Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$, and let $g$ be the Hermitian metric associated to $\omega$. Let $dV_g$ denote the Riemannian volume measure of $g$, and equip smooth complex-valued differential forms on $X$ with the $L^2$ [inner product](/page/Inner%20Product) induced by $g$ and $dV_g$, linear in the first argument and conjugate-linear in the second. For every pair of integers $p,q\geq 0$, let $A^{p,q}(X)$ denote the complex [vector space](/page/Vector%20Space) of smooth complex-valued differential forms of type $(p,q)$ on $X$, with $A^{p,q}(X)=0$ if $p>n$ or $q>n$. Let $A^k(X;\mathbb C)$ denote the complex vector space of smooth complex-valued differential $k$-forms on $X$. Let $d:A^k(X;\mathbb C)\to A^{k+1}(X;\mathbb C)$ be the [exterior derivative](/theorems/1525), and let $d^*:A^{k+1}(X;\mathbb C)\to A^k(X;\mathbb C)$ be its formal adjoint for the fixed $L^2$ inner product. Let $\partial:A^{p,q}(X)\to A^{p+1,q}(X)$ and $\bar\partial:A^{p,q}(X)\to A^{p,q+1}(X)$ denote the Dolbeault components of $d$, and let $\partial^*:A^{p+1,q}(X)\to A^{p,q}(X)$ and $\bar\partial^*:A^{p,q+1}(X)\to A^{p,q}(X)$ be their formal adjoints for the same metric. Let $\Delta_d=d d^*+d^*d$ and $\Delta_{\bar\partial}=\bar\partial\bar\partial^*+\bar\partial^*\bar\partial$ denote the associated Laplacians. Let $H^{p,q}_{\bar\partial}(X)$ denote Dolbeault cohomology, and define the Hodge number by $h^{p,q}(X)=\dim_{\mathbb C}H^{p,q}_{\bar\partial}(X)$. Let $\mathcal H^{p,q}_{\bar\partial}(X)=\{\alpha\in A^{p,q}(X):\Delta_{\bar\partial}\alpha=0\}$ denote the space of $\bar\partial$-harmonic forms for the metric $g$. Then, for all integers $p,q\geq 0$, one has $h^{p,q}(X)=h^{q,p}(X)$.