= Solution
Choose a constant positive <Hermitian form> on $V$. Its underlying flat <Kähler metric> is translation-invariant and descends through the <Euclidean lattice> to the compact <complex torus> $X$. The <Dolbeault Hodge decomposition> gives a unique <Dolbeault Laplacian>-harmonic representative of every <Dolbeault cohomology> class. By the <Kähler Laplacian identity>, these are also <Hodge Laplacian>-harmonic.
Choose a constant coframe on $V$. A lifted harmonic form is a sum of constant wedge basis forms with smooth periodic coefficients. For the flat metric, the <Hodge Laplacian> acts coefficientwise by the scalar constant-coefficient Laplacian. Each coefficient is a <harmonic function> on a compact flat real torus, hence is constant: <integration by parts> gives $\int|Da|^2=0$. Conversely every constant-coefficient form is harmonic. Thus the harmonic representatives are exactly the translation-invariant <differential forms of type (p, q)>.
The constant $(1,0)$ covectors form $V^*=\operatorname{Hom}_{\mathbb C}(V,\mathbb C)$; the constant $(0,1)$ covectors form the space of <antilinear maps> $\overline{V^*}=\operatorname{Hom}_{\overline{\mathbb C}}(V,\mathbb C)$. Sending a constant wedge form to its cohomology class gives
$$
\boxed{H_{\bar\partial}^{p,q}(X)\cong\bigwedge^pV^*\otimes\bigwedge^q\overline{V^*}.}
$$
The map is intrinsic and does not depend on the auxiliary flat metric: that metric proves every class has exactly one constant representative, while the inclusion of constant forms is defined directly by the quotient. \b[The Dolbeault cohomology of a complex torus consists of constant forms], and in particular
$$
h^{p,q}(X)=\binom gp\binom gq,\qquad g=\dim_{\mathbb C}V.
$$
For indices outside $0,\ldots,g$, both the exterior-power expression and the cohomology group vanish.
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