# Dolbeault cohomology of complex tori.

Let $T=\mathbb C^n/\Lambda$ a complex torus. It is completely elementary to prove that the de Rham cohomology of $T$ in degree $q$ is isomorphic to the $q$-th exterior power of the dual of $\mathbb C^n$.

Indeed, if $\omega$ is a $d$-closed $q$-form on $T$ then for every translation $\tau\colon T\to T$ the form $\tau^*\omega-\omega$ is $d$-exact: one can explicitly determine a $(q-1)$-form $\eta$ such that $d\eta=\tau^*\omega-\omega$ simply by integration of $\omega$ along the segment $[x,x+a]$ if $\tau$ is the translation by $a$ (mimicking the homotopy formula for proving the Poincaré lemma). This shows that the average $\widetilde\omega$ of $\omega$ on $T$ (which has constant coefficients) is again cohomologous to $\omega$. In this way we have a surjective map form the $q$-th exterior power of the dual of $\mathbb C^n$ to $H^q(T,\mathbb C)$. This map is then straightforwardly seen to be injective.

Of course, this proof has nothing to do with the fact that $T$ is complex, but

is there an analogous elementary argument for the Dolbeault cohomology?

Everything goes through, except for the statement

if $\omega$ is a $\overline\partial$-closed $(p,q)$-form on $T$ then $\tau^*\omega-\omega$ is $\overline\partial$-exact.

Of course, I am not looking for a proof which passes through harmonic theory or any other non elementary argument.