First let us mention an interesting question that arises from Theorem 1.1. We know that on $X\backslash E$ the Ricci-flat metrics converge smoothly on compact sets to an incomplete Ricci-flat metric $\omega_{1}$. Its metric completion is a metric space $(X_{\infty},d_{\infty})$. Do the original metrics $(X,\omega_{t})$ actually converge to $(X_{\infty},d_{\infty})$ in the Gromov-Hausdorff topology? We can prove this in the case when $X$ is a $K3$ surface and $\alpha=c_{1}(L)$. In fact, from section 5 we know that $E$ is a union of $(-2)$-curves and the contraction map $f:X\to Y$ maps them to orbifold points. The results of [An], [BKN], [Ti] give that a subsequence of $(X,\omega_{t})$ converges to $Y$ with its orbifold Ricci-flat metric $d_{\infty}$ in the Gromov-Hausdorff topology. But on the smooth part of $Y$ we have that $\omega_{1}$ and $d_{\infty}$ coincide, because they are both singular Ricci-flat metrics on the whole of $Y$. Hence the metric completion of $\omega_{1}$ is $d_{\infty}$.

Also, when $X$ admits a birational Calabi-Yau model $Y$, which has a singular Ricci-flat metric by [EGZ], what is the relation between $Y$ and $X_{\infty}$? In dimension $3$ or more it’s hard to see how a singular Ricci-flat metric on $Y$ should define a metric space structure.

There are two possible directions where it would be desirable to extend Theorem 1.1. The first case is when we look at the whole Kähler cone, instead of just the ample cone, and possibly drop the projectiveness assumption. Suppose $X$ is a compact Calabi-Yau $n$-fold and fix $\omega_{0}$ a Ricci-flat metric on $X$. The Neron-Severi space $N^{1}(X)_{\mathbb{R}}$ embeds into

$$H^{1,1}_{\mathbb{R}}(X)=H^{2}(X,\mathbb{R})\cap H^{1,1}(X),$$

but in general it is a proper subspace (for example a generic projective K3 has $\rho(X)=1<20=\dim H^{1,1}_{\mathbb{R}}(X)$). Inside $H^{1,1}_{\mathbb{R}}(X)$ we have $\mathcal{K}$, the Kähler cone, and its closure $\overline{\mathcal{K}}$, the nef cone. We have that

$$\mathcal{K}_{NS}=\mathcal{K}\cap N^{1}(X)_{\mathbb{R}},$$

and similarly for the nef cone. Given a nonzero class $\alpha\in\overline{\mathcal{K}}\backslash\mathcal{K}$, and a smooth path $\alpha_{t}:[0,1]\to\overline{\mathcal{K}}$ that ends at $\alpha$, Yau’s Theorem gives a path $\omega_{t}$ of Ricci-flat metrics in $\alpha_{t}$ and we can analyze their behaviour as $t$ approaches $1$. Let’s assume that $\alpha$ is big, which again means that $\alpha^{n}>0.$ We would like to repeat the construction we did in the algebraic case. There are two main points where we used the assumption that $X$ was projective and that $\alpha$ belonged to the Neron-Severi space: Proposition 4.1 and Kodaira’s lemma. We conjecture that the analogue of Proposition 4.1 still holds, namely we propose the

Notice that the proof of this conjecture would have to use the fact the $X$ is Calabi-Yau, since in general a nef class cannot be represented by a smooth nonnegative form [DPS2]. If this conjecture were proved, we could then write $\omega$ as the smooth limit of Kähler forms in $\alpha_{t}$, as in Proposition 4.1. The correct substitute for Kodaira’s lemma would then be given by the theory of closed positive currents: following [P2], which relies on the fundamental [DP], we know that there would exist a modification $\pi:\tilde{X}\to X$ such that

$$\pi^{*}\omega=\tilde{\omega}+[E]-\sqrt{-1}\partial\overline{\partial}\eta,$$

where $\tilde{\omega}$ is a Kähler form on $\tilde{X}$, $E$ is an effective $\mathbb{Q}$-divisor on $\tilde{X}$ and $\eta$ is quasi-psh, smooth of $E$ and has only log poles along $E$. Then we could just work on $\tilde{X}$, and get the same estimates as above, outside $E$, thus proving the Kähler analogue of Theorem 1.1.

The second direction is to look at the case when the class $\alpha$ is nef but not necessarily big. A guiding example is the following: let $X$ be an elliptically fibered $K3$ surface, so $X$ comes equipped with a morphism $f:X\to\mathbb{P}^{1}$ with 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 $\mathbb{P}^{1}$. This metric on $\mathbb{P}^{1}$ was first studied by McLean [McL]. In a recent paper, Song-Tian [ST] gave a more direct proof of the result of Gross-Wilson. Moreover they noticed that McLean’s metric satisfies an elliptic equation outside the images of the singular fibers, namely its Ricci curvature equals the pullback of the Weil-Petersson metric from the moduli space of elliptic curves, that comes from the variation of the complex structure of the fibers of $f$.

We believe that in higher dimensions a similar picture should be true, when $\alpha=c_{1}(L)$. In this case Conjecture 2.1 would imply the existence of a morphism $f:X\to Y$ with connected fibers, where $\dim Y=\kappa(X,L)<n$. Then we expect that outside a proper subvariety $E\subset X$, a sequence of Ricci-flat metrics with class approaching $\alpha$ should converge in $C^{\infty}$ on compact sets of $X\backslash E$ to the pullback of a metric on $Y$. It is readily verified that, up to a subsequence, the Ricci-flat metrics converge weakly as currents to the pullback of a metric on $Y$. When $Y$ is a curve, the fibers of $f$ are again Calabi-Yau’s, and a computation as in [ST] shows that limit metric on $Y$ will satisfy the same equation as McLean’s metric (in this case the potentials of the Ricci-flat metrics have a uniform $C^{0}$ bound [ST]). It might be possible to construct higher-dimensional examples of this behaviour using the results of Section 8 in [Fi], where the equation of McLean’s metric appears in his condition (C).

The situation is different when $\alpha$ is not $c_{1}(L)$, and $X$ possibly not projective. Then an example of McMullen [McM] shows that the Ricci-flat metrics can converge smoothly to zero on an open set of $X$. Also easy examples on tori show that the fibration structure as above cannot be expected when the limiting class $\alpha$ is not rational. Instead we still expect the Ricci-flat metrics to converge smoothly on compact sets outside a subvariety $E$ to a limit nonnegative form $\omega$, whose determinant vanishes identically. The kernel of $\omega$ would then define a complex foliation with singularities on $X\backslash E$, whose leaves might be dense in $X$. The leaves of the foliation are always complex submanifolds, but they might not vary holomorphically and the rank of the foliation might change on different open sets (as in McMullen’s example). Notice that if the curvature is uniformly bounded, then Ruan’s result [Ru] implies that this picture is basically true and moreover that the foliation is holomorphic, so that its rank is constant on a Zariski open set. In McMullen’s example the curvature blows up, and the resulting foliation is not holomorphic, thus showing that Ruan’s result doesn’t hold if the curvature is unbounded.

Let us mention that the results of [BKN], [Ba] also give a description of the behaviour of the Ricci-flat metrics near the singularities, where some bubbling occurs. Unfortunately our methods don’t seem to give results of this kind and it would be very interesting to study this in higher dimensions when the limit Calabi-Yau model doesn’t have orbifold singularities.

Finally let us notice that some of the results here generalize to the following setting: $X$ is a compact Kähler manifold, and we fix a smooth volume form $\Omega$. If $\alpha_{t}$ is a path of Kähler classes as in the beginning of this section, then for each $t<1$ Yau’s theorem [Y2] gives a unique Kähler form $\omega_{t}$ in $\alpha_{t}$ such that

$$\omega_{t}^{n}=\frac{\alpha_{t}^{n}}{\int_{X}\Omega}\Omega.$$

We can then study the behaviour of the metrics $\omega_{t}$ as $t$ approaches $1$. If the image of $\alpha_{t}$ lies in $N^{1}(X)_{\mathbb{R}}$ and the limit class $\alpha=c_{1}(L)$ where $L$ is a nef, big and semiample line bundle, then the argument of Theorem 1.1 goes through, and we get smooth convergence on compact sets outside the null locus of $L$.