Example 10.26.3.87A3AA3B13FD On the same square torus let $\theta _0=\frac{\mathrm i}{2}\mathrm dz_2\wedge \mathrm d\bar z_2$ and, for $t>0$, put
\[ \beta _ t=\frac{\mathrm i}{2}\left[ (t\, \mathrm dz_1+\mathrm dz_2) \wedge (t\, \mathrm d\bar z_1+\mathrm d\bar z_2) +t^2\mathrm dz_2\wedge \mathrm d\bar z_2\right]. \]
These are flat Ricci-flat Kähler forms, $\beta _ t\to \theta _0$ smoothly, and
\[ \operatorname {diam}(X,\beta _ t)\leq t. \]
The limiting class $\{ \theta _0\} _{\mathrm{BC}}$ is not rigid.
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
\[ |t(z_1+k)+z_2|^2+t^2|z_2|^2 \leq t^2/2+t^2/2=t^2. \]
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.
$\square$
Pome
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