Dolbeault Theory

Exterior Form on Complexification of the Tangent Bundle

Let $(M,J)$ be an almost complex manifold. The complexified tangent bundle of $M$ is the bundle $\begin{array}{c}TM\otimes \mathbb{C}\\ \downarrow \\ M\end{array}$ with fiber $(TM\otimes \mathbb{C}{)}_p=T_{p}M\otimes \mathbb{C}$ at $p\in M$. If $\begin{array}{ll}{T}_{p}M& \text{is a }2n\text{-dimensional vector space over }\mathbb{R}\text{ , then}\\ T_{p}M\otimes \mathbb{C}& \text{is a }2n\text{-dimensional vector space over }\mathbb{C}\text{ .}\end{array}$ We may extend $J$ linearly to $TM\otimes \mathbb{C}$: $J(v\otimes c)=Jv\otimes c\text{ ,}v\in TM\text{ ,}c\in \mathbb{C}\text{ .}$ Since $J^2=-\text{Id}$, on the complex vector space $(TM\otimes \mathbb{C}{)}_p$, the linear map $J_p$ has eigenvalues $\pm i$. Let $\begin{array}{rl}{T}_{1,0}& =\{v\in TM\otimes \mathbb{C}\mid Jv=+iv\}=(+i)\text{-eigenspace of }J\\ & =\{v\otimes 1-Jv\otimes i\mid v\in TM\}\\ & =\text{(J-)holomorphic tangent vectors ;}\\ & \\ T_{0,1}& =\{v\in TM\otimes \mathbb{C}\mid Jv=-iv\}=(-i)\text{-eigenspace of }J\\ & =\{v\otimes 1+Jv\otimes i\mid v\in TM\}\\ & =\text{(J-)anti-holomorphic tangent vectors .}\end{array}$ Since $\begin{array}{rl}{\pi }_{1,0}:TM& ⟶T_{1,0}\\ v& ⟼\frac{1}{2}(v\otimes 1-Jv\otimes i)\end{array}$ is a (real) bundle isomorphism such that ${\pi }_{1,0}\circ J=i{\pi }_{1,0}$, and $\begin{array}{rl}{\pi }_{0,1}:TM& ⟶T_{0,1}\\ v& ⟼\frac{1}{2}(v\otimes 1+Jv\otimes i)\end{array}$ is also a (real) bundle isomorphism such that ${\pi }_{0,1}\circ J=-i{\pi }_{0,1}$, we conclude that we have isomorphisms of complex vector bundles $(TM,J)\simeq T_{1,0}\simeq \overline{T_{0,1}}\text{ ,}$ where $\overline{T_{0,1}}$ denotes the complex conjugate bundle of $T_{0,1}$. Extending ${\pi }_{1,0}$ and ${\pi }_{0,1}$ to projections of $TM\otimes \mathbb{C}$, we obtain an isomorphism $({\pi }_{1,0},{\pi }_{0,1}):TM\otimes \mathbb{C}\stackrel{\cong }{\to }{T}_{1,0}\oplus T_{0,1}\text{ .}$Similarly, the complexified cotangent bundle splits as $({\pi }^{1,0},{\pi }^{0,1}):T^{\ast }M\otimes \mathbb{C}\stackrel{\cong }{\to }{T}^{1,0}\oplus T^{0,1}$ where \begin{align*} T^{1,0} &= (T_{1,0})^* = \{\eta \in T^* \otimes \mathbb{C} \mid \eta(J\omega) = i\eta(\omega) \text{ , } \forall \omega \in TM \otimes \mathbb{C}\} \\ &= \{\xi \otimes 1 - (\xi \circ J) \otimes i \mid \xi \in T^*M\} \\ &= \textbf{complex-linear cotangent vectors ,} \\ \\ T^{0,1} &= (T_{0,1})^* = \{\eta \in T^* \otimes \mathbb{C} \mid \eta(J\omega) = -i\eta(\omega) \text{ , } \forall \omega \in TM \otimes \mathbb{C}\} \\ &= \{\xi \otimes 1 + (\xi \circ J) \otimes i \mid \xi \in T^*M\} \\ &= \textbf{complex-antilinear cotangent vectors ,} \end{align*} and ${\pi }^{1,0},{\pi }^{0,1}$ are the two natural projections $\begin{array}{rl}{\pi }^{1,0}:T^{\ast }M\otimes \mathbb{C}& ⟶T^{1,0}\\ \eta & ⟼{\eta }^{1,0}:=\frac{1}{2}(\eta -i\eta \circ J)\text{ ;}\\ & \\ {\pi }^{0,1}:T^{\ast }M\otimes \mathbb{C}& ⟶T^{0,1}\\ \eta & ⟼{\eta }^{0,1}:=\frac{1}{2}(\eta +i\eta \circ J)\text{.}\end{array}$

Forms of Type $(l,m)$

For an almost complex manifold $(M,J)$, let ${\Lambda }^{\ell ,m}=({\Lambda }^{\ell }{T}^{1,0})\wedge ({\Lambda }^m{T}^{0,1})$ In particular, ${\Lambda }^{1,0}=T^{1,0}$ and ${\Lambda }^{0,1}=T^{0,1}$.

Complex-Valued k-Forms

For an almost complex manifold $(M,J)$ \begin{align*} \Omega^k(M; \mathbb{C}) \quad &:= \quad \text{sections of } \Lambda^k(T^*M \otimes \mathbb{C}) \\ &= \quad \textbf{complex-valued k-forms on } M \text{ , where} \\ \\ \Lambda^k(T^*M \otimes \mathbb{C}) \quad &:= \quad \Lambda^k(T^{1,0} \oplus T^{0,1}) \\ &= \quad \bigoplus_{\ell+m=k} (\Lambda^\ell T^{1,0}) \wedge (\Lambda^m T^{0,1}) \\ &= \quad \bigoplus_{\ell+m=k} \Lambda^{\ell,m} \text{ .} \end{align*}

Differential Form of Type $(l,m)$

The differential forms of type $(\ell ,m)$ on $(M,J)$ are the sections of ${\Lambda }^{\ell ,m}$: ${\Omega }^{\ell ,m}:=\text{ sections of }{\Lambda }^{\ell ,m}.$ Then ${\Omega }^k(M;\mathbb{C})={\oplus }_{\ell +m=k}{\Omega }^{\ell ,m}.$

Let ${\pi }^{\ell ,m}:{\Lambda }^k(T^{\ast }M\otimes \mathbb{C})\to {\Lambda }^{\ell ,m}$ be the projection map, where $\ell +m=k$. The usual exterior derivative $d$ composed with two of these projections induces differential operators $\partial$ and $\bar{\partial }$ on forms of type $(\ell ,m)$: \begin{align*} \partial \quad &:= \quad \pi^{\ell+1,m} \circ d : \Omega^{\ell,m}(M) \longrightarrow \Omega^{\ell+1,m}(M) \\ \bar{\partial} \quad &:= \quad \pi^{\ell,m+1} \circ d : \Omega^{\ell,m}(M) \longrightarrow \Omega^{\ell,m+1}(M). \end{align*}

J-holomorphic Function

Let $f:M\to \mathbb{C}$ be a smooth complex-valued function on $M$. The exterior derivative $d$extends linearly to $\mathbb{C}$-valued functions as $df=d(Ref)+id(Imf).$ A function f is (J-)holomorphic at $x\in M$ if $d{f}_p$ is complex linear, i.e., $d{f}_p\circ J=id{f}_p$. A function f is (J-)holomorphic if it is holomorphic at all $p\in M$.

(J-)anti-holomorphic Function

A function f is (J-)anti-holomorphic at $p\in M$ if $d{f}_p$ is complex antilinear, i.e., $d{f}_p\circ J=-id{f}_p$.