Example 11.1.12F336FEED9D87

Example 11.1.12.F336FEED9D87 Let $X=(\mathbb C/(\mathbb Z+\mathrm i\mathbb Z))^2$. For $j=1,2$, let $z_ j=x_ j+\mathrm i y_ j$ be the standard coordinates on the universal cover $\mathbb C^2$, with $x_ j,y_ j$ real, and put

The forms $\beta _ j$ descend to $X$. For a real parameter $t$, define

Since $1\leq b\leq 3$, the form $\omega $ is Hermitian. It is not closed, because $\mathrm{d}\omega =\mathrm{d}b\wedge \beta _2$ is not identically zero.

For a function $f$ of $x_1$ alone, one has $\mathrm{dd}^{\mathrm{c}}f=(f''/4)\beta _1$, where the prime denotes differentiation in $x_1$. Hence

Writing $B=\int _ X\beta _1\wedge \beta _2>0$, we have

Indeed,

and over a period the average of $\cos (2\pi x_1)$ is zero while that of its square is $1/2$.