We will first begin with differential forms in Riemann surfaces then use the Poincare and Dolbeault's lemmas to obtain the fine resolutions for the locally constant sheaf $\mathbb{C}$, and $\Omega^1$. These help us easily deduce the theorems of de Rham for $d$-operator and of Dolbeault for $\overline{\partial}$-operator.
This note will become very long if we continue our discussion about differential forms on higher dimensional differentiable manifolds, and the generalizations of the two theorems above ( mainly based on the book of K. Kodaira "Complex Manifolds and Deformation of Complex Structures"). So, it will be mentioned later in our Part III (Our old Part III about Serre's duality should be changed to Part IV).
1. Differential forms on Riemann surfaces.
We will give a quick survey in this section. For further reference, one can take a look on the book of R. Miranda "Algebraic Curves and Riemann Surfaces".
Let $V_1\subset\mathbb{C}$ be an open subset with coordinate $z$, then a holomorphic 1-form on $V_1$ is of the form $\omega_1:=f(z)dz$, where $f$ is a holomorphic function on $U$. Assume that $V_2\subset \mathbb{C}$ is another subset with $V_1\cap V_2\ne \emptyset$, and $\omega_2:=g(w)dw$ be a holomorphic 1-form on $V_2$, then $\omega_1$ and $\omega_2$ is called compatible with each other if $f(z)dz=g(w)dw$ on $V_1\cap V_2$, i.e. in this case, we have $f(z)dz=g(w)dw$. This is equivalent to $f(z)\frac{\partial z}{\partial w}=g(w).$
Let $M$ be a 1-dimensional complex manifold (i.e. a Riemann surface), with charts $\phi_j: U_j\to V_j$. Then a holomorphic 1-form on $M$ is given by a collection of $\{\omega_j\}$, where $\omega_j$ is holomorphic 1-form on $V_j$, which are compatible with each other. Holomorphic $1$-form on $M$ forms a sheaf, which is often denoted by $\Omega^1$.
As we know in complex analysis. the Cauchy-Riemann (C-R) equation gives us the criterion when is a differentiable function $f(z)=f(x+iy)=u(x,y)+iv(z,y)$ on an open subset $U\subset\mathbb{C}$ is holomorphic. If we denote
$$f_x=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}, f_y=\frac{\partial u}{\partial y}+i\frac{\partial v}{\partial y}$$
Then C-R's criterion lets us know that $f$ is holomorphic iff $f_x+if_y=0$. Using this, we will investigate the differential 1-form on a Riemann surface $M$.
A differential 1-form in an open subset $U\subset \mathbb{C}$ is given by the form $f(z)dx+g(z)dy$, where $f,g$ are differentiable functions on $U$, and $z=x+iy$. We see $\bar{z}=x-iy$, and $x=\frac{1}{2}(z+\bar{z}), y=\frac{1}{2i}(z-\bar{z})$. Then
$$\frac{\partial x}{\partial z}=\frac{\partial x}{\partial \bar{z}}=\frac{1}{2},\frac{\partial y}{\partial z}=-\frac{\partial y}{\partial \bar{z}}=\frac{1}{2i}$$
Hence, we can represent a differential 1-form on $U$ as $f(z,\bar{z})dz+g(z,\bar{z})d\bar{z}$. Then for any differentiable function $f$, we have
$$\frac{\partial f}{\partial z}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial z}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial z}=\frac{1}{2}(f_x-if_y), \frac{\partial f}{\partial \bar{z}}=\frac{1}{2}(f_x+if_y)$$
Due to the C-R equation, one can see $f$ is holomorphic iff $\frac{\partial f}{\partial \bar{z}}=0$. For any differentiable function $f$ on $U$, we define three important operators
i. $\partial f=\frac{\partial f}{\partial z}$
ii. $\bar{\partial} f=\frac{\partial f}{\partial\bar{ z}}$
iii. $df = \partial f+\bar{\partial }f$
Remark 1.1. Due to what we have explained, a differentiable function $f$ on $U$ is holomorphic iff $\bar{\partial }f=0$.
From the definition of differentialbe 1-form on an open subset of $U$, we can generalize this to define differential 1-form on a Riemann surfaces $M$, which is a collection of differential 1-form on charts with compatible condition.
For a differential $1$-form $\omega$ on $M$, we say it is of type $(0,1)$ if it is locally represented by $g(z,\bar{z})d\bar{z}$. And it is of type $(1,0)$ if it is locally represented by $f(z,\bar{z})dz$. Because these types do not change under the compatible condition, they are well-defined definition. From this, we have four kinds of sheaves on a Riemann surface $M$
i. $\mathscr{E}$; the sheaf of differentiable functions on $M$.
ii. $\mathscr{E}^{1,0}$: the sheaf of differential 1-form on $M$ of type $(1,0)$.
iii. $\mathscr{E}^{0,1}$: the sheaf of differential 1-form on $M$ f type $(0,1)$.
iv. $\mathscr{E}^1$: the sheaf of differential $1$-form on $M$.
The natural question is that how the sheaf of type $(1,1)$ looks like? For this, we need to introduce the notions of differential $2$-form. A differential $2$-form on $M$ locally is represented by $fdz\wedge d\bar{z}$, where $f$ is a differentiable function on a chart of $M$, with $dz\wedge d\bar{z}=-d\bar{z}\wedge dz$, $dz\wedge dz=d\bar{z}\wedge d\bar{z}=0$.
Notation 1.2. We denote $\mathscr{E}^2$ the sheaf of differential $2$-form on $M$ (i.e. it is the sheaf of type $(1,1)$ on M). Also, because a Riemann surface is a 1-dimensional complex manifold, i.e. a two dimensional real manifold, for all $n\ge 3$, $\mathscr{E}^n$ is the actually the zero sheaf.
From a differential 1-form $\omega:=fdz+gd\bar{z}$, we can obtain a differential $2$-form by applying to $\omega$ one of three kinds of operators we have just defined above
i. $\partial \omega:= \frac{\partial g}{\partial z}dz\wedge d\bar{z}$
ii. $\bar{\partial}\omega:=\frac{\partial f}{\partial \bar{z}}d\bar{z}\wedge dz=-\frac{\partial f}{\partial \bar{z}}dz\wedge d\bar{z}$
iii. $d\omega:=\partial \omega +\bar{\partial}\omega=(\frac{\partial g}{\partial z}-\frac{\partial f}{\partial \bar{z}})dz\wedge d\bar{z}$
Remark 1.3. By Remark 1.1, $\bar{\partial}\omega=0$ iff $\omega$ is a holomorphic $1$-form.
Now, let $\alpha$ be a differential $2$-form (resp. $1$-form) on $M$, we say $\alpha$ is $d$-exact (resp. $\partial$-exact,$\bar{\partial}$-exact) if there exists a differential 1-form $\omega$ (resp. a differential function $\omega$) on $M$ such that $d\omega=\alpha$ (resp. $\partial\omega=\alpha, \bar{\partial}\omega=\alpha$). And a $1$-form $\omega$ is $d$-closed if $d\omega=0$ (similarly for $\partial$-closed, $\bar{\partial}$-closed). And one can see that any exact $1$-form is closed.
2. The lemmas of Poincare and Dolbeault and their consequences for short exact sequences of sheaves.
We will state without proof for the two lemmas, and deduce their consequences.
Proposition 2.1 (Poincare's Lemma). Let $\omega$ be a differential $1$-form on a neighbor hood of $p$ on a Riemann surface $M$ with $d\omega=0$, then there exists a neighborhood $U$ at $p$ and a differentiable function $f$ defined on $U$ such that $df=\omega$ on $U$.
If we denote $\mathscr{K}:=\ker: \mathscr{E}^1\xrightarrow{d} \mathscr{E}^2$ the kernel sheaf of the operator $d$ from $\mathscr{E}^1\to \mathscr{E}^2$. Then by the Poincare's lemma, the following exact sequence is exact of sheaves
$$\mathscr{E}\xrightarrow{d}\mathscr{K}\to 0$$
i.e. the sheaf map $d$ is onto. This is equivalent to say the $Im(d)=\ker:\mathscr{E}^1\xrightarrow{d} \mathscr{E}^2$, and hence, we obtain the following exact sequence\
$$\mathscr{E}\xrightarrow{d} \mathscr{E}^1\xrightarrow{d} \mathscr{E}^2\to 0$$
For the kernel of the first map, one can see that a $df=0$ locally at $p$ iff $f$ is locally constant function at $p$. If we denote $\mathbb{C}$-the locally constant sheaf on $M$, then we obtain the following exact sequence of sheaves
$$0\to \mathbb{C}\to \mathscr{E}\xrightarrow{d} \mathscr{E}^1\xrightarrow{d} \mathscr{E}^2\to 0$$
This exact sequence will play an important role in the proof of the de Rham's theorem.
Proposition 2.2 (Dolbeault's Lemma). Let $\omega$ be a differential 1-form defined on a neighborhood of a point $p$ on a Riemann surface $M$ of type $(0,1)$. Then on some neighborhoods $U$ of $p\in M$, there exists a differentiable function $f$ defined on $U$ such that $\bar{\partial }f=\omega$
The theorem is equivalent to the sheaf map $\mathscr{E}\xrightarrow{\bar{\partial}}\mathscr{E}^{0,1}$ is onto. And due to Remark 1.1. we have the following short exact sequences of sheaves
$$0\to \mathscr{O}\to \mathscr{E}\xrightarrow{\bar{\partial}} \mathscr{E}^{0,1}\to 0$$
Also, if we view $\mathscr{E}^2$ as $\mathscr{E}^{1,1}$, then the sheaf map $\bar{\partial }$ will map $\mathscr{E}^{1,0}$ to $\mathscr{E}^{1,1}$. And due to the Doulbeault's lemma, $\mathscr{E}^{1,0}\xrightarrow{\bar{\partial }}\mathscr{E}^{1,1}$ is onto. And for any differential form $\omega$ of type $(1,0)$, we have $\bar{\partial }\omega =0$ iff $\omega$ is the holomorphic $1$-form (Remark 1.3). We finally obtain the following short exact sequence of sheaves
$$0\to \Omega^1\to \mathscr{E}^{1,0}\xrightarrow{\bar{\partial}}\mathscr{E}^{1,1}\to 0$$
The two later short exact sequences play an important role in the proof of the Dolbeault's theorem.
3. Fine resolution of sheaves and the theorem of de Rham.
Definition 3.1. Let $\mathscr{F}$ be a sheaf on a differentiable manifold $M$. Then $\mathscr{F}$ is a fine sheaf if for any locally finite open covering $\{U_j\}$ of an open subset $U\subset M$, there is a family of homomorphism $h_j: \mathscr{F}(U_j)\to \mathscr{F}(U)$ such that:
i. $Supp(h_j)\subset U_j$
ii. $\sum_{j}h_j(s|_{U_j})=s$ for all $s\in \mathscr{F}(U)$
Theorem 3.9 in the book of Kodaira states that if $\mathscr{F}$ is a fine sheaf on $M$, then $H^q(M,\mathscr{F})=0$, for all $q\ge 1$. And in example 3.2, he shows that $\mathscr{E}, \mathscr{E}^{p,q}$ are fine sheaves, where $\mathscr{E}^{p,q}$ is the sheaf of differential $(p,q)$-form on $M$ (In the case $M$ is a Riemann surface, $(p,q)$ can be $(0,1),(1,0), (1,1)$).
Now, if $\mathscr{F}^0,...,\mathscr{F}^n$ be sheaves on a differentiable manifold $M$, where $\mathscr{F}^1,...,\mathscr{F}^n$ is fine, and there exists an exact sequence of sheaves
$$0\to \mathscr{F}^0\to \mathscr{F}^1\to...\to \mathscr{F}^n\to 0$$
then this exact sequence is called the fine resolution of $\mathscr{F}^0$. Back to our case, where $M$ is a Riemann surface, we have the fine resolution for the locally constant sheaf $\mathbb{C}$
$$0\to \mathbb{C}\to \mathscr{E}\xrightarrow{d} \mathscr{E}^1\xrightarrow{d}\mathscr{E}^2\to 0$$
This induces the following short exact sequences
$$0\to \mathbb{C}\to \mathscr{E}\xrightarrow{d} d\mathscr{E}\to 0$$
And
$$0\to d\mathscr{E}\to \mathscr{E}^1\to \mathscr{E}^2\to 0$$
The first short exact sequence yields the following long exact sequence (note that $\mathscr{E}$ is a fine sheaf)
$$0\to \mathbb{C}\to H^0(M,\mathscr{E})\xrightarrow{d} H^0(M,d\mathscr{E})\to H^1(M,\mathbb{C})\to 0\to H^1(M,d\mathscr{E})\to H^2(M,\mathbb{C})\to 0...$$
In particular, due to the ker-coker exact sequence, we have
$$H^1(M,\mathbb{C})\cong H^0(M,d\mathscr{E})/dH^0(M,\mathscr{E})$$
One can see that $H^0(M,d\mathscr{E})$ consists all $d$-closed $1$-form on $M$, and $dH^0(M,d\mathscr{E})$ consists of $d$-exact $1$-form on $M$. And hence, $H^0(M,\mathbb{C})$ measures how large the difference of these objects are. It is called the $0$-th de Rham's cohomology group. And also from the long exact sequence, for $n\ge 1$, we have $H^{n+1}(M,\mathbb{C})\cong H^n(M,d\mathscr{E})$.
Now, the second short exact sequence induces the following long exact sequence
$$0\to H^0(M,d\mathscr{E})\to H^0(M,\mathscr{E}^1)\xrightarrow{d} H^0(M,\mathscr{E}^2)\to H^1(M,d\mathscr{E})\to 0$$
And due to the ker-coker exact sequence, we have $H^1(M,d\mathscr{E})\cong H^0(M,\mathscr{E}^2)/dH^0(M,\mathscr{E}^1)$, and $H^n(M,\mathscr{E})$ vanishes for $n>1$. But due to the Poincare's lemma, we have $\mathscr{E}^2=d\mathscr{E}^1$, i.e. $H^1(M,d\mathscr{E})\cong H^0(M,d\mathscr{E}^1)/dH^0(M,\mathscr{E}^1)$, which is called the $1$-st de Rham cohomology group of $M$. We can see that, this quotient, again, is actually measure how large the difference $\{\text{closed 1-form on M}\}/\{\text{exact 1-form on M}\}$.
Combining things together, we have $H^1(M,\mathbb{C})\cong H^0(M,d\mathscr{E}^0)/dH^0(M,\mathscr{E}^0)$, and $H^2(M,\mathbb{C})\cong H^0(M,d\mathscr{E}^1)/dH^0(M,\mathscr{E}^1)$. And an important consequence is that the $k$-th de Rham cohomology group is independent on the differentiable structure of $M$, it is just dependent on the topological structure of $M$.
Theorem 3.2 (de Rham). Let $M$ be a differentiable manifold. We denote $\mathscr{E}^p$ the sheaf of differential $p$-forms on $M$. Then for all $n\ge 1$, we have
$$H^n(M,\mathbb{C})\cong H^0(M,d\mathscr{E}^{p-1})/dH^0(M,\mathscr{E}^{p-1})$$
So, to compute the $k$-th de Rham's cohomology group, we need to compute $H^{k+1}(M,\mathbb{C})$. For example, if $M$ is a Riemann surface, and $M$ is simply connected, then $H^n(M,\mathbb{C})=0$ for $n\ge 1$, and in this case, the following de Rham's complex is exact.
$$0\to \mathscr{E}(M)\xrightarrow{d} \mathscr{E}^1(M)\xrightarrow{d} \mathscr{E}^2(M)\to 0$$
4. The theorem of Dolbeault.
Recall that in Section 2, we obtain the fine resolution for $\mathscr{O}$ and $\Omega^1$. We first consider the fine resolution for $\mathscr{O}$,
$$0\to \mathscr{O}\to \mathscr{E}\xrightarrow{\bar{\partial}} \mathscr{E}^{0,1}\to 0$$
It then induces the following long exact sequence
$$0\to \mathscr{O}(M)\to \mathscr{E}(M)\xrightarrow{\bar{\partial}} \mathscr{E}^{0,1}(M)\to H^1(M,\mathscr{O})\to 0$$
Due to the ker-coker exact sequence, we have
$$H^1(M,\mathscr{O})\cong coker(\bar{\partial})=\mathscr{E}^{0,1}(M)/\bar{\partial }\mathscr{E}(M)$$
One can see that $\mathscr{E}^{0,1}(M)$ consists of all $\bar{\partial }-closed $ $(0,1)$-form, and $\bar{\partial }\mathscr{E}(M)$ consists of all exact $(0,1)$-form of $M$. Hence $H^1(M,\mathscr{O})$ measure how far for a $\bar{\partial }$-closed $(0,1)$-form to be exact. If we consider $\mathscr{O}$ as $\Omega^0$, and $\mathscr{E}(M)$ as $\mathscr{E}^{0,0}(M)$, we then have
$$H^1(M,\Omega^0)\cong \mathscr{E}^{0,1}(M)/\bar{\partial }\mathscr{E}^{0,0}(M)$$
Now, let us investigate the fine resolution of $\Omega^1$. Recall that we have the following exact sequence of sheaves
$$0\to \Omega^1\to \mathscr{E}^{1,0}\xrightarrow{\bar{\partial}}\mathscr{E}^{1,1}\to 0$$
This induces the long exact sequence below
$$0\to \Omega^1(M)\to \mathscr{E}^{1,0}(M)\xrightarrow{\bar{\partial }}\mathscr{E}^{1,1}(M)\to H^1(M,\Omega^1)\to 0$$
Again, due to the ker-coker exact sequence, we have
$$H^1(M,\Omega^1)\cong \mathscr{E}^{1,1}(M)/\bar{\partial }\mathscr{E}^{1,0}(M)$$
We finally obtain the theorem of Dolbeault.
Theorem 4.1 (Dolbeault). Let $M$ be a differentiable manifold, and $\mathscr{E}^{p,q}$ the sheaf of $(p,q)$-form on $M$. Then we have $H^q(M,\mathscr{E}^p)\cong H^0(M,\bar{\partial }\mathscr{E}^{p,q})/\bar{\partial }H^0(M,\mathscr{E}^{p,q-1})$.
The differential form on higher dimensional complex manifolds as well as the two theorems in this note will be discussed in our next part.
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