Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 118 / 2

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.

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The forms $d{z}_I\wedge d{\overline{z}}_J$ are parallel for the flat metric. Consequently the Hodge Laplacian acts coefficientwise:If every coefficient is constant, this vanishes. Conversely, if $\Delta \alpha =0$, orthogonality of the constant frame gives $\Delta {\alpha }_{I,J}=0$ for every $I,J$. Each coefficient is a harmonic function on a compact connected manifold and hence is constant by the Strong maximum principle for harmonic functions.

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The Hodge decomposition theorem for compact Kähler manifolds stateswith each summand represented uniquely by harmonic forms of type $(p,q)$. Part b shows that these are precisely the constant-coefficient forms $d{z}_I\wedge d{\overline{z}}_J$. Therefore the Hodge numbers of a complex $n$-torus are

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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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