[proofplan] We prove the decomposition locally in holomorphic coordinates, where the complexified cotangent bundle has the frame $dz_1,\dots,dz_n,d\bar z_1,\dots,d\bar z_n$. In that frame, every complex-valued $k$-form has a unique expansion into wedge monomials containing exactly $p$ factors of the form $dz_i$ and exactly $q$ factors of the form $d\bar z_j$, with $p+q=k$. We then check that holomorphic coordinate changes preserve the holomorphic and antiholomorphic spans separately, so the local type projections are independent of the chart. These local projections therefore glue to global projections, and uniqueness of the local frame expansion gives directness. [/proofplan] [step:Decompose a complex-valued form in one holomorphic coordinate chart] Let $(U,\varphi)$ be a holomorphic coordinate chart on $X$, where \begin{align*} \varphi: U \to V \subseteq \mathbb{C}^n \end{align*} is a biholomorphism onto an [open set](/page/Open%20Set) $V$. Let $z_i: U \to \mathbb{C}$ denote the $i$th coordinate function, defined by $z_i=\pi_i \circ \varphi$, where $\pi_i:\mathbb{C}^n\to\mathbb{C}$ is projection onto the $i$th coordinate. The complex-valued one-forms $dz_1,\dots,dz_n,d\bar z_1,\dots,d\bar z_n$ form a local frame for the complexified cotangent bundle $T_{\mathbb{C}}^*X$ over $U$. For an increasing multi-index $I=(i_1,\dots,i_p)$ with $1\leq i_1<\cdots<i_p\leq n$, define \begin{align*} dz_I := dz_{i_1}\wedge \cdots \wedge dz_{i_p}. \end{align*} For an increasing multi-index $J=(j_1,\dots,j_q)$ with $1\leq j_1<\cdots<j_q\leq n$, define \begin{align*} d\bar z_J := d\bar z_{j_1}\wedge \cdots \wedge d\bar z_{j_q}. \end{align*} For the empty multi-index, set $dz_\varnothing=d\bar z_\varnothing=1$. Since exterior powers of a free module have the wedge products of increasing frame elements as a basis, the forms \begin{align*} dz_I \wedge d\bar z_J \end{align*} with $|I|+|J|=k$ form a local frame for $\Lambda^k T_{\mathbb{C}}^*X$ over $U$. Hence every $\omega \in A^k(X)$ has a unique local expansion on $U$ of the form \begin{align*} \omega|_U=\sum_{|I|+|J|=k} f_{I,J}\, dz_I\wedge d\bar z_J, \end{align*} where each coefficient is a smooth function $f_{I,J}:U\to\mathbb{C}$. For each pair $(p,q)$ with $p+q=k$, define the local type component of $\omega$ on $U$ by \begin{align*} \omega_U^{p,q}:=\sum_{|I|=p, |J|=q} f_{I,J}\, dz_I\wedge d\bar z_J. \end{align*} Then \begin{align*} \omega|_U=\sum_{p+q=k}\omega_U^{p,q}, \end{align*} and the uniqueness of the coefficients $f_{I,J}$ makes the local decomposition unique. [guided] The goal in one chart is to reduce the statement to linear algebra. A holomorphic coordinate chart is a map \begin{align*} \varphi: U \to V \subseteq \mathbb{C}^n \end{align*} from an open subset $U\subseteq X$ onto an open subset $V\subseteq \mathbb{C}^n$. Let $z_i:U\to\mathbb{C}$ be the coordinate function $z_i=\pi_i\circ\varphi$, where $\pi_i:\mathbb{C}^n\to\mathbb{C}$ is the $i$th coordinate projection. The complex-valued one-forms $dz_1,\dots,dz_n,d\bar z_1,\dots,d\bar z_n$ form a local frame of the complexified cotangent bundle $T_{\mathbb{C}}^*X$ over $U$. Now take an increasing multi-index $I=(i_1,\dots,i_p)$ and define \begin{align*} dz_I := dz_{i_1}\wedge \cdots \wedge dz_{i_p}. \end{align*} Similarly, for an increasing multi-index $J=(j_1,\dots,j_q)$, define \begin{align*} d\bar z_J := d\bar z_{j_1}\wedge \cdots \wedge d\bar z_{j_q}. \end{align*} For the empty multi-index, we set $dz_\varnothing=d\bar z_\varnothing=1$, so that the notation also covers pure $(p,0)$-forms and pure $(0,q)$-forms. Since the $dz_i$ and $d\bar z_j$ form a frame for $T_{\mathbb{C}}^*X$, the exterior algebra basis for $\Lambda^k T_{\mathbb{C}}^*X$ is obtained by wedging $k$ distinct frame elements in increasing order. Grouping those basis elements according to how many $dz$ factors and how many $d\bar z$ factors they contain gives the frame elements \begin{align*} dz_I \wedge d\bar z_J \end{align*} with $|I|+|J|=k$. Therefore, for any smooth complex-valued $k$-form $\omega\in A^k(X)$, there are unique smooth coefficient functions $f_{I,J}:U\to\mathbb{C}$ such that \begin{align*} \omega|_U=\sum_{|I|+|J|=k} f_{I,J}\, dz_I\wedge d\bar z_J. \end{align*} The uniqueness is important: it means that the coefficient of each basis wedge is not a choice but is determined by $\omega$. For fixed integers $p,q$ with $p+q=k$, define \begin{align*} \omega_U^{p,q}:=\sum_{|I|=p, |J|=q} f_{I,J}\, dz_I\wedge d\bar z_J. \end{align*} This is exactly the part of $\omega|_U$ containing $p$ holomorphic factors and $q$ antiholomorphic factors. Summing over all pairs with $p+q=k$ recovers the full local expansion: \begin{align*} \omega|_U=\sum_{p+q=k}\omega_U^{p,q}. \end{align*} Thus the decomposition exists and is unique inside a single holomorphic coordinate chart. [/guided] [/step] [step:Show that type is preserved by holomorphic coordinate changes] Let $(U,\varphi)$ and $(U',\psi)$ be holomorphic coordinate charts with nonempty overlap $W:=U\cap U'$. Let $z_1,\dots,z_n$ be the coordinate functions on $U$, and let $w_1,\dots,w_n$ be the coordinate functions on $U'$. Define the transition map \begin{align*} F:=\psi\circ\varphi^{-1}:\varphi(W)\to\psi(W). \end{align*} Since $X$ is a complex manifold, $F$ is holomorphic. Writing $F=(F_1,\dots,F_n)$, the coordinate identity on $W$ is $w_a=F_a(z_1,\dots,z_n)$ for each $a\in\{1,\dots,n\}$. The complex chain rule and holomorphicity of $F_a$ give \begin{align*} dw_a=\sum_{i=1}^n \frac{\partial F_a}{\partial z_i}\, dz_i. \end{align*} Taking complex conjugates gives \begin{align*} d\bar w_a=\sum_{i=1}^n \overline{\frac{\partial F_a}{\partial z_i}}\, d\bar z_i. \end{align*} Thus each $dw_a$ lies in the span of $dz_1,\dots,dz_n$, and each $d\bar w_a$ lies in the span of $d\bar z_1,\dots,d\bar z_n$. Therefore a wedge monomial containing $p$ factors of the form $dw_a$ and $q$ factors of the form $d\bar w_b$ expands only into wedge monomials containing $p$ factors of the form $dz_i$ and $q$ factors of the form $d\bar z_j$. Hence the subspace of local forms of type $(p,q)$ is the same whether computed in the $z$-coordinates or in the $w$-coordinates. [/step] [step:Glue the local type projections to global forms] For each chart $U$ and each pair $(p,q)$ with $p+q=k$, the previous step shows that the local component $\omega_U^{p,q}$ agrees on overlaps with the corresponding local component computed in any other holomorphic chart. Hence the family of local forms $\{\omega_U^{p,q}\}$ glues to a unique global smooth form \begin{align*} \omega^{p,q}\in A^{p,q}(X). \end{align*} Since the equality \begin{align*} \omega|_U=\sum_{p+q=k}\omega_U^{p,q} \end{align*} holds in every chart $U$, the glued global forms satisfy \begin{align*} \omega=\sum_{p+q=k}\omega^{p,q}. \end{align*} Thus every element of $A^k(X)$ is a sum of global forms of types $(p,q)$ with $p+q=k$. [/step] [step:Use local uniqueness to prove the sum is direct] Suppose that a finite sum of type components vanishes: \begin{align*} \sum_{p+q=k}\alpha^{p,q}=0, \end{align*} where $\alpha^{p,q}\in A^{p,q}(X)$ for each pair $(p,q)$ with $p+q=k$. Restrict this equality to an arbitrary holomorphic coordinate chart $U$. In the local frame $dz_I\wedge d\bar z_J$, the form $\alpha^{p,q}|_U$ is a sum only of basis elements with $|I|=p$ and $|J|=q$. The families of basis elements corresponding to distinct pairs $(p,q)$ are disjoint. Since expansion in a local frame is unique, all local coefficients of every $\alpha^{p,q}|_U$ vanish. Therefore $\alpha^{p,q}|_U=0$ for every chart $U$ and every pair $(p,q)$. Because holomorphic coordinate charts cover $X$, each global form $\alpha^{p,q}$ vanishes on all of $X$. Hence the sum is direct. Combining existence with directness gives \begin{align*} A^k(X)=\bigoplus_{p+q=k}A^{p,q}(X). \end{align*} This is the desired type decomposition. [/step]