Dolbeault cohomology of a complex torus ($H_{\bar{\partial }}^{p,q}(V/\Lambda )\cong {\bigwedge }^p{V}^{\ast }\otimes {\bigwedge }^q\overline{V^{\ast }}$)

For a complex torus $X=V/\Lambda$ of complex dimension $g$, constant forms give a natural isomorphism $H_{\bar{\partial }}^{p,q}(X)\cong {\bigwedge }^p{V}^{\ast }\otimes {\bigwedge }^q\overline{V^{\ast }}$. A flat Kähler metric and the Dolbeault Hodge decomposition prove that these constant forms are precisely the harmonic representatives. Thus $h^{p,q}=\left(\genfrac{}{}{0ex}{}{g}{p}\right)\left(\genfrac{}{}{0ex}{}{g}{q}\right)$ for $0\le p,q\le g$, and zero otherwise.