In Part I of this series, we have constructed the tangent bundle, which is the essential object in algebraic geometry and complex manifolds. In this note, we will see why it is important. The main reason is that one can construct the dual bundle as well as tensor product of tangent bundle. These constructions allow us to define the notion of "sheaves with coefficients in a vector bundle", and via this, we surprisingly get two versions for the generalization of the Doulbeault's theorem discussed in Part III, and to understand the statement of the Serre's duality.
1. Dual bundles and tensor product of tangent bundles.
Definition 1.1. Let $E,F$ be two vector bundles of rank $n$ over a differentiable manifold $M$. Then $E$ is called the dual bundle of $F$ if
i. For every point $p\in M$, $E_p$ is dual to $F_p$, i.e. the inner product is defined for every $u\in E_p$, and $v\in F_p$.
ii. If we take a sufficiently fine open cover $\{U_j\}$ of $M$, then for all $p\in U_j$, and $u=(p,e_j^1,...,e_j^n)\in E$, and $v=(p,f_j^1,...,f_j^n)\in F$, where $(e_j^1,...,e_j^n), (f_j^1,...,f_j^n)$ are fiber coordinates of $E,F$, respectively, then the inner product between $u,v$ is given by
$$\langle u,v\rangle=\sum_{i=1}^ne_j^if_j^i$$
From the definition. one can see that the dual bundle of $TM$-the tangent bundle of $M$-an $n$-dimensional manifold is exactly the sheaf of all $1$-form on $M$: locally at a point $p$, with coordinate $(x_j^1,...,x_j^n)$, a $1$-form $\omega$ is represented by $u = e_1dx_j^1+...+e_ndx_j^n$, and also, a derivative at $p$ is of the form $v=f_1\frac{\partial}{\partial x_j^1}+...+f_n\frac{\partial}{\partial x_j^n}$. And one can define $\langle u,v\rangle =\sum_{i=1}^n e_if_i$. Hence, the set of $1$-form at $p$ is a dual space of $T_pM$, which is denoted $T_p^*M$. If we define $T^*M=\cup_{p\in M}T_p^*M$, and equip the differentiable structure for it as what we did with $TM$, we have $T^*M$ is a dual bundle of $TM$.
We next construct the tensor product of tangent bundles and its dual.
Definition 1.2. Let $E, F$ be two vector bundles of rank $m,n$, respectively over a differentiable manifold $M$, and $G$ a vector bundle over $M$ of rank $mn$. Then $G$ is called the tensor product of $E, F$ if
i. For every point $p\in M$, $G_p=E_p\otimes F_p$.
ii. For any sufficiently fine open cover $\{U_j\}$ of $M$, and $u=(p,e_j^1,...,e_j^m)\in E$, $v=(p,f_j^1,...,f_j^n)\in F$, the fiber coordinate of $u\otimes v$ of $H$ over $U_j$ is given by $g_j^{\alpha\beta}=e_j^\alpha f_j^\beta$.
This definition again allows us to construct $\otimes^r TM$, which is called the contravariant tensor field of $M$, and $\otimes^r T^*M$-the covariant tensor field of $M$. Let us take a look on elements of $\otimes^r T^*M$, which is of the form on $U_j$, a chart of $M$.
$$\sum_{\alpha,..,\gamma}\varphi_{\alpha...\gamma}^jdx_j^\alpha\otimes...\otimes dx_j^\gamma$$
An element of $\otimes^r T^*M$ is symmetric if $\varphi_{\alpha\beta}=\varphi_{\beta\alpha}$, and skew-symmetric if $\varphi_{\alpha\beta}=-\varphi_{\beta\alpha}$. Hence, one can see that a differential $r$-form
$$\varphi=\sum_{\alpha,...\gamma}\varphi_{\alpha...\gamma}dx_j^\alpha\wedge...\wedge dx_j^\gamma$$
is a skew-symmetric $r$-form on $\otimes^r T^*M$. Hence, the sheaf of differential $r$-forms on $M$ forms a subbundle of $\otimes^r T^*M$, i.e. a subsheaf of $T^*M$, which is also a vector bundle over $M$. This is denoted by $\bigwedge^r T^*M:=\mathscr{E}^r$.
Notation 1.3. By a complex vector bundle $E$ over $M$, we mean that $E, M$ are complex manifolds. If the projection map $\pi: E\to M$ is holomorphic, then $E$ is called a holomorphic vector bundle over $M$.
Now, if $E$ is a complex vector bundle over $M$ with transition functions $\{f_{jk\beta}^\alpha\}$ (coefficients in the invertible matrix of coordinate change between charts). We denote $\overline{E}$ the vector bundle over $M$ with transition function $\{\overline{f_{jk\beta}^\alpha}\}$. And one can construct a vector bundle of the form $\bigwedge^p T^*M\otimes\bigwedge^q \overline{T^*M}$, which is exactly the sheaf of $(p,q)$-form on $M$, i.e. $\bigwedge^p T^*M\otimes\bigwedge^q \overline{T^*M}:=\mathscr{E}^{p,q}$. Take a closer look on this vector bundle, we can see that its basis over a chart $U_j$ is
$$dz_j^\alpha\wedge...dz_j^\gamma\otimes d\bar{z_j}^\mu\wedge...\wedge d \bar{z_j}^\nu$$
2. Sheaves with coefficients in a vector bundle and the first generalization of the Dolbeault's theorem.
Construction/Definition 2.1. Let $E$ be a complex vector bundle over a complex manifold $M$. By $\mathscr{O}(E)$, we mean that it is a sheaf of holomorphic sections of $E$ over $M$. $\mathscr{O}(E)$ is called the sheaf of holomorphic functions with coefficient in $M$.
If $E=TM$, $\mathscr{O}(TM)$ is the sheaf of holomorphic vector fields on $M$. We often denote this sheaf as $\Theta$, which is sometimes called tangent sheaf (Note: It is NOT the tangent bundle!). If $E=\bigwedge^p T^*M$, one can see $\mathscr{O}(\bigwedge^p T^*M)$ is the sheaf of holomorphic section on $\bigwedge^p T^*M$. Locally on a chart $U_j$, a section $s_j$ can be represented by $\sum_{\alpha,...\gamma}\varphi_{\alpha...\gamma}dx_j^\alpha\wedge...\wedge dx_j^\gamma$, where all $\varphi_{\alpha...\gamma}$ are holomorphic functions. This means, $\mathscr{O}(\bigwedge^p T^*M)=\Omega^p$-the sheaf of holomorphic $p$-forms on $M$.
Construction/Definition 2.2. Let $E, M$ be defined as above. We define $\mathscr{E}(E)$ the sheaf of differentiable sections of $E$ over $M$. $\mathscr{E}(E)$ is called the sheaf of differentiable function with coefficients in $E$.
In the case $E=\bigwedge^p T^*M\otimes \bigwedge^q \overline{T^*M}$, then we have $\mathscr{E}(\bigwedge^p T^*M\otimes\bigwedge^q\overline{T^*M})=\mathscr{E}^{p,q}$-the sheaf of $(p,q)$-forms on $M$.
Now, let us take a look again on the fine resolution for the sheaf $\Omega^p$
$$0\to \Omega^p\to \mathscr{E}^{p,0}\xrightarrow{\bar\partial} \mathscr{E}^{p,1}\xrightarrow{\bar\partial}...\xrightarrow{\bar\partial} \mathscr{E}^{p,n}\to 0$$
We can realize $\Omega^p$ as $\mathscr{O}(\bigwedge^pT^*M)$. Also, the sheaf of $\mathscr{E}^{p,q}$ now becomes $\mathscr{E}(\bigwedge^p T^*M\otimes\bigwedge^q\overline{T^*M})$. So if we let $E$ denote the vector bundle $\bigwedge^p T^*M$, we obtain the following exact sequence of sheaves
$$0\to \mathscr{O}(F)\to \mathscr{E}(E)\xrightarrow{\bar\partial} \mathscr{E}(E\otimes \overline{T^*M})\xrightarrow{\bar\partial}...\xrightarrow{\bar\partial} \mathscr{E}(E\otimes \bigwedge^{n}\overline{T^*M})\to 0$$
Further, if we denote $\mathscr{E}^{0,q}(E):= \mathscr{E}(E\otimes \bigwedge^q\overline{T^*M})$, the exact sequence of sheaves above can write again as
$$0\to \mathscr{O}(E)\to \mathscr{E}^{0,0}(E)\xrightarrow{\bar\partial}\mathscr{E}^{0,1}(E)\xrightarrow{\bar\partial}...\xrightarrow{\bar\partial}\mathscr{E}^{0,n}(E)\to 0$$
And it looks very similar to the fine resolution for the sheaf $\mathscr{O}$,
$$0\to \mathscr{O}\to \mathscr{E}^{0,0}\xrightarrow{\bar\partial} \mathscr{E}^{0,1}\xrightarrow{\bar\partial}...\xrightarrow{\bar\partial} \mathscr{E}^{0,n}\to 0$$
Now, we want to generalize the situation, when we define $E$ is any holomorphic vector bundle over $M$ of rank $\nu$. We will take a look on sections of $E\otimes \bigwedge^q\overline{T^*M}$, and then define the suitable $\bar\partial$-operator so that we again have the fine resolution of $\mathscr{O}(E)$, given by
$$0\to \mathscr{O}(E)\to \mathscr{E}^{0,0}(E)\xrightarrow{\bar\partial}\mathscr{E}^{0,1}(E)\xrightarrow{\bar\partial}...\xrightarrow{\bar\partial}\mathscr{E}^{0,n}(E)\to 0$$
Let $U$ be any open subset of $M$, and $\{U_j\}$ its open covering. If $\{e_{j1},...,e_{j\nu}\}$ is the basis for $E$ over $U_j$, and $\{f_{jk\mu}^\lambda\}$ the transition functions, where $f_{jk\mu}^\lambda$ is holomorphic, i.e.
$$e_{k\mu}=\sum_{\lambda=1}^\nu f_{jk\mu}^\lambda e_{j\lambda}$$
on $U_j\cap U_k$. Then the basis for $E\otimes \bigwedge^r\overline{T^*M}$ can be given by
$$e_{j\lambda}\otimes d\bar{z}_j^\alpha\wedge...\wedge d\bar{z}_j^\gamma$$
So, for any differentiable section $\varphi_j$ on $U_j$, it can be expressed as
$$\varphi_j=\sum_{\lambda=1}^\nu e_{j\lambda}\otimes(\sum_{\alpha...\gamma}\varphi_{j\alpha...\gamma}^\lambda d\bar{z}_j^\alpha\wedge...\wedge d\bar{z}_j^\gamma)$$
If for fixed $j$ and $\lambda$, if we let $\varphi_j^\lambda:=\sum_{\alpha...\gamma}\varphi_{j\alpha...\gamma}^\lambda d\bar{z}_j^\alpha\wedge...\wedge d\bar{z}_j^\gamma$, then $\varphi_j^\lambda$ is a differential $p$-form on $U_j$, and
$$\varphi_j=\sum_{\lambda=1}^\mu e_{j\mu}\otimes \varphi_j^\lambda$$
And due to the formula related to transition functions, we have on $U_j\cap U_k$
$$\varphi_k^\mu=\sum_{\lambda=1}^\nu f_{jk\mu}^\lambda \varphi_j^\lambda$$
That means, if we are given a differentiable section on $U$, then on each $U_j$, we have a $\nu$-tuple of differential $p$-forms $(\varphi_j^1,...,\varphi_j^\mu)$ that compatible with the transition functions. Conversely, it can if we are give a $\nu$-tupe of differential $p$-form on each $U_j$ compatible with the transition functions of line bundles, i.e. they agree on the overlap $U_j\cap U_k$. So we can glue them to get a unique section on $U$.
In short, if $E$ is any holomorphic vector bundle over $M$ of rank $\nu$, then the section of $E$ is characterized by the $\nu$-tuples of differential form, and they are compatible with transition functions. And hence, we can define
$$\bar\partial: \mathscr{E}^{0,q}(E)\to \mathscr{E}^{0,q+1}$$
by sending $(\varphi_j^1,...,\varphi_j^\nu)\mapsto (\bar\partial \varphi_j^1,...,\bar\partial\varphi_j^\nu)$. Because each $f_{jk\mu}^\lambda$ is holomorphic, $\bar\partial f_{jk\mu}^\lambda$ vanishes. Hence, this definition is well-defined. Because the sheaf $\mathscr{E}^{0,q}$ on $M$ is fine, we have $\mathscr{E}^{0,q}(E)$ is also fine. And finally, we obtain the following fine resolution for the sheaf $\mathscr{O}(E)$
$$0\to \mathscr{O}(E)\to \mathscr{E}^{0,0}(E)\xrightarrow{\bar\partial}\mathscr{E}^{0,1}(E)\xrightarrow{\bar\partial}...\xrightarrow{\bar\partial}\mathscr{E}^{0,n}(E)\to 0$$
With information from the exact sequence of sheaves above as in our previous part, we can easily formulate the first form of the generalization of the Dolbeault's theorem
Theorem 2.3. Let $M$ be a complex manifold and $E$ is a holomorphic vector bundle over $M$, then $$H^{m+1}(M, \mathscr{O}(E))\cong H^0(M,\bar\partial\mathscr{E}^{0,m}(E))/\bar\partial H^0(M,\mathscr{E}^{0,m}(E))$$.
3. The second generalization of Doulbeault's theorem.
Let us take a look on the fine resolution of the sheaf $\mathscr{O}(E)$
$$0\to \mathscr{O}(E)\to \mathscr{E}^{0,0}(E)\xrightarrow{\bar\partial}\mathscr{E}^{0,1}(E)\xrightarrow{\bar\partial}...\xrightarrow{\bar\partial}\mathscr{E}^{0,n}(E)\to 0$$
In particular, we want to generalize this more, if we observe that $\mathscr{O}(E)$ is somehow $\Omega^0(E)$. And what we want to construct is the perfect generalization of the resolution for the sheaf $\Omega^p(E)$, as in the original form
$$0\to \Omega^p\to \mathscr{E}^{p,0}\xrightarrow{\bar\partial} \mathscr{E}^{p,1}\xrightarrow{\bar\partial}...\xrightarrow{\bar\partial} \mathscr{E}^{p,n}\to 0$$
So, what is the suitable definition for $\Omega^p(E)$ so that we have the following fine resolution
$$0\to \Omega^p(E)\to \mathscr{E}^{p,0}(E)\xrightarrow{\bar\partial} \mathscr{E}^{p,1}(E)\xrightarrow{\bar\partial}...\xrightarrow{\bar\partial} \mathscr{E}^{p,n}(E)\to 0$$
Actually, if we look back, we will see $\bigwedge^pT^*M\otimes \bigwedge^p \overline{T^*M}$ is realized as $\mathscr{E}^{p,q}$-the sheaf of differential $(p,q)$-form on $M$. So the perfect construction is that we tensor $E$ with $\bigwedge^pT^*M\otimes \bigwedge^p \overline{T^*M}$ to define the $\mathscr{E}^{p,q}(E)$-the sheaf of $(p,q)$-form with coefficients on $E$.
Construction/Definition 3.3. Let $E,M$ be defined as above. We define $\mathscr{E}^{p,q}:=\mathscr{E}(E\otimes \bigwedge^pT*M\otimes \bigwedge^p \overline{T^*M})$-the sheaf of differentiable section on the holomorphic vector bundle $E\otimes \bigwedge^pT*M\otimes \bigwedge^p \overline{T^*M}$. This is called the sheaf of $(p,q)$-form with coefficients in $E$. Also, we define $\Omega^p(E):=\mathscr{O}(E\otimes \bigwedge^pT^*M)$-the sheaf of holomorphic section on the holomorphic vector bundle $E\otimes \bigwedge^pT^*M$. It is called the sheaf of holomorphic $p$-form with coefficient in $E$.
Now, by the very similar method to Section 2, we obtain the fine resolution for the sheaf $\Omega^{p}(E)$
$$0\to \Omega^p(E)\to \mathscr{E}^{p,0}(E)\xrightarrow{\bar\partial} \mathscr{E}^{p,1}(E)\xrightarrow{\bar\partial}...\xrightarrow{\bar\partial} \mathscr{E}^{p,n}(E)\to 0$$
And the second generalization of the Dolbeault's theorem now follows
Theorem 3.2. Let $M$ be a complex manifold and $E$ is a holomorphic vector bundle over $M$, then $$H^{m+1}(M, \Omega^p(E))\cong H^0(M,\bar\partial\mathscr{E}^{p,m}(E))/\bar\partial H^0(M,\mathscr{E}^{p,m}(E))$$.
4. Serre's duality and Riemann-Roch's theorem revisited.
The Serre's duality is also proved in the book of Kodaira, with the use of harmonic forms. We will state it here
Theorem 4.1. (Serre's duality). Let $M$ is a compact complex manifold of dimension $n$, and $E$ a holomorphic vector bundle over $M$. Then
$$H^q(M,\Omega^p(E))\cong H^{n-q}(M, \Omega^{n-p}(E^*))$$
where $E^*$ is the dual bundle of $E$.
Assume that $M$ is an $n$-dimensional compact complex manifold. We define the canonical bundle of $M$ as $K:=\bigwedge^n T^*M$. It can be seen that $\bigwedge^n T^*M$ have basis over a chart $U_j$ as $dx_j^1\wedge...\wedge dx_j^n$. Hence, $K$ is a line bundle over $M$. In the case $p=0$, we have $\Omega^p(E)=\mathscr{O}(E)$, and $\Omega^n(E^*)$, due to Definition 3.1, is $\mathscr{O}(E^*\otimes \bigwedge^nT*M)=\mathscr{O}(E^*\otimes K)$. And the Serre's duality becomes
$$H^q(M,\mathscr{O}(E))\cong H^{n-q}(M, \mathscr{O}(E^*\otimes K))$$
In the case $M$ is a compact Riemann surface, i.e. 1-dimensional complex manifold (NOTE: All non-singular algebraic curves over $\mathbb{C}$ are Riemann surfaces), one can see that $K=T^*M=\Omega^1$, the dual bundle of $TM$. If $E$ is any line bundle over $M$, note that $E^*$ is just $E^{-1}$, because all line bundle over $M$ forms a group structure with tensor product. So, Serre's duality becomes
$$H^1(M,\mathscr{O}(E))\cong H^{0}(M, \mathscr{O}(E^*\otimes K))\cong \mathscr{O}(K\otimes E^{-1})$$
Therefore, the case $\deg L-\deg K<0$, we have $h^0(E^{-1}\otimes K)=0$, and hence, we have $h^1(L)=0$. This implies $h^1(L)$ will vanish when $\deg L$ is sufficient large. Using this, we can easily obtain the cohomology version for Riemann-Roch's theorem (For proof, see Gathmann's note, version 2014, Chapter VIII)
$$h^0(D)-h^1(D)=\deg D+1-g$$
Because for any divisor $D$, we can realize $\mathscr{O}(D)$ as a line bundle $E$. Hence, the Riemann-Roch's theorem can be restated as
$$h^0(E)-h^1(E)=\deg E+1-g$$
By the corollary of Serre's duality theorem that we have discussed, one obtains
$$h^0(E)-h^0(K\otimes E^{-1})=\deg E+1-g$$
This is a Riemann-Roch's theorem for line bundles over a compact Riemann surface.
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