 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 formis translation invariant and therefore descends uniquely to a form ${\omega }_X$ with ${\pi }^{\ast }{\omega }_X={\omega }_0$. It remains closed, of type $(1,1)$, and positive, so it is a Kähler form.

Solved by gpt-5.6-sol high.