Example 10.26.2.5B00F62ACA6B On $X=(\mathbb C/(\mathbb Z+\mathrm i\mathbb Z))^2$, for $a\in \mathbb R$ set
\[ \theta _ a=\frac{\mathrm i}{2} (\mathrm dz_2-a\, \mathrm dz_1) \wedge (\mathrm d\bar z_2-a\, \mathrm d\bar z_1). \]
Its class is rigid exactly when $a$ is irrational. In that case its ray is irrational, that is, no nonzero real multiple of its de Rham class is rational, and every family $\theta _ a+t\kappa $ as above collapses to a point in diameter. For rational $a$ the class is not rigid, and these diameters have a positive lower bound.
Proof.
The kernel is the complex line $z_2=a z_1$. Write $z_ j=x_ j+\mathrm i y_ j$. An integral character has the form
\[ \ell =m_1x_1+m_2x_2+n_1y_1+n_2y_2, \qquad m_ j,n_ j\in \mathbb Z, \]
and it vanishes on the kernel exactly when $m_1+a m_2=n_1+a n_2=0$. For irrational $a$, only the zero character does so, hence the image of the kernel is dense. For rational $a$, a nonzero such character exists. Now apply theorem 10.26.1. The periods of $\theta _ a$ on the coordinate two-tori in the $(x_2,y_2)$ and $(x_1,y_2)$ directions are $1$ and $-a$, respectively. If a nonzero multiple of its class were rational, the first period would make that multiple rational, and the second would then make $a$ rational. This proves the assertion about the ray.
$\square$
Pome
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