In this section we give some examples where Theorem 1.2 applies.

The easiest example is a complex torus $X$ of dimension $n$ fibering over another torus $Y$ of lower dimension $m$. The fibers are also tori and they are all biholomorphic. In this case Ricci-flat metrics are just flat, and they can be identified with constant positive definite Hermitian $n\times n$ matrices. If we degenerate the Kähler class on $X$ to the pullback of a Kähler class from $Y$, the matrices converge to a nonnegative definite matrix whose kernel generates the tangent space to the fibers. So the fibers are shrunk to points and the flat metrics on $X$ converge to the flat metric on $Y$ in the given class. This is of course compatible with Theorem 1.2, because in this case the Weil-Petersson metric is identically zero, and the set $S$ of singular fibers is empty.

To see a more interesting example, let $X$ be an elliptically fibered $K3$ surface, so $X$ comes equipped with a morphism $f:X\to\mathbb{P}^{1}$ with generic fibers elliptic curves. Then the pullback of an ample line bundle on $\mathbb{P}^{1}$ gives a nef line bundle $L$ on $X$ with Iitaka dimension $1$. In the case when all the singular fibers of $f$ are of Kodaira type $I_{1}$, Gross-Wilson have shown in [GW] that sequences of Ricci-flat metrics on $X$ whose class approaches $c_{1}(L)$ converge in $C^{\infty}$ on compact sets of the complement of the singular fibers to the pullback of a Kähler metric on $\mathbb{P}^{1}$ (minus the $24$ points which correspond to the singular fibers). Their argument relies on explicit model metrics that are almost Ricci-flat, and it is not well-suited to generalization to higher dimensions. More recently Song-Tian [ST1] gave a more direct proof of the result of Gross-Wilson and they noticed that the limit metric has Ricci curvature equal to the Weil-Petersson metric. Our Theorem 1.2 applies in this example, as well as in higher dimensions.

One can easily construct examples of higher-dimensional Calabi-Yau manifolds that are algebraic fiber spaces, to which Theorem 1.2 applies. For example the case of Calabi-Yau threefolds is studied extensively in [O], where many examples are given.