= Solution
Choose a small ball $B$ whose translates by distinct lattice elements are disjoint. The restrictions of the quotient map $\pi:\mathbb C^n\to\mathbb C^n/\Lambda$ to translates of $B$ supply holomorphic charts, because every transition map is a complex translation. A closed fundamental parallelepiped is compact and surjects onto the quotient, so the resulting <complex torus> $X$ is compact.
The standard form
$$
\omega_0=\frac i2\sum_{j=1}^ndz_j\wedge d\overline z_j
$$
is translation invariant and therefore descends uniquely to a form $\omega_X$ with $\pi^*\omega_X=\omega_0$. It remains closed, of type $(1,1)$, and positive, so it is a <Kähler manifold>[Kähler form].
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