[proofplan] The proof is local, so we work inside a holomorphic coordinate ball. A Kähler form is closed and of type $(1,1)$, and the local $\partial\bar\partial$-lemma on coordinate balls gives a real-valued smooth potential $\varphi$ with $\omega=i\partial\bar\partial\varphi$. We then compute the coordinate expression of $i\partial\bar\partial\varphi$ and compare it with the usual Hermitian coefficient matrix of the Kähler form. Finally, if two potentials give the same form, their difference has vanishing $i\partial\bar\partial$, which is exactly the local pluriharmonicity condition. [/proofplan] [step:Choose a coordinate ball and apply the local $\partial\bar\partial$-lemma] Fix $p\in X$. Choose a holomorphic coordinate neighbourhood $(V,z)$ of $p$, where \begin{align*} z=(z_1,\dots,z_n):V\to z(V)\subset\mathbb{C}^n \end{align*} is a biholomorphism onto an open subset. Shrinking $V$ if necessary, choose an open coordinate ball $U\subset V$ with $p\in U$ and $z(U)=B(z(p),r)$ for some $r>0$. Since $\omega$ is a Kähler form, $\omega$ is a real smooth closed $(1,1)$-form. Hence $\omega|_U$ is a real smooth closed $(1,1)$-form on the coordinate ball $U$. By the local $\partial\bar\partial$-lemma on coordinate balls, applied to the form $\omega|_U$, there exists a real-valued function $\varphi\in C^\infty(U;\mathbb{R})$ such that \begin{align*} \omega|_U=i\partial\bar\partial\varphi. \end{align*} Here we are citing a result not yet in the wiki: local $\partial\bar\partial$-lemma on coordinate balls. [guided] Fix a point $p\in X$. Because $X$ is a complex manifold of complex dimension $n$, there is a holomorphic coordinate chart \begin{align*} z=(z_1,\dots,z_n):V\to z(V)\subset\mathbb{C}^n \end{align*} defined on an open neighbourhood $V$ of $p$. We shrink this chart domain to a smaller [open set](/page/Open%20Set) $U\subset V$ such that $p\in U$ and $z(U)=B(z(p),r)$ is an ordinary Euclidean ball in $\mathbb{C}^n$ for some $r>0$. This shrinking is harmless because the theorem only asks for a neighbourhood of $p$. The key input is the local $\partial\bar\partial$-lemma on coordinate balls. Its hypotheses are exactly the hypotheses now in force: $\omega|_U$ is smooth, real-valued as a differential form, closed because $d\omega=0$, and of type $(1,1)$ because $\omega$ is a Kähler form. The local $\partial\bar\partial$-lemma therefore gives a real-valued smooth function \begin{align*} \varphi:U\to\mathbb{R} \end{align*} such that \begin{align*} \omega|_U=i\partial\bar\partial\varphi. \end{align*} This is precisely the local potential asserted in the theorem. Here we are citing a result not yet in the wiki: local $\partial\bar\partial$-lemma on coordinate balls. [/guided] [/step] [step:Identify the coefficient matrix of $i\partial\bar\partial\varphi$] In the coordinates $(z_1,\dots,z_n)$, the operators $\partial$ and $\bar\partial$ act on $\varphi$ by \begin{align*} \partial\varphi=\sum_{j=1}^n\frac{\partial\varphi}{\partial z_j}\,dz_j, \qquad \bar\partial\varphi=\sum_{k=1}^n\frac{\partial\varphi}{\partial\bar z_k}\,d\bar z_k. \end{align*} Therefore \begin{align*} \partial\bar\partial\varphi &=\partial\left(\sum_{k=1}^n\frac{\partial\varphi}{\partial\bar z_k}\,d\bar z_k\right)\\ &=\sum_{j,k=1}^n\frac{\partial^2\varphi}{\partial z_j\partial\bar z_k}\,dz_j\wedge d\bar z_k. \end{align*} Multiplying by $i$, we obtain \begin{align*} i\partial\bar\partial\varphi = i\sum_{j,k=1}^n \frac{\partial^2\varphi}{\partial z_j\partial\bar z_k}\,dz_j\wedge d\bar z_k. \end{align*} Thus, if \begin{align*} \omega|_U=i\sum_{j,k=1}^n g_{j\bar{k}}\,dz_j\wedge d\bar z_k, \end{align*} then equality of differential forms gives \begin{align*} g_{j\bar{k}}=\frac{\partial^2\varphi}{\partial z_j\partial\bar z_k} \end{align*} for every $1\le j,k\le n$. [guided] The coordinate computation explains why the complex Hessian appears. Since $\varphi:U\to\mathbb{R}$ is smooth, its $(1,0)$- and $(0,1)$-differentials are \begin{align*} \partial\varphi=\sum_{j=1}^n\frac{\partial\varphi}{\partial z_j}\,dz_j, \qquad \bar\partial\varphi=\sum_{k=1}^n\frac{\partial\varphi}{\partial\bar z_k}\,d\bar z_k. \end{align*} Applying $\partial$ to $\bar\partial\varphi$ gives \begin{align*} \partial\bar\partial\varphi &=\partial\left(\sum_{k=1}^n\frac{\partial\varphi}{\partial\bar z_k}\,d\bar z_k\right)\\ &=\sum_{j,k=1}^n\frac{\partial}{\partial z_j}\left(\frac{\partial\varphi}{\partial\bar z_k}\right)\,dz_j\wedge d\bar z_k\\ &=\sum_{j,k=1}^n\frac{\partial^2\varphi}{\partial z_j\partial\bar z_k}\,dz_j\wedge d\bar z_k. \end{align*} Multiplication by $i$ gives \begin{align*} i\partial\bar\partial\varphi = i\sum_{j,k=1}^n \frac{\partial^2\varphi}{\partial z_j\partial\bar z_k}\,dz_j\wedge d\bar z_k. \end{align*} On the other hand, every real $(1,1)$-form in these coordinates has a unique expression \begin{align*} \omega|_U=i\sum_{j,k=1}^n g_{j\bar{k}}\,dz_j\wedge d\bar z_k \end{align*} with smooth coefficient functions $g_{j\bar{k}}:U\to\mathbb{C}$ satisfying the Hermitian symmetry $g_{j\bar{k}}=\overline{g_{k\bar{j}}}$. Comparing the coefficients of the basis forms $dz_j\wedge d\bar z_k$ gives \begin{align*} g_{j\bar{k}}=\frac{\partial^2\varphi}{\partial z_j\partial\bar z_k} \end{align*} for every $1\le j,k\le n$. [/guided] [/step] [step:Translate positivity of the form into positivity of the complex Hessian] For each point $q\in U$ and each tangent vector \begin{align*} \xi=\sum_{j=1}^n \xi_j\frac{\partial}{\partial z_j}\Big|_q\in T_q^{1,0}X, \end{align*} the Hermitian form associated to $\omega$ is \begin{align*} h_q(\xi,\xi)=\sum_{j,k=1}^n g_{j\bar{k}}(q)\,\xi_j\overline{\xi_k}. \end{align*} Using the coefficient identity from the previous step, \begin{align*} h_q(\xi,\xi) = \sum_{j,k=1}^n \frac{\partial^2\varphi}{\partial z_j\partial\bar z_k}(q)\, \xi_j\overline{\xi_k}. \end{align*} Therefore $\omega$ is positive at $q$ exactly when the Hermitian matrix \begin{align*} \left(\frac{\partial^2\varphi}{\partial z_j\partial\bar z_k}(q)\right)_{j,k=1}^n \end{align*} is positive definite. Since $q\in U$ was arbitrary, positivity of $\omega$ on $U$ is equivalent to pointwise positive definiteness of the complex Hessian of $\varphi$ on $U$. [/step] [step:Show that the difference of two potentials is pluriharmonic] Let $\varphi,\psi\in C^\infty(U;\mathbb{R})$ satisfy \begin{align*} \omega|_U=i\partial\bar\partial\varphi \qquad\text{and}\qquad \omega|_U=i\partial\bar\partial\psi. \end{align*} Define the smooth real-valued function \begin{align*} \rho:U&\to\mathbb{R}\\ q&\mapsto \varphi(q)-\psi(q). \end{align*} Subtracting the two identities gives \begin{align*} i\partial\bar\partial\rho = i\partial\bar\partial\varphi-i\partial\bar\partial\psi = \omega|_U-\omega|_U = 0. \end{align*} Since multiplication by $i$ is injective on complex-valued differential forms, this is equivalent to \begin{align*} \partial\bar\partial\rho=0. \end{align*} By the local characterization of pluriharmonic functions, a real-valued smooth function is pluriharmonic exactly when its complex Hessian vanishes, equivalently when $\partial\bar\partial\rho=0$. Hence $\rho=\varphi-\psi$ is pluriharmonic on $U$. This proves the asserted uniqueness up to pluriharmonic functions and completes the proof. [/step]