10.26 Rigid boundary classes on a torus5A68768CC1C3

Proof. Rigidity when $H=X$. Suppose first that $H=X$. By lemma 20.4.10, a positive current in $\alpha $ can be written $T=\theta +\mathrm{dd}^{\mathrm{c}}u$ with $u$ quasi-plurisubharmonic, in particular $u\in L^1(X)$. For a constant real vector $v\in K$, the coefficient $T(v,Jv)$ is a positive Radon measure. Its total integral against Haar measure is zero, because $\theta (v,Jv)=0$ and the remaining part of the coefficient is a combination of constant-coefficient derivatives of $u$. Hence this coefficient vanishes. Writing $D_ v$ for differentiation along $v$ and using that $K$ is complex, we obtain

For a real linear functional $\ell $ with $\ell (\Lambda )\subset \mathbb Z$, let $\widehat u(\ell )$ be the Fourier coefficient of $u$ at the character $\exp (2\pi \mathrm i\ell )$. The identity above gives

Since the image of $K$ is dense, a nonzero character cannot vanish identically on $K$. Thus every nonconstant Fourier coefficient of $u$ is zero, and convolution with a smooth approximate identity shows that $u$ is constant. Hence $T=\theta $.

Nonrigidity when $H\ne X$. If $H\ne X$, a nontrivial character of the real torus $X/H$ gives a nonzero $\ell $ that vanishes on $K$. Put $\phi =\cos (2\pi \ell )$. Since $K$ is complex, the form $\mathrm{dd}^{\mathrm{c}}\phi $ vanishes whenever one of its arguments lies in $K$. On a fixed complement of $K$, $\theta $ is positive definite and $\mathrm{dd}^{\mathrm{c}}\phi $ is uniformly bounded. Consequently, the forms $\theta \pm \varepsilon \mathrm{dd}^{\mathrm{c}}\phi $ are semipositive for sufficiently small $\varepsilon >0$. They represent the same class, and they are distinct because a nonconstant smooth function on a compact torus cannot have zero $\mathrm{dd}^{\mathrm{c}}$. Hence $\alpha $ is not rigid.

Collapse when $H=X$. Suppose again that $H=X$, and fix $\delta >0$. By density and compactness, there are finitely many vectors $k_1,\ldots ,k_ N\in K$ whose images form a $\delta $-net in the flat metric of $\kappa $. Put $R=\max _ j|k_ j|_\kappa $. Since $\theta (k_ j,Jk_ j)=0$, the straight path from $0$ to the image of $k_ j$ has $(\theta +t\kappa )$-length $\sqrt t\, |k_ j|_\kappa $. For $0<t\leq 1$ these metrics are bounded above by a fixed multiple of $\kappa $. Joining this path to a point within $\delta $ therefore gives

since by translation invariance the diameter equals the supremum of the distances from $0$. Letting first $t\downarrow 0$ for this fixed net and then $\delta \downarrow 0$ proves the third condition.

No collapse when $H\ne X$. If $H\ne X$, choose $\ell $ as above. Its differential vanishes on $K$, so $|\mathrm d\ell (v)|\leq L|v|_\theta $ for some $L>0$, where $|v|_\theta $ is the seminorm defined by $\theta $. Since $\theta +t\kappa \geq \theta $, the surjective map $\ell \bmod \mathbb Z$ onto the circle of length one is $L$-Lipschitz for every $t>0$. Two points mapping to opposite points of the circle therefore have distance at least $1/(2L)$, and the diameters cannot tend to zero. This completes the proof of the equivalences. Since the second condition does not involve $\kappa $, neither does the third. $\square$

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

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. $\square$

Proof. The coefficient matrix of $\beta _ t$ is $\left(\begin{smallmatrix}t^2& t\\ t& 1+t^2\end{smallmatrix}\right)$, with determinant $t^4>0$. Its coefficients are constant, which gives flatness and Ricci-flatness. Represent a point by $(z_1,z_2)$ with real and imaginary parts in $[-1/2,1/2]$, and choose a Gaussian integer $k$ nearest to $-z_1-z_2/t$. The displacement $(z_1+k,z_2)$ represents the same point of the torus and has squared metric norm

The straight path along this displacement proves the bound for the distance from zero, and translation invariance gives the diameter bound. For $0<|s|<1/\pi $, the form $\theta _0+\mathrm{dd}^{\mathrm{c}}(s\cos (2\pi x_2))=(1-\pi s\cos (2\pi x_2))\theta _0$ is another semipositive representative, so the limiting class is not rigid. Finally, the coefficient matrix of $\beta _ t-\theta _0$ has eigenvalues $t^2\pm t$, so this difference is indefinite for $0<t<1$: the forms $\beta _ t$ do not approach $\theta _0$ from above. $\square$