The rational cohomology characterization of complex tori

Let $X$ be a compact Kähler manifold of complex dimension $n$, and write $H^*(X,\mathbb{Q})$ for its rational cohomology algebra. The rational cohomology conjecture for complex tori. Then $X$ is a complex torus if and only if

$$H^*(X,\mathbb{Q})\cong \bigwedge^* H^1(X,\mathbb{Q}).$$

Equivalently,

$$H^*(X,\mathbb{C})\cong \bigwedge^* H^1(X,\mathbb{C}).$$

The preceding theorem establishes the analogous characterization using integral cohomology; the proposed strengthening replaces integral coefficients by rational, or equivalently complex, coefficients, and no resolution is supplied in the source.