# Review (Monday, March 15)

:::{.remark}
Recall that a *sheaf of rings* \( \sheaf{F} \) on $X\in \Top$ is an assignment of a ring $\sheaf{F}(U)$ to each open set $U\subseteq X$ and restriction maps $\sheaf{F}(U) \mapsvia{\rho_{UV}} \sheaf{F}(V)$ for \( V \subseteq U \) that is a presheaf, so

1. This diagram commutes:

\begin{tikzcd}
	U && V && W
	\arrow["{\rho_{UV}}", from=1-1, to=1-3]
	\arrow["{\rho_{VW}}", from=1-3, to=1-5]
	\arrow["{\rho_{UW}}"', curve={height=30pt}, from=1-1, to=1-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJVIl0sWzIsMCwiViJdLFs0LDAsIlciXSxbMCwxLCJcXHJob197VVZ9Il0sWzEsMiwiXFxyaG9fe1ZXfSJdLFswLDIsIlxccmhvX3tVV30iLDIseyJjdXJ2ZSI6NX1dXQ==)

2. $\phi_{UU} = \one_{\sheaf{F}(U)}$ and \( \sheaf{F}(\emptyset) = 0 \).

That additionally satisfies unique gluing on double overlaps.
:::

:::{.example title="?"}
Any reasonable class of functions whose behavior is only locally restricted.
Examples are being smooth or continuous, but e.g. being constant is a global condition.
Other examples include
$X\in \Mfd^n(C^\infty(\wait, \RR))$, denoting $\OO$ the sheaf of smooth functions.
This also carries a sheaf of *abelian groups* \( \Omega^p \).
In the special case where $U$ is a coordinate chart, we have functions \( \varphi_U: U\to \RR^n \).
Writing \( S \da \varphi_U(U) \), we can define 
\[ 
\Omega^p(U) \cong \Omega^p(S) \da \ts{ \sum f_I(\vector x) dx_I \st f_I \in C^\infty(\RR^n, \RR)}
.\]
:::

:::{.remark}
More generally, for an arbitrary open $U$, cover it by coordinate charts \( \ts{ U_i } \covers U \).
Then we want \( \omega_i \in \Omega^p(U_i) \) which are compatible on double overlaps, so such a collection defines a section \( \ts{ \omega_i \st i\in I } \in \Gamma( \Omega^p(U) ) \).
The compatibility is given by taking coordinate charts \( \varphi_i: U_i \to \RR^n \)  with \( \omega_i \in \Omega^p(U_i) \), we consider
\[
t_{ij}: \varphi_i \circ \varphi_2 ^{-1} : \varphi_j(U_i \intersect U_j) \to \varphi_i( U_i \intersect U_j)
,\]
and we require that the pullback satisfies $t_{ij}^*(\omega_1) = \omega_2$
This pullback can be thought of as a coordinate change for the forms.
Writing \( x_I \) as coordinates on $U_i$ and \( y_J \) on $U_j$, we can write 
\[
x_1 &= h_1(y_J) \\
x_2 &= h_2(y_J) \\
\vdots& \\
x_n &= h_n(y_J) \\
\]
which expresses $t_{ij}$ in coordinates.
This allows us to give meaning to the formal symbols $dx_I$:
\[
dx_1 &\da \sum_{i=1}^n \dd{h_1}{y_i} dy_i \\
dx_2 &\da \sum_{i=1}^n \dd{h_2}{y_i} dy_i \\
\vdots& \\
dx_k &\da \sum_{i=1}^n \dd{h_k}{y_i} dy_i \\
,\]
and under these substitutions in the original expression we obtain 
\[ 
\omega_1 = \sum_{\abs I = p} f_I(\vector x) dx_I \mapsto \omega_2
.\]

:::{.remark}
For \( X \in \Mfd(\Hol(\wait, \CC)) \) such that \( \varphi_V \circ \varphi_U ^{-1} : \varphi_U( U \intersect V) \to \varphi_V(U \intersect V)\) is holomorphic, so $\delbar z_i = 0$.
Then \( \Omega^p(U) = \ts{ \sum_{\abs I = p} f_I( \vector z) dz_I }  \), and the *key difference* is that the $f_I$ be holomorphic.
This matters since POUs exist in the smooth setting but not the complex setting.
Note that $\OO, \Omega^p$ denote smooth/holomorphic functions and smooth/holomorphic $p\dash$forms in the smooth/complex settings.
So we need a new notation for *smooth holomorphic* $p\dash$forms in the complex setting.
We defined $A^{p, 0}$ to be the smooth $p\dash$forms, and $A^{p, q}$ the smooth $(p, q)\dash$forms.
In local coordinates, these look like
\[
A^{p, q}(U) = \ts{ \sum_{\abs I = p, \abs J = q} f_{I, J} (\vector z) dz_I \wedge d\bar{z}_J } 
.\]
:::

:::{.example title="?"}
\envlist

- \( \Re(z) \dz \in A^{1, 0}(\CC)\) is a smooth $(1, 0)\dash$form.
- \( z\dw - w\dz \in \Omega^1(\CC^2) \) is a holomorphic 1-form.
- On $\CC^3$, \( z_1 dz_2 \wedge d\bar{z}_3 - \Re(z_3) dz_1 d\bar{z}_1 \in A^{1, 1}(\CC^3) \).
:::

:::{.remark}
Why are these $A^{p, q}$ useful?
They give a resolution of \( \Omega^p \) on a complex manifold.
There are maps of sheaves
\[
0 \to \Omega^p \mapsvia{i} A^{p, 0}
,\]
where being a map of sheaves means there are maps \( \Omega^p(U) \to A^{p, 0}(U) \) for all opens $U$ which are compatible with restriction:

\begin{tikzcd}
	{\Omega^p(U)} && {A^{p, 0}(U)} \\
	\\
	{\Omega^p(V)} && {A^{p, 0}(U)}
	\arrow["{i_U}", from=1-1, to=1-3]
	\arrow["{i_V}", from=3-1, to=3-3]
	\arrow["{\rho_{UV}^*}"{description}, no head, from=1-1, to=3-1]
	\arrow["{\rho_{UV}^*}", no head, from=1-3, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXE9tZWdhXnAoVSkiXSxbMCwyLCJcXE9tZWdhXnAoVikiXSxbMiwwLCJBXntwLCAwfShVKSJdLFsyLDIsIkFee3AsIDB9KFUpIl0sWzAsMiwiaV9VIl0sWzEsMywiaV9WIl0sWzAsMSwiXFxyaG9fe1VWfV4qIiwxLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dLFsyLDMsIlxccmhvX3tVVn1eKiIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XV0=)

It's clear that this works for $i$, since any holomorphic function simply *is* smooth.
We could continue this resolution:


\[
0 \to \Omega^p \mapsvia{i} A^{p, 0} \mapsvia{\delbar} A^{p, 1}
\]
where
\[
\delbar \qty{ \sum_{I, J} f_{I, J} dz_I \wedge d\bar{z}_J } 
\da 
\sum_{I, J, K} \dd{f_{I, J}}{z_k} d \bar{z}_k \wedge dz_I \wedge d\bar{z}_J
.\]
We then defined Dolbeaut cohomology, $H^q(X, \Omega^p) = \ker \delbar_{p, q} / \im \delbar_{p, q-1}$.


:::