指标定理（五） (9-3)

Let .𝓜 be the sheaf of meromorphic functions on M，and 𝓞 be the sheaf of holomorphic functions on M. The presheaf ∪ ↦ 𝓜 (∪)/𝓞 (∪) can be viewed as the presheaf of principle parts, since we say that two meromorphic functions define the same principle part if their difference is holomorphic. Similarly we consider the sheaf. 𝓜 * of meromorphic functions that are never identically zero in any open set，and thus . 𝓜 * (∪) forms an abelian group under multiplication. Also define 𝓞 * to be the sheaf of holomorphic functions that never vanishes， and 𝓞 * (∪) also forms an abelian group under multiplication. The presheaf ∪↦𝓜 *(∪)/𝓞 *(∪) is the presheaf of divisors.

In general，some divisors cannot be represented by a globally defined meromorphic function. We say that two divisors D，E are equiυαlent if they differ by a principle divisor，i.e. D – E＝(f) for some meromorphic function f. Principle divisors form a group Prin(X). If the surface is compact，we can define deg(D)＝∑z∈X D(x). Thus if two divisors are equivalent，they have the same degree. We define a sheaf 𝓞 ᴅ as

𝓞 ᴅ(∪)＝{f ∈ 𝓜 (∪)：(f)(x) ≥ –D(x)}.

If D is represented by a meromorphic function g， then (f)(x) ≥ –D(x) says that fg is holomorphic.

Let P denotes a divisor taking the value 1 at P ∈ X and zero otherwise. We then have an exact sequence

0 → 𝓞 ᴅ → 𝓞 ᴅ₊ᴘ → 𝓞 ᴅ₊ᴘ/𝓞 ᴅ → 0.

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