫ᴄ(1＋c₁(L)＋ . . . )

c₁(C)

(1＋── ＋ . . . )

2 c₁(C)

＝∫ᴄ c₁(L)＋──

2

deg(K*ᴄ)

＝deg(L)＋────.

2

25

Here K*ᴄ is isomorphic to the holomorphic

tangent bundle. Thus χ(C，𝓞 )

deg(K*ᴄ)

＝───

2 ＝h⁰(C，𝓞 ) – h¹(C，𝓞 )＝1 – g. This gives us the previous result.

6 Further Developments

There are various generalizations and applications. We only mention some of them.

Originally Atiyah-Singer index theorem was proved using K-theory.

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