$h^{p,q}$ of a complex torus.

Question

As we know, a compact Kähler surface with trivial canonical bundle is a K3 surface or a torus of dimension 2. I know $h^{0,2}$ of a K3 surface is 1, and I know $h^{0,2}$ of a torus must not be zero (otherwise it is always algebraic), but I don't know how to compute $h^{0,2}$ of a complex torus of dimension 2.

By the way, is there a general method to compute all the Hodge numbers of a complex torus of dimension $n$? Any comments are welcome!

Answer 1

Since you use the tag Hodge theory: One can give $\mathbb C^n/\Gamma$ the flat metric from that of $\mathbb C^n$. Thus it suffices to find all harmonic forms.

Since the complexified cotangent bundle are trivial and spanned by the global forms $$dz^1, \cdots dz^n, d\bar z^1\cdots, d\bar z^n,$$ all $(p, q)$-forms on $\mathbb C^n/\Gamma$ are globally given by $$ \alpha = \alpha_{i_1\cdots i_p j_1\cdots j_q} dz^{i_1}\wedge \cdots \wedge dz^{i_p} \wedge d\bar z ^{j_1} \wedge \cdots \wedge d\bar z^{j_q}.$$

where $\alpha_{i_1\cdots i_p j_1\cdots j_q} \in C^\infty (\mathbb C^n/\Gamma)$. Since

$$\Delta \alpha = (\Delta \alpha_{i_1\cdots i_p j_1\cdots j_q}) dz^{i_1}\wedge \cdots \wedge dz^{i_p} \wedge d\bar z ^{j_1} \wedge \cdots \wedge d\bar z^{j_q},$$

if $\alpha$ is harmonic, $\alpha_{i_1\cdots i_p j_1\cdots j_q}$ must be harmonic functions. Since $\mathbb C^n /\Gamma$ is compact, $\alpha_{i_1\cdots i_p j_1\cdots j_q}$ are constants by maximum principle. Thus $H^{p,q}$ is spanned by $$\{ dz^{i_1}\wedge \cdots \wedge dz^{i_p} \wedge d\bar z ^{j_1} \wedge \cdots \wedge d\bar z^{j_q}\}_{i_1<\cdots<i_p, j_1<\cdots <j_q}.$$

and

$$h^{p,q} = C^n_p C^n_q.$$