Example 11.23.528F39F2B7C2C

Example 11.23.5.28F39F2B7C2C Let $E=\mathbb C/(\mathbb Z+\mathrm i\mathbb Z)$ and $X=E^3$. For $1\leq j\leq 3$, write $z_ j=x_ j+\mathrm i y_ j$ for the coordinates on the covering space, and let $\eta _ j$ be the form on $X$ induced by $(\mathrm i/2)\mathrm dz_ j\wedge \mathrm d\bar z_ j$. The periodic functions

define a Hermitian form

Since $bc=1$,

The functions $b$ and $c$ depend only on $x_1$, so $\mathrm{dd}^{\mathrm{c}}b$ and $\mathrm{dd}^{\mathrm{c}}c$ are multiples of $\eta _1$, and $\mathrm{dd}^{\mathrm{c}}(\omega ^2)=0$. Thus $\omega $ is Gauduchon.

Put $v=\cos (2\pi x_2)$ and $h=-\pi \cos (2\pi x_2)$, so that $\mathrm{dd}^{\mathrm{c}}v=h\eta _2$. For a real $s$ with $0<|s|<1/(\pi \mathrm e)$, the form

is positive, since $b\geq \mathrm e^{-1}$ and $|h|\leq \pi $. However,

Indeed, $\mathrm{dd}^{\mathrm{c}}c=(c''/(4\pi ))\eta _1$, where the prime denotes differentiation in $x_1$, and $c''(0)=4\pi ^2\mathrm e^{-1}$, while $h=-\pi $ at $x_2=0$. Thus a Gauduchon form can have arbitrarily small positive potential perturbations that are not Gauduchon.