Lemma 10.19.1464DA68EB4DA

Proof. The measure $T\wedge \alpha \wedge \beta $ is positive. Since $\mathrm{dd}^{\mathrm{c}}T=0$, Stokes' theorem for currents gives

so this measure vanishes. All products in this identity pair a current with smooth forms. Now apply lemma 11.22.4 to $\pi $, with the base Hermitian form $\eta =\sigma _1+\sigma _2$. Since $\eta ^2=2\sigma _1\wedge \sigma _2$, we have $T\wedge \pi ^*\eta ^2=0$, and since the fiber is a compact connected elliptic curve, the lemma gives the unique positive $\mathrm{dd}^{\mathrm{c}}$-closed current $S$. The same lemma shows that $S$ is closed when $T$ is; conversely, pullback commutes with $\mathrm{d}$. $\square$