Line bundles and divisors

For a divisor $D\in\mathrm{Div}(X)$, define a sheaf $\mathcal{O}_D$ on $X$

$$\mathcal{O}_D(U):=\{f\in\mathcal{M}_X(U):(f)\geq -D\}.$$

Prop. $\mathcal{O}_D$ is a locally free sheaf of $\mathcal{O}_X$-module of rank $1$.

Recall the exponential sequence

$$0\rightarrow\mathbb{Z}\rightarrow\mathcal{O}_X\rightarrow\mathcal{O}_X^*\rightarrow 0.$$

Given an open cover $\mathcal{U}=\{U_i\}_{i\in I}$, such that $U_i, U_{ij}$ are simply connected. There is a long exact sequence of groups induced by the short exact sequence:

Here we shall define the Chern class (named after 陈省身)

$$c_1:H^1(\mathcal{U},\mathcal{O}_X^*)\rightarrow H^2(\mathcal{U},\mathbb{Z})$$

and prove that $c_1$ fits into the long exact sequence.

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Definition of Chern class

Given $g\in Z^1(\mathcal{U},\mathcal{O}_X^*)$. Since $U_{ij}$ is simply connected, then $\exists h_{ij}\in\mathcal{O}_X(U_{ij})$, s.t.

$$g_{ij}=\exp(2\pi ih_{ij}).$$

Hence $(h_{ij})$ defines an element in $C^1(\mathcal{U},\mathcal{O}_X)$. Notice that $g$ satisfies cocycle condition. Then

$$g_{ij}g_{jk}g_{ik}^{-1}=\exp(2\pi i(h_{ij}+h_{jk}-h_{ik}))=1$$

Namely $h_{ij}+h_{ji}-h_{ik}\in\mathbb{Z}$, which implies that$(\delta h)_{ijk}=h_{ij}+h_{ji}-h_{ik}$, and $\delta h\in Z^1(\mathcal{U},\mathbb{Z})$. Therefore, we can define

$$c_1([g]):=[\delta h]\in H^2(\mathcal{U},\mathbb{Z}).$$

In this process, the definition relies on the choice of $h$. If another $h'$ is given, then

$$h'-h=(2\pi ia_{ij})\in C^1(\mathcal{U},\mathbb{Z}).$$

Hence $[\delta h-\delta h']=[\delta a]\in B^1(\mathcal{U},\mathbb{Z})$​.

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Exactness at $H^1(\mathcal{U},\mathcal{O}_X^*)$

Case I: Let $[g]=e([f])\in\mathrm{im}(e)$, where $f=(f_{ij})\in Z^1(\mathcal{U},\mathcal{O}_X)$. Then by cocycle condition, $\exists h\in C^0(\mathcal{U},\mathcal{O}_X^*)$, s.t.

$$g_{ik}=h_i\exp(2\pi if_{ik})h_k^{-1}.$$

Since $U_i$ is simply-connected, then $\exists k\in C^0(\mathcal{U},\mathcal{O}_X)$, s.t.

$$\exp(2\pi ik_i)=h_i, \quad\forall i\in I$$

$$\Rightarrow g_{jl}=\exp\left(2\pi i(k_j+f_{jl}-k_l)\right), \quad\forall j,k\in I$$

By definition of Chern class above, $c_1([g])=[\delta (k_j+f_{jl}-k_l)]=[\delta f]$, and $e([f])=[g]$. Therefore,

$$c_1\circ e(f)=\delta f=:b\in B^2(\mathcal{U},\mathbb{Z}).$$

This implies that $c_1([g])=0\in H^2(\mathcal{U},\mathbb{Z})$​.

Case II: Let $[g]\in\ker(c_1)$, namely $c_1([g])=[b]=0\in H^2(\mathcal{U},\mathbb{Z})$. By the construction of Chern class, there exists $h\in C^1(\mathcal{U},\mathcal{O}_X)$, s.t.

$$g_{ij}=\exp(2\pi ih_{ij}),\quad b=\delta h$$

The condition $[b]=0$ means that $[b]=[\delta h]\in H^2(\mathcal{U},\mathbb{Z})$ is a coboundary. Namely, $\exists a\in C^1(\mathcal{U},\mathbb{Z})\subset C^1(\mathcal{U},\mathcal{O}_X)$, s.t.

$$\delta g=\delta a =b.$$

Set $f_{ij}=h_{ij}-a_{ij}\in C^1(\mathcal{U},\mathcal{O}_X)$. Then

$$\delta f=\delta h-\delta a=b-b=0,$$

which implies that $f\in Z^1(\mathcal{U},\mathcal{O}_X)$. In addition, since $a_{ik}\in\mathbb{Z}$, then

$$\exp(2\pi if_{ik})=\exp(2\pi ih_{ik})=g_{ik}\in\mathcal{O}_X(U_{ik}).$$

Therefore, $[g]=e[f]$. This completes the proof of exactness at $H^1(\mathcal{U},\mathcal{O}_X^*)$.

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Exactness at $H^2(\mathcal{U},\mathbb{Z})$

Assume that we do not know $H^2(\mathcal{U},\mathcal{O}_X)=0$.

(1) $\mathrm{im} c_1\subset \ker i$: suppose that $[g]\in H^1(X,\mathcal{O}_X^*)$. $\exists b\in Z^2(\mathcal{U},\mathbb{Z})$, s.t.

$$b=c_1(g)=\delta h$$

for some $h\in C^1(\mathcal{U},\mathcal{O}_X)$. Hence $b\in B^2(\mathcal{U},\mathcal{O}_X)$. In other words,

$$i(c_1([g]))=[\delta h]=0\in H^2(\mathcal{U},\mathcal{O}_X).$$

(2) $\ker i\subset\mathrm{im}c_1$: let $[b]\in H^2(\mathcal{U},\mathbb{Z})$ such that $i[b]=0\in H^2(\mathcal{U},\mathcal{O}_X)$. Then $b\in B^2(\mathcal{U},\mathcal{O}_X)$, which means that $\exists h\in C^1(\mathcal{U},\mathcal{O}_X)$, such that

$$b=\delta h.$$

Define $g'_{ij}=\exp(2\pi i h_{ij})$. Then $g'\in C^1(\mathcal{U},\mathcal{O}_X^*)$, and

$$(\delta g')_{ijk}=\exp(2\pi i(h_{ij}+h_{jk}-h_{ik}))=\exp(2\pi ia_{ijk})=1.$$

Thus $\delta g'=0\in C^2(\mathcal{U},\mathcal{O}_X^*)$, and $g'\in Z^1(\mathcal{U},\mathcal{O}_X^*)$. In this case, by definition of Chern class, $c_1(g')=[\delta h]=[b]$, which means that $b$ has a preimage under $c_1$.

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Divisors corresponds to line bundles

Cor. Let $X$ be a Riemann surface. Then there is a natural isomorphism of groups

$$\mathrm{Pic}(X)\cong H^1(X,\mathcal{O}_X^*).$$

Prop. Let $X$ be a Riemann surface, $D\in\mathrm{Div}(X)$ be a divisor on $X$. Then there exists a holomorphic line bundle $L$ with a meromorphic section $s:X\rightarrow L$, s.t.

$$(s)=D.$$

Denote the line bundle and meromorphic section of $D$ by $L_D$ and $s_D$.