Proposition 3.1 . [030X]

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Proposition 3.1.

Let $\omega$ be any Kähler metric on $U$ cohomologous to $\omega_{SF}$ in $H^{2}(U,\mathbb{R})$ . Then there exist a holomorphic section $\sigma:B\to U$ of $f$ and a smooth real function $\xi$ on $U$ such that

(3.3)

T_{\sigma}^{*}\omega_{SF}-\omega=\sqrt{-1}\partial\overline{\partial}\xi

on $U$ .

If in addition $\omega$ is also semi-flat, then $\xi$ is constant on each fiber $M_{y}$ and is therefore the pullback of a function from $B$ .