指标定理（五） (9-5)

H*(X，𝓞 ᴅ) is finite dimensional because by compactness it has a finite good covering， and sheaf cobomology coincides with ˉCech cohomology，which is finite dimensional.

5.2 Divisors and line bundles

We briefly explain how divisors and line bundles are related. We have an exact sequence

0 → 𝓞 * → 𝓜 * → 𝓜 */𝓞 * →0，

which induces

· · · → H⁰(M，𝓜 *) → H⁰(M，𝓜 */𝓞 *) → H¹(M，𝓞 *) → . . . .

Set δ：H⁰(M，𝓜 */𝓞 *) → H¹(M，𝓞 *).Suppose divisor D ∈ H⁰(M，𝓜 */𝓞 *) is represent-ed by (αᵢ)，where αᵢ is defined on ∪ᵢ in an open cover 𝓤 . Then αᵢ/αⱼ is holomorphic，and δ：(αᵢ) ↦ (αᵢ/αⱼ) ∈ H¹(M，𝓞 *). (αᵢ/αⱼ) satisfies the cocycle condition and defines a holomorphic line bundle L. One verifies that this is well-defined. This is a group homo-morphism，since addition of divisors corresponds to multiplication of (αᵢ)，(αᵢ/αⱼ)，and corresponds to tensor product of holomorphic line bundles. The image of H⁰(M，𝓜 *) → H⁰(M，𝓜 */𝓞 *) is by definition Prin(M). However，H⁰(M，𝓜 */𝓞 *) → H¹(M，𝓞 *) is not surjective in the most general case.

Now we briefly recall Dolbeault cohomology for a holomorphie vector bundle. Given a holomorphic vector bundle π：E → X，for fixed p define

＿＿

∧ᵖ，q(E)＝∧ᵖT* ⨂ ∧qT* ⨂ E，Aᵖ，q(X，E)：

＿＿

＝Γ(∧ᵖT* ⨂ ∧qT* ⨂ E).

Define

ˉ∂ᴇ：Aᵖ，q(X，E) → Aᵖ，q⁺¹(X，E)，α ⨂ s ↦ ˉ∂α ⨂ s.

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This is independent of trivializations chosen for E because transition functions of E are holomorphic. Thus

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