= Solution
All quotients $\mathbb C^n/\Lambda$ are diffeomorphic to the real torus $\mathbb R^{2n}/\mathbb Z^{2n}$ after choosing a real basis of the lattice. The complex structure is nevertheless visible in cohomology: under the cap-product period pairing, $H^{1,0}(X)$ is an $n$-dimensional subspace of
$$
H^1(X;\mathbb C)=\operatorname{Hom}(H_1(X;\mathbb Z),\mathbb C),
$$
and its elements are exactly the period homomorphisms of holomorphic one-forms.
A biholomorphism pulls $H^{1,0}$ onto $H^{1,0}$ and induces an element of $GL(2n,\mathbb Z)$ on integral first homology. The group $GL(2n,\mathbb Z)$ is countable, so the orbit of any one period subspace is countable. On the other hand, varying a period parameter $\tau$ in the upper half-plane in lattices generated by $e_1,\ldots,e_n$ and $\tau e_1,ie_2,\ldots,ie_n$ produces uncountably many such subspaces. Two choices lying in distinct $GL(2n,\mathbb Z)$-orbits therefore give diffeomorphic but nonbiholomorphic complex $n$-tori.
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