# Friday, April 02

## When Line Bundles are $\OO$ of a Divisor

:::{.remark}
Last time: if we have such a Hodge diamond, can we solve for $h^{1, 1}$?

\begin{tikzcd}
	&& 1 \\
	& q && q \\
	p && {h^{1, 1}} && p \\
	& q && q \\
	&& 1
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOSxbMiwyLCJoXnsxLCAxfSJdLFswLDIsInAiXSxbNCwyLCJwIl0sWzEsMywicSJdLFszLDMsInEiXSxbMiw0LCIxIl0sWzEsMSwicSJdLFszLDEsInEiXSxbMiwwLCIxIl1d)

Recall Noether's formula
\[
\chi(S, \OO_S)
&= \int \ch(\OO_S) \td(S) \\
&= \int_S {x_1 \over 1-e^{-x_1} } {x_2 \over 1- e^{-x_2} } \\
&= {K^2 + \chi_\Top(S) \over 12}
,\]
where $c_1(TS) = - K$ and $\chi_\Top$ is due to the Chern-Gauss-Bonet formula.
We have
\[
\chi(\OO_S) = h^0(\OO_S) - h_1(\OO_S) + h^2(\OO_S) = 1-q+p
.\]

On the other hand,
\[
\chi_\Top(S) = 1 -2q + (2p + h^{1, 1}) -4q = 1-4q + 2^p + h^{1, 1}
,\]
so 
\[
12(1-q+p) = K^2 + 2-4g + 2p + h^{1, 1} \implies h^{1, 1} = 110 - 8q + 10p-K^2
.\]

:::

:::{.remark}
Recall the extraordinarily important exact sequence
\[
0 \to \OO(-p) \to \OO \to \OO_p \to 0
,\]
where the right-hand side is the sheaf of holomorphic functions vanishing at $p$ and this is an inclusion into the sheaf of holomorphic functions, and the right-hand term is the skyscraper sheaf.
There is a similar exact sequence for an embedded curve $C\embeds S$ in a surface:
\[
0 \to \OO_S(-C) \to \OO_S \to \OO_C \to 0
,\]
where the left term is the sheaf of holomorphic functions vanishing on $C$. 
Note that this has no global sections!
Any function vanishing along a compact subset (?) are constant (?).
Locally on an open set $U$, one can write $C \intersect U = V(f_u)$, since algebraically this ring is locally a PID.
So this is a line bundle, where we can map into the trivial bundle by $\phi \mapsto \phi/f_u$.
Thus
\[
\OO_S(U) / \OO_S(-C)(U) \cong \OO_C(C \intersect U )
.\]
We then get surjectivity since every holomorphic function on $C$ extends to a holomorphic function on $S$.

Now letting $\bundle{E} \in\Vect(\Hol)$, we can tensor this exact sequence to get
\[
0 \to \bundle{E}(-C) \to \bundle{E} \to \ro{\bundle{E}}{C} \to 0
,\]
which is also exact since locally we have the splitting principle.
:::

:::{.proposition title="Every line bundle over a smooth projective complex manifold is O of a divisor"}
Let $X$ be a smooth projective
[^projective_Def-Reminder]
complex manifold.
Then every line bundle over $X$ is of the form $L = \OO_X(D)$ for some divisor $D = \sum n_i D_i \in \ZZ[\submfds(\codim_1)]$.

[^projective_Def-Reminder]: 
So $X$ admits an embedding into some $\CP^N$.

:::

## Proof

:::{.proof title="?"}
Let $H$ be a **hyperplane section**, i.e. an intersection of $X$ with a generic hyperplane in $\CP^N$.

:::{.lemma title="Serre Vanishing Theorem"}
For any vector bundle $\bundle{E}$ and all $i>0$, for $k\gg 0$ we have
\[
h^i(X, \bundle{E} \tensor \OO(kH) ) = 0
.\]
:::


:::{.remark}
We'll not prove this!
It requires some heavy analysis and the Kähler identities, see Huybrechts complex geometry Prop 5.27.
:::

We can write
\[
\chi(L\tensor \OO(kH)) 
&= \int_X \ch(L\tensor \OO(kH)) \td(X) \\
&= \int_X \ch(L) \ch(H)^k \td(X) \\
&= \int_X \qty{ 1 + c_1(L) + {c_1(L)^2 \over 2} + \cdots } 
\cdot \qty{1+ kh + {(kh)^2 \over 2 } + \cdots + {(kh)^{\dim X} \over (\dim X)!} }
\cdot \qty{1 + \td_1(X) + \td_2(X) + \cdots }
.\]

where $h$ is the restriction of the generator of $H^2(\CP^N; \ZZ)$ to $X$.
Note that for $k$ large, the dominating term grows like $(kh)^{\dim X}$, so asymptotically we have
\[
\cdots \sim \int_X { k^{\dim X} h^{\dim X} \over (\dim X)! }
.\]

What is this $\dim(X)\dash$fold intersection?

\begin{tikzpicture}
\fontsize{43pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Spring/FourManifolds/sections/figures}{2021-04-02_14-25.pdf_tex} };
\end{tikzpicture}

We can slice $X$ by multiple hyperplanes, each homologically perturbed, and so $\int_X h^{\dim X}$ is the number of points where $\dim X$ generic hyperplanes intersect $X$, which is called the **degree** $\deg X$.
This roughly follows from $\int_X \omega_\FS^{\dim X} > 0$.
Alternatively, suppose $X \intersect H = \emptyset$, then $X \embeds H^c = \AA^N$.
Then each holomorphic coordinate restricts to a constant on $X$ by the maximal principle.

Back to what we were proving: we have
\[
\chi(L \tensor \OO(kH) ) \sim ck^{\dim X}
,\]
for $c$ some constant.
By Serre Vanishing, $h^i(L\tensor \OO(kH)) = 0$ for $k\gg 0$, and so we obtain
\[
h^0(L\tensor \OO(kH)) \sim ck^{\dim X} \implies \exists k \text{ s.t. } h^0(L\tensor \OO(kH)) > 0
.\]
We conclude that there is some nonzero section $s\in \Hsh^0(X; L\tensor \OO(kH))$ for which $\OO(\div s) \cong L\tensor \OO(kH)$.
Thus $L \cong \OO(\div s - kH)$, where $\div s- kH$ is some divisor.


:::

:::{.remark}
With some more work, one can show $L\cong \OO(C-D)$ for $C,D$ *smooth* divisors.
:::

## Aside

:::{.remark}
Felix Klein has a "proof" of the existence of a meromorphic function on a Riemann surface.
The argument roughly goes as follows:
take your Riemann surface and make it out of metal.
Attach it to a battery:

\begin{tikzpicture}
\fontsize{44pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Spring/FourManifolds/sections/figures}{2021-04-02_14-39.pdf_tex} };
\end{tikzpicture}

This induces an electric potential function $V: C\to \RR$, where $V$ is the real part of the meromorphic function.
Here $V$ is a harmonic function away from $p$ and $q$.
:::