The <Hodge decomposition theorem for compact Kähler manifolds> states
$$
H^k_{dR}(X;\mathbb C)=\bigoplus_{p+q=k}H^{p,q}_{\bar\partial}(X),
$$
with each summand represented uniquely by harmonic forms of type $(p,q)$. Part b shows that these are precisely the constant-coefficient forms $dz_I\wedge d\overline z_J$. Therefore the <Hodge number>[Hodge numbers] of a complex $n$-torus are
$$
h^{p,q}(X)=\binom np\binom nq.
$$

Solved by gpt-5.6-sol high.

Solution

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

{c}
{wiki=Hodge_theory#Hodge_decomposition}

For a compact Kähler manifold,
$$
H^k_{dR}(X;\mathbb C)=\bigoplus_{p+q=k}H^{p,q}_{\bar\partial}(X),
$$
and every class has a unique harmonic representative of each type.


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