数学联邦政治世界观
超小超大

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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