[proofplan] The complex structure gives a pointwise splitting of the complexified cotangent bundle into its $(1,0)$ and $(0,1)$ cotangent subbundles. In a holomorphic coordinate chart, this splitting is exhibited by the local coframe $dz_1,\dots,dz_n,d\bar z_1,\dots,d\bar z_n$. Taking the $k$th exterior power of this direct sum decomposes a local basis into wedges with exactly $p$ holomorphic and $q$ antiholomorphic factors, and these are precisely the local frames of $\Lambda^{p,q}T^*M$. Passing from vector bundles to smooth sections gives the decomposition of complex-valued differential forms, and uniqueness follows from the directness of the local frame decomposition. [/proofplan]

[step:Split the complexified cotangent bundle into holomorphic and antiholomorphic parts]Let $T^*_{\mathbb C}M:=T^*M\otimes_{\mathbb R}\mathbb C$ denote the complexified cotangent bundle, regarded as a complex vector bundle over $M$. The complex structure on $M$ determines complex subbundles $T^{*1,0}M\subset T^*_{\mathbb C}M$ and $T^{*0,1}M\subset T^*_{\mathbb C}M$, where $T^{*1,0}M$ is the bundle locally spanned by holomorphic coordinate differentials and $T^{*0,1}M$ is the bundle locally spanned by antiholomorphic coordinate differentials. Fix a point $x\in M$. Choose a holomorphic coordinate chart $(U,z)$ around $x$, where \begin{align*} z:U\to z(U)\subseteq\mathbb C^n \end{align*} is a holomorphic coordinate map. Write $z_j=x_j+i y_j$ for the complex coordinate functions on $U$, where $x_j:U\to\mathbb R$ and $y_j:U\to\mathbb R$ are smooth real-valued functions. Define complex-valued smooth one-forms $dz_j\in\Omega^1(U;\mathbb C)$ and $d\bar z_j\in\Omega^1(U;\mathbb C)$ by \begin{align*} dz_j:=dx_j+i\,dy_j. \end{align*} \begin{align*} d\bar z_j:=dx_j-i\,dy_j. \end{align*} For each $u\in U$, the set \begin{align*} \{dz_1|_u,\dots,dz_n|_u,d\bar z_1|_u,\dots,d\bar z_n|_u\} \end{align*} is a complex basis of $T^*_{\mathbb C,u}M$, because it is obtained from the real coframe \begin{align*} \{dx_1|_u,\dots,dx_n|_u,dy_1|_u,\dots,dy_n|_u\} \end{align*} by an invertible complex-linear change of basis. Thus over $U$ we have a direct sum decomposition \begin{align*} T^*_{\mathbb C}M|_U=T^{*1,0}M|_U\oplus T^{*0,1}M|_U. \end{align*} Since the subbundles $T^{*1,0}M$ and $T^{*0,1}M$ are defined intrinsically by the complex structure and the above local description is valid in every holomorphic coordinate chart, these local decompositions glue to a global direct sum decomposition \begin{align*} T^*_{\mathbb C}M=T^{*1,0}M\oplus T^{*0,1}M. \end{align*}[/step]

[guided]Let $T^*_{\mathbb C}M:=T^*M\otimes_{\mathbb R}\mathbb C$ be the complexified cotangent bundle. The point of complexifying is that the complex structure separates cotangent vectors into two types: those locally generated by differentials of holomorphic coordinates and those locally generated by differentials of antiholomorphic coordinates. Fix a point $x\in M$ and choose a holomorphic coordinate chart $(U,z)$ around $x$, with \begin{align*} z:U\to z(U)\subseteq\mathbb C^n. \end{align*} Write the coordinate functions as $z_j=x_j+i y_j$, where $x_j:U\to\mathbb R$ and $y_j:U\to\mathbb R$ are smooth real-valued functions. The real one-forms $dx_1,\dots,dx_n,dy_1,\dots,dy_n$ form a real local coframe for $T^*U$. After tensoring with $\mathbb C$, they also form a complex local frame for $T^*_{\mathbb C}M|_U$. Now define complex-valued one-forms \begin{align*} dz_j:=dx_j+i\,dy_j. \end{align*} \begin{align*} d\bar z_j:=dx_j-i\,dy_j. \end{align*} These are not additional independent real one-forms; they are a complex-linear recombination of the real coframe. The inverse recombination is \begin{align*} dx_j=\frac{1}{2}(dz_j+d\bar z_j). \end{align*} \begin{align*} dy_j=\frac{1}{2i}(dz_j-d\bar z_j). \end{align*} Therefore $dz_1,\dots,dz_n,d\bar z_1,\dots,d\bar z_n$ form a complex local frame for $T^*_{\mathbb C}M|_U$. By definition, $T^{*1,0}M|_U$ is locally spanned by $dz_1,\dots,dz_n$, and $T^{*0,1}M|_U$ is locally spanned by $d\bar z_1,\dots,d\bar z_n$. Since the combined list is a complex basis at every point of $U$, the sum is direct and exhausts the complexified cotangent space: \begin{align*} T^*_{\mathbb C}M|_U=T^{*1,0}M|_U\oplus T^{*0,1}M|_U. \end{align*} Because holomorphic coordinate changes preserve the span of the $dz_j$ forms and separately preserve the span of the $d\bar z_j$ forms, these locally defined summands agree on overlaps of holomorphic coordinate charts. Hence the local decompositions assemble into the global complex vector bundle decomposition \begin{align*} T^*_{\mathbb C}M=T^{*1,0}M\oplus T^{*0,1}M. \end{align*}[/guided]

[step:Decompose the exterior power by counting holomorphic and antiholomorphic factors] Fix an integer $k$ with $0\le k\le 2n$. For each pair of integers $(p,q)$ with $0\le p,q\le n$, define \begin{align*} \Lambda^{p,q}T^*M:=\Lambda^p T^{*1,0}M\otimes_{\mathbb C}\Lambda^q T^{*0,1}M. \end{align*} Here $\Lambda^p$ and $\Lambda^q$ denote exterior powers over $\mathbb C$. Using the local frame from the previous step, a local frame for $\Lambda^kT^*_{\mathbb C}M|_U$ is given by wedge products \begin{align*} dz_{i_1}\wedge\cdots\wedge dz_{i_p}\wedge d\bar z_{j_1}\wedge\cdots\wedge d\bar z_{j_q}, \end{align*} where $1\le i_1<\cdots<i_p\le n$, $1\le j_1<\cdots<j_q\le n$, and $p+q=k$. For fixed $(p,q)$, these wedge products form a local frame of $\Lambda^{p,q}T^*M|_U$. Wedge products with distinct pairs $(p,q)$ contain different numbers of $dz$ factors and $d\bar z$ factors, so their spans intersect only in the zero section. Therefore, on $U$, \begin{align*} \Lambda^kT^*_{\mathbb C}M|_U=\bigoplus_{\substack{0\le p,q\le n,\ p+q=k}}\Lambda^{p,q}T^*M|_U. \end{align*} Since this decomposition is obtained from the intrinsic splitting $T^*_{\mathbb C}M=T^{*1,0}M\oplus T^{*0,1}M$, it is compatible on overlaps of holomorphic coordinate charts. Hence it gives the global direct sum decomposition \begin{align*} \Lambda^kT^*_{\mathbb C}M=\bigoplus_{\substack{0\le p,q\le n,\ p+q=k}}\Lambda^{p,q}T^*M. \end{align*} [/step]

[step:Pass from the bundle decomposition to smooth differential forms] Let $\Omega^k(M;\mathbb C):=\Gamma(M,\Lambda^kT^*_{\mathbb C}M)$ denote the complex [vector space](/page/Vector%20Space) of smooth complex-valued $k$-forms on $M$. For each pair $(p,q)$ with $0\le p,q\le n$, let \begin{align*} \Omega^{p,q}(M):=\Gamma(M,\Lambda^{p,q}T^*M) \end{align*} be the complex vector space of smooth forms of type $(p,q)$. Let $\alpha\in\Omega^k(M;\mathbb C)$. The direct sum decomposition of vector bundles from the previous step gives, at every point $x\in M$, a unique decomposition \begin{align*} \alpha_x=\sum_{\substack{0\le p,q\le n,\ p+q=k}}\alpha_x^{p,q}, \end{align*} with $\alpha_x^{p,q}\in\Lambda^{p,q}T_x^*M$. In any holomorphic coordinate chart, the components $\alpha_x^{p,q}$ are obtained by grouping the smooth local coefficient functions of $\alpha$ according to the number of $dz$ and $d\bar z$ factors. Hence $x\mapsto \alpha_x^{p,q}$ is a smooth section of $\Lambda^{p,q}T^*M$, so $\alpha^{p,q}\in\Omega^{p,q}(M)$. Thus \begin{align*} \alpha=\sum_{\substack{0\le p,q\le n,\ p+q=k}}\alpha^{p,q}. \end{align*} If also \begin{align*} \alpha=\sum_{\substack{0\le p,q\le n,\ p+q=k}}\beta^{p,q} \end{align*} with $\beta^{p,q}\in\Omega^{p,q}(M)$, then pointwise directness gives $\alpha_x^{p,q}=\beta_x^{p,q}$ for every $x\in M$ and every admissible pair $(p,q)$. Therefore $\alpha^{p,q}=\beta^{p,q}$ as smooth sections. The decomposition is unique, completing the proof. [/step]