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

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!

## 1 Answer

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

and

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

• What do you mean by "harmonic $(p,q)$ forms must be constant"? after the Laplacian, isn't it still be a funtion of the point in the torus?
– Tom
Aug 18, 2020 at 17:12
• @Tom I have clarified a bit. Aug 18, 2020 at 17:16
• What distinguish a torus from a general compact Kähler manifold? It seems a general manifold can still apply your method and get the same answer?
– Tom
Aug 18, 2020 at 17:29
• The point is on the torus, I can represent $\alpha$ globally. On another compact Kahler manifolds, I cannot apply maximum principle, since $\alpha_{i_1\cdots i_p j_1\cdots j_q}$ would be just local functions. @Tom Aug 18, 2020 at 17:32
• everything is clear and I think it's perfect answer, thanks a lot:)
– Tom
Aug 18, 2020 at 17:37