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
\[ \beta _ j=\frac{\mathrm i}{2\pi }\, \mathrm{d}z_ j\wedge \mathrm{d}\bar z_ j. \]
The forms $\beta _ j$ descend to $X$. For a real parameter $t$, define
\[ b(x_1)=2+\cos (2\pi x_1),\qquad \omega =\beta _1+b(x_1)\beta _2,\qquad u_ t=t\cos (2\pi x_1). \]
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
\[ \omega _{u_ t} =\bigl (1-\pi ^2t\cos (2\pi x_1)\bigr )\beta _1+b\beta _2>0 \quad \text{if}\quad |t|<\pi ^{-2}. \]
Writing $B=\int _ X\beta _1\wedge \beta _2>0$, we have
914CB776FFB8
11.1.13
\begin{equation} \mathrm M_\omega (u_ t)=(4-\pi ^2t)B. \end{equation}
Indeed,
\[ \omega _{u_ t}^2 =2\bigl (1-\pi ^2t\cos (2\pi x_1)\bigr ) \bigl (2+\cos (2\pi x_1)\bigr )\beta _1\wedge \beta _2, \]
and over a period the average of $\cos (2\pi x_1)$ is zero while that of its square is $1/2$.
Although the complex torus is Kähler, the background $\omega $ chosen above is not closed, and this already allows the mass to vary. On non-Kähler manifolds even an upper bound for the mass can fail; the Hopf manifolds of section 23.1 give examples.
Proof.
The potentials $u_ t$ of example 11.1.12 are smooth on the compact manifold $X$, so any two of them differ by a bounded function and have the same singularity type in the sense of section 5.33. For the same reason, their multiplier ideals are the structure sheaf for every positive multiple of the weight. By (11.1.13), their masses differ.
$\square$
Pome
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