Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 118 / 2 / d / 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 ofand 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 $e_1,\dots ,e_n$ and $\tau e_1,i{e}_2,\dots ,i{e}_n$ 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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