指标定理（五） (9-4)

𝓞 ᴅ₊ᴘ/𝓞 ᴅ is merely a presheaf. But indeed，the long exact sequence associated to the sequence still holds. For ∪ such that P ∉ ∪，𝓞 ᴅ₊ᴘ/𝓞 ᴅ(∪) ＝0. For ∪ such that P ∈ ∪. 𝓞 ᴅ₊ᴘ/𝓞 ᴅ(∪) is one-dimensional. H⁰ (M，𝓞 ᴅ₊ᴘ/𝓞 ᴅ)＝ℂ since it is equal to the space of global sections, and now we compute H¹(M，𝓞 ᴅ₊ᴘ/𝓞 ᴅ).

Suppose ξ ∈ H¹(M，𝓞 ᴅ₊ᴘ/𝓞 ᴅ) is represented by ф ∈ Z¹(𝓤 ，𝓞 ᴅ₊ᴘ/𝓞 ᴅ). The cover 𝓤 has a refinement 𝓥 such that P is covered by only one open set Vᵢ in 𝓥 . Thus 0＝δф(i，i，i)＝ф(i，i) –ф(i，i)＋ф(i，i)＝ф(i，i)，and ф＝0. Consequently H¹(M，𝓞 ᴅ₊ᴘ/𝓞 ᴅ)＝0.

23

Consider the long exact sequence

0 → H⁰(M，𝓞 ᴅ) → H⁰(M，𝓞 ᴅ₊ᴘ) → H⁰(M，𝓞 ᴅ₊ᴘ/𝓞 ᴅ)＝ℂ

→ H¹(M，𝓞 ᴅ) → H¹(M，𝓞 ᴅ₊ᴘ) → H¹(M，𝓞 ᴅ₊ᴘ/𝓞 ᴅ)＝0.

It shows that H*(M，𝓞 ᴅ) is finite dimensional iff H*(M，𝓞 ᴅ₊ᴘ) is. Suppose this is the case， and now the alternating sum of the dimensions in the exact sequence is zero. That is to say，if χ(H*(M，𝓞 ᴅ))：＝dim(H⁰(M，𝓞 ᴅ)) – dim(H¹(M，𝓞 ᴅ))，we have

χ(H*(M，𝓞 ᴅ)) – χ(H*(M，𝓞 ᴅ₊ᴘ))＋1=0，

i.e.

χ(H*(M，𝓞 ᴅ)) – deg(D)＝χ(H*(M，𝓞 ᴅ₊ᴘ)) – deg(D＋P).

In particular for a compact Riemann surface， χ(H*(M，𝓞 ᴅ)) – deg(D) should be a con-stant. Recall that the genus g is defined to be dimH¹(X，𝓞 ) where 𝓞 ＝𝓞 ₀ is the sheaf of holomorphic functions，Taking D to be zero we find the constant is χ(H*(M，𝓞 ᴅ))＝1–g.Thus we have proved that

Theorem 13 Suppose D is α diυisor on α compαct Riemαnn surfαce X of genus g. Then H⁰(X，𝓞 ᴅ) αnd H¹(X，𝓞 ᴅ) αre finite dimensionαl υector spαces αnd

dimH⁰(X，𝓞 ᴅ) – dimH¹(X，𝓞 ᴅ)＝1 – 9 ＋degD.

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