Assume the same setting as in the Introduction, except that $M$ need not be projective and $M_{y}$ need not be a torus. If we have that

(4.22)

$$\tilde{\omega}_{t}\to f^{*}\omega$$

in $C^{\infty}_{loc}(M\backslash S,\omega_{M})$, where $\omega$ is as before, then on each fiber $M_{y}$ with $y\in N\backslash f(S)$ we have

(4.23)

$$\frac{\tilde{\omega}_{t}|_{M_{y}}}{t}\to\omega_{SF,y},$$

where $\omega_{SF,y}$ is the unique Ricci–flat metric on $M_{y}$ cohomologous to $\omega_{M}|_{M_{y}}$ and the convergence is smooth and uniform as $y$ varies on a compact subset of $N\backslash f(S)$.