[Vector Bundles III] Differential Forms on Differentiable Manifolds and the Dolbeault's Theorem

We will introduce the notion of differential forms on differentiable manifolds in this note. Also, we restrict ourselves on the higher dimensional analogue of the Dolbeault's theorem for our further purposes, rather than the de Rham's theorem, which easily follows from the generalization of the Poincare's lemma for $d$-operator with highly similar argument for the proof of the Doulbeault's theorem. We will continue to generalize this theorem to the sheaves with coefficients in a vector bundles in Part IV.

1. Differential forms on differentiable manifolds.  Remember that a complex manifold of dimensional $n$ can be considered as a differentiable manifold of dimension $2n$. In the case of Riemann surfaces, i.e. complex manifolds of dimension 1, we realize it as differentiable manifolds of dimension , where we can define at most the sheaf of differential $2$-forms. Similarly, for a differentiable manifold of dimension $n$, we can define at most the sheaf of differential $n$-forms.

Definition 1.1. Let $M$ be a differentiable manifold of dimension $n$. A differential $r$-form $\varphi$ over $M$ is defined locally on a chart $\phi_j; U_j\to U_j'^\subset \mathbb{R}^n$ as
$$\varphi_j=\sum_{\alpha...\gamma}\varphi_{\alpha...\gamma}dx_j^\alpha\wedge...\wedge dx_j^\gamma$$
such that $\varphi_j=\varphi_k$ on $U_j\cap U_k$, where $(x_j^1,...,x_j^n)$ is local coordinate of the chart $U_j$, and $\sum_{\alpha...\gamma}$ represents the all $r$-tuples chosen from the set $\{1,...,n\}$, and $\varphi_{\alpha...\gamma}$ is a differentiable function on $U_j$. The sheaf of differentiable $r$-forms is denoted by $\mathscr{E}$.

Now, if $M$ is a complex manifold, similar to the case of Riemann surfaces, we can introduce a differential $(p,q)$-form $\varphi$ on $M$, which is locally represented by
$$\varphi_j=\sum_{\alpha...\gamma,\mu...\nu}\varphi_{\alpha...\beta,\mu...\nu}dz_j^\alpha\wedge...\wedge dz_j^\gamma\wedge d\bar{z}_j^\mu...\wedge d\bar{z}_j^\nu$$
where $\varphi_{\alpha...\beta,\mu...\nu}$ are differentiable on $U_j$ and $\varphi_j=\varphi_k$ on $U_j\cap U_k$. The sheaf of $(p,q)$ forms on $M$ is denoted by $\mathscr{E}^{p,q}$. In the case $q=0$, and $\varphi_{\alpha...\beta}$ are holomorphic functions, then $\varphi$ is a homomorphic $p$-form. The sheaf of holomorphic $p$-form is denoted by $\Omega^p$.

Similar to the case of Riemann surfaces, one can define the $\bar{\partial}$-operator for each term on a $(p,q)$-form locally represented as above
$$\bar{\partial}(\varphi dz_j^\alpha\wedge...\wedge dz_j^\gamma\wedge d\bar{z}_j^\mu...\wedge d\bar{z}_j^\nu)=\sum_{\lambda=1}^n\frac{\partial \varphi}{\partial\bar{z}_j^\lambda}d\bar{z}_j^\lambda \wedge dz_j^\alpha\wedge...\wedge dz_j^\gamma\wedge d\bar{z}_j^\mu...\wedge d\bar{z}_j^\nu$$
$$=(-1)^p\sum_{\lambda=1}^n\frac{\partial \varphi}{\partial \bar{z}_j^\lambda} dz_j^\alpha\wedge...\wedge dz_j^\gamma\wedge d\bar{z}_j^\lambda  \wedge d\bar{z}_j^\mu...\wedge d\bar{z}_j^\nu$$
And similarly, one can define the $\partial$-operator
$$\partial(\varphi dz_j^\alpha\wedge...\wedge dz_j^\gamma\wedge d\bar{z}_j^\mu...\wedge d\bar{z}_j^\nu)=\sum_{\lambda=1}^n\frac{\partial \varphi}{\partial\bar{z}_j^\lambda}dz_j^\lambda \wedge dz_j^\alpha\wedge...\wedge dz_j^\gamma\wedge d\bar{z}_j^\mu...\wedge d\bar{z}_j^\nu$$
And finally, the $d$-operator, $d\varphi:=\partial \varphi+\bar{\partial}\varphi$. One can see that $\bar{\partial}$ sends a $(p,q)$ forms to a $(p,q+1)$-forms, $\partial$-operator sends a $(p,q)$-form to a $(p+1,q)$-form, and $d$-operator sends a differential $r$-form to a $(r+1)$-form.

Note. We have the following decomposition
$$\mathscr{E}^r(M)=\bigoplus_{p+q=r}\mathscr{E}^{p,q}(M)$$

2. The Dolbeault's lemma and the fine resolution of the sheaf $\Omega^p$.

Theorem 2.1 (Dolbeault's Lemma). Let $\varphi$ be a $(p,q)$-form on a neighborhood of a point $p$ in $M$ such that $\bar{\partial}\varphi=0$ . Then there exists a $(p,q-1)$-form $\omega$ on an open neighborhood $U$ of $p$ such that $\bar{\partial}\omega=\varphi$ on $U$.

In particular. we have the following exact sequence of sheaves (note that the second arrows exact due to the Cauchy-Riemann equation for several complex variables).
$$0\to \mathscr{O}\to \mathscr{E}^{0,0}\xrightarrow{\bar\partial} \mathscr{E}^{0,1}\xrightarrow{\bar\partial}...\xrightarrow{\bar\partial} \mathscr{E}^{0,n}\to 0$$
This is the fine resolution for sheaf $\mathscr{O}$. More generally, due to the C-R criterion for several complex variables, we also have the following fine resolution for the sheaf $\Omega^p$
$$0\to \Omega^p\to \mathscr{E}^{p,0}\xrightarrow{\bar\partial} \mathscr{E}^{p,1}\xrightarrow{\bar\partial}...\xrightarrow{\bar\partial} \mathscr{E}^{p,n}\to 0$$

From the fine resolution above, we obtain the following short exact sequence of sheaves
$$0\to \Omega^p\to \mathscr{E}^{p,0}\xrightarrow{\bar\partial}\bar\partial \mathscr{E}^{p,1}\to 0$$
And we have these information from the induced long exact sequence
$$0\to H^0(M,\Omega^p)\to H^0(M,\mathscr{E}^{p,0})\xrightarrow{\bar\partial}H^0(M,\bar\partial\mathscr{E}^{p,0})\to H^1(X,\Omega^p)\to 0$$
$$H^{m+1}(M,\Omega^p)\cong H^m(M,\bar\partial\mathscr{E}^{p,0})(m\ge 1)$$
From the first one, we have $H^1(X,\Omega^p)\cong H^0(M,\bar\partial\mathscr{E}^{p,0})/\bar\partial H^0(M,\mathscr{E}^{p,0})$. Also, from the fine resolution, we also have the following short exact sequence of sheaves for all $q\ge 0$
$$0\to \bar\partial \mathscr{E}^{p,q}\to \mathscr{E}^{p,q+1}\xrightarrow{\bar\partial} \bar\partial \mathscr{E}^{p,q+1}\to 0$$
Similarly, from the induced long exact sequence, we have the following information
$$H^1(M,\bar\partial\mathscr{E}^{p,q})\cong H^0(M,\bar\partial \mathscr{E}^{p,q+1})/\bar\partial H^0(M, \mathscr{E}^{p,q})$$
$$H^{m+1}(M,\bar\partial\mathscr{E}^{p,q})\cong H^m(M,\bar\partial \mathscr{E}^{p,m+1})(m\ge 1)$$
So, we have for any differentiable manifold $M$,
$$H^{m+1}(M,\Omega^p)\cong H^m(M,\bar\partial\mathscr{E}^{p,0})\cong H^{m-1}(M,\bar\partial\mathscr{E}^{p,1})\cong...\cong H^1(M,\bar\partial\mathscr{E}^{p,m-1})\cong $$
$$\cong H^0(M,\bar\partial\mathscr{E}^{p,m})/\bar\partial H^0(M,\mathscr{E}^{p,m})$$
This is the generalization of the Dolbeault's theorem in Part 2. The more general version will be discussed in Part IV, where we introduce the notion of sheaves with coefficients in a vector bundle.