# 2025-01-29-10-23-36

:::{.remark}
Let $X$ be a manifold of dimension $n$, $\T_X$ its tangent bundle, $\Omega_X$ its cotangent bundle (both of rank $n$), and let $A^1(X)$ be the space of sections of $\Omega_X$.
Write $x_1,\cdots, x_n$ for local coordinates, and let $\alpha = f_1 \dx_1 + \cdots + f_n \dx_n$.
Write $\Omega^k(X) = \wedge^k \Omega_X$, which is a vector bundle of rank $n$, and let $A^k(X)$ be its space of sections.
Such a form can be written as 
\[
\alpha = \sum_{1\leq i_1\leq \cdots \leq i_k \leq n} \dx_{i_1}\wedge\cdots\wedge \dx_{i_k} = \sum_{I \subseteq [1, n]} f_I \dx_I
.\]
There is a differential
\begin{align*}
d: A^k(X) &\to A^{k+1}(X) \\
\alpha = \sum_I f_I \dx_I &\mapsto d\alpha = \sum_I df_I \wedge \dx_I = \sum_I \sum_{1\leq i \leq n} \dd{f_I}{x_i}\dx_i \wedge \dx_I
\end{align*}

Recall Stokes theorem: 
\[
\int_M d\alpha = \int_{\boundary M} \alpha
.\]
:::

:::{.remark}
Now let $X$ be a complex manifold of complex dimension $n$ with local coordinates $z_1,\cdots, z_n$.
An almost complex structure is an endomorphism $I: \T_X\to\T_X$ with $I^2 = -\id$ -- this is locally multiplication by $i$.
Write $\T_{X, \CC} \da \T_X \tensor_\RR \CC$, which is of complex rank $2n$, then $I: \T_{X, \CC}\selfmap$ becomes a $\CC\dash$linear endomorphism.
We get an eigenspace decomposition $\T_{X, \CC} = \T_X^{1, 0} \oplus \T_X^{0, 1}$, the $i$ and $-i$ eigenspaces respectively.
We saw that $\T_X^{1, 0}$ is a holomorphic vector bundle of rank $n$.
There is a dual decomposition $\Omega_{X, \CC} \da \Omega_X \tensor_\RR \CC = \Omega_X^{1, 0} \oplus \Omega_X^{0, 1}$.
A $(1, 0)$ form is of the form $\alpha = \sum_{j\leq n} f_j \dz_j$, and a $(0, 1)$ form is of the form $\alpha = \sum_{j\leq n} f_j \dzbar_j$ where $f_j$ is a $\CC\dash$valued smooth function.
We have
\[
\Omega^k_{X, \CC} 
= \wedge^k \Omega_{X, \CC} 
= \wedge^k (\Omega_X^{1, 0} \oplus \Omega_X^{0, 1}) 
= \bigoplus _{k=p+q} \wedge^p \Omega_X^{1, 0} \tensor \wedge^q \Omega_X^{0, 1} 
\da \bigoplus _{k=p+q} \Omega_X^{p, q}
.\]
We can write a $(p, q)$ form as 
\[
\alpha = \sum_{I = 1\leq i_1 < \cdots < i_p\leq n, J = 1\leq j_1 < \cdots < j_q \leq n } f_{IJ} \dz_I \wedge \dzbar_J
.\]
where $\dz_I = \dz_{i_1} \wedge \cdots \wedge \dz_{i_p}$ and $\dzbar_J = \dzbar_{j_1}\wedge \cdots \wedge \dzbar_{j_q}$.
Write $A^{p, q}(X)$ for the space of sections, and $A^k_\CC(X) \da \bigoplus _{p+q=k} A^{p, q}(X)$.
There is a differential $d: A^k_\CC(X)\to A^{k+1}_\CC(X)$ where $\alpha = \sum_{I, J} f_{IJ} \dz_I \wedge \dzbar_J$ is sent to
\begin{align*}
d\alpha = \sum_{I, J} df_{I, J} \wedge \dz_I \wedge \dzbar_J \\
= \sum_{I, J} (df_{IJ})^{1, 0} \wedge \dz_I \wedge \dzbar_J
+ \sum_{I, J} (df_{IJ})^{0, 1} \wedge \dz_I \wedge \dzbar_J \\
\da \sum_{i\leq n} \dd{f_{IJ}}{z_i}\dz_i
+ \sum_{i\leq n} \dd{f_{IJ}}{\zbar_i}\dzbar_i \\
\da \del \alpha + \delbar \alpha,
\end{align*}
a $(p+1, q)$ form plus a $(p, q+1)$ form.
We have $d = \del + \delbar$ where $\del: A^{p, q}(X) \to A^{p+1, q}(X)$ and $\delbar: A^{p, q}(X) \to A^{p, q+1}(X)$.
For any manifold, we have $d^2 = 0$, which implies that
\[
0 = d^2 = (\del + \delbar)^2 = \del^2 + (\del\delbar + \delbar\del) + \delbar^2 \in A^{p+2, q} \oplus  A^{p+1, q+1} \oplus  A^{p, q+2}
,\]
and so $\del^2 = 0, \delbar^2 = 0$, and $\del\delbar + \delbar\del = 0$ by degree considerations in the direct sum.
:::