PomeProposition 10.27.2.6055AD1EB4F6 The forms $\theta _ h$ descend and extend by zero to smooth semipositive closed forms $T_ h$ on $X$. They all represent one nonzero nef Bott–Chern class $\alpha $ with $\alpha ^2=0$. This class is not rigid, that is, it contains more than one closed positive current, and its image in real de Rham cohomology spans an irrational ray. Each $T_ h$ vanishes on a neighborhood of the exceptional set.
Proposition 10.27.26055AD1EB4F6
Proof. The form $\theta _ h$ is closed, because $\mathrm dh(y_1)$ is proportional to $\mathrm dy_1$, and it is semipositive. Since $h$ is even, $\theta _ h$ is invariant under the involution. At every fixed point, $y_1$ belongs to $\frac12\mathbb Z$ modulo $\mathbb Z$, so the form vanishes on a neighborhood of that point. Its descent to the complement of the singular points of $Y$ therefore extends by zero across their resolved neighborhoods. This proves smoothness and the asserted vanishing.
Let $h_1,h_2$ be admissible and put $k=h_1-h_2$. This function is even, has mean zero and vanishes near $\frac12\mathbb Z$. Choose $f$ with
\[ f'(y)=4\pi \int _0^ y k(s)\, \mathrm ds, \qquad f''=4\pi k. \]
The derivative $f'$ is periodic and odd, so its mean is zero, and $f$ can be chosen periodic and even. Evenness and the zero mean of $k$ give $\int _0^{1/2}k=0$, hence $f'=0$ near $\frac12\mathbb Z$. Therefore $f(y_1)$ descends, and it extends by constants near each exceptional curve to a smooth function $G$ on $X$. Since
\[ \mathrm{dd}^{\mathrm{c}}f(y_1)=\frac{f''(y_1)}{4\pi } \, \mathrm dx_1\wedge \mathrm dy_1 \quad \text{on }A, \qquad \text{we have}\qquad \mathrm{dd}^{\mathrm{c}}G=T_{h_1}-T_{h_2}. \]
Thus all the forms $T_ h$ have the same Bott–Chern class $\alpha $. Distinct admissible functions give distinct forms, so $\alpha $ is not rigid. The forms are semipositive and have square zero, so $\alpha ^2=0$. By lemma 10.27.1, adding any positive multiple of a Kähler form gives a Kähler representative of a nearby class; therefore $\alpha $ is nef.
By the mean-one condition, the periods of $\theta _ h$ on the real two-tori generated by $(\lambda _1,\lambda _3)$ and by $(\lambda _2,\lambda _3)$ are $1$ and $a$, respectively. Translate both cycles by $(0,\mathrm i/4)$. The translated cycles lie in $\{ y_2=1/4\} $ modulo $\mathbb Z$ and avoid the sixteen fixed points, and their periods are unchanged because the form is closed. Their images in $X$ lie in the complement of the exceptional curves, which is doubly covered without ramification by $A\setminus A[2]$. The first period shows that $\alpha \ne 0$. If a nonzero multiple $b\alpha $ had a rational de Rham class, then evaluating it on these cycles would make both $b$ and $ba$ rational, contradicting $a\notin \mathbb Q$. Finally, since $\alpha ^2=0$, Stokes' theorem shows that $\alpha $ itself has no Kähler representative, so it lies on the boundary of the nef cone. $\square$
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