ScalingStacks

In this section we will prove a uniform diameter bound and use this to compare our results to the existing literature.

Let the setting be as in the Introduction, namely let $X$ be a projective Calabi-Yau $n-$fold and $\alpha\in N^{1}(X)_{\mathbb{R}}$ a big and nef class that is not ample. Given $\alpha_{t}:[0,1]\to\overline{\mathcal{K}}_{NS}$ a smooth path such that $\alpha_{t}\in\mathcal{K}_{NS}$ for $t<1$ and $\alpha_{1}=\alpha$, Yau’s Theorem [Y2] gives us a unique Ricci-flat Kähler metric $\omega_{t}\in\alpha_{t}$ for any $t<1$. Then

implies that $\omega_{t}^{n}=C_{t}\omega_{0}^{n}$ for some constant $C_{t}$, which is easily computed by

In particular $C_{t}>0$ and $0<\lim_{t\to 1}C_{t}<\infty$, which means that the volume form of $\omega_{t}$ is uniformly equivalent to the volume form of $\omega_{0}$. We claim that the diameter of $(X,\omega_{t})$ is uniformly bounded above when $t$ approaches $1$. First we need a lemma, which appears as Lemma 1.3 in [DPS1]. For the reader’s convenience, we include a proof here.

Choose $\delta\leq\min(C_{1}^{2},1/2\int_{X}\omega_{0}^{n})$, and for any $t<1$ pick any $p_{t}\in U_{t,\delta}$. If we denote the metric ball of $\omega_{t}$ centered at $p$ and with radius $r$ by $B_{t}(p,r)$, then we get that $U_{t,\delta}\subset B_{t}(p_{t},C_{2})$, where $C_{2}=C_{1}\delta^{-1/2}\geq 1$. Then

and so

for some constant $C_{3}$ independent of $t$. Since $\mathrm{Ric}(\omega_{t})=0$, the Bishop volume comparison Theorem and (3.2) give that

The following lemma is due to Yau (see e.g. Theorem I.4.1 in [SY]), but we provide a proof for completeness.

Lemma 3.2 gives that for any $1<R<\mathrm{diam}(\omega_{t})$ we have

Choosing $R=\mathrm{diam}(\omega_{t})-1$ and using (3.3) we get

which is bounded independent of $t<1$, and proves our claim (a similar argument can be found in [P1]). Once we have the diameter bound, we can apply the Bishop volume comparison Theorem again and get that for any point $p\in X$ and any $r>0$, $t<1$

where $c>0$ is a uniform constant. A well-known computation in Chern-Weil theory gives

where $\textrm{Rm}_{t}$ is the Riemann curvature tensor of $\omega_{t}$ and $c_{2}(X,\omega_{t})$ is the second Chern form of $\omega_{t}$. If $n=2$ we can thus apply Theorem C of [An], Theorem 5.5 of [BKN] or Proposition 3.2 of [Ti] and get that a subsequence of $(X,\omega_{t})$ converges to an Einstein orbifold with isolated singularities in the Gromov-Hausdorff topology, and also in the $C^{\infty}$ topology on compact sets outside the orbifold points. If $n>2$ these theorems require a uniform bound on

which in general can not be expressed in terms of topological data as above. Instead when $n>2$ we apply a general theorem of Gromov [Gr] that says that any sequence of compact Riemannian manifolds of dimension $2n$ with diameter bounded above and Ricci curvature bounded below, has a subsequence that converges in the Gromov-Hausdorff topology to a compact length space. Thus a subsequence of $(X,\omega_{t})$ converges to a compact metric space $Y$, and Theorem 1.15 in [CCT] says that $Y$ is a complex manifold outside a rectifiable set $R\subset Y$ of real Hausdorff codimension at least $4$. Moreover their Theorem 9.1 gives supporting evidence that $R$ should in fact be a complex subvariety of $Y$.

On the other hand our Theorem 1.1 gives the convergence of the whole sequence of metrics, and not just of a subsequence, and the limit metric is uniquely determined by the class $\alpha$. When $n>2$ the convergence we get is stronger than Gromov-Hausdorff convergence, but it only happens outside the singular set $E$. Also, when $\alpha=c_{1}(L)$, we see precisely who the limit space $Y$ is, namely the Calabi-Yau model of $X$ associated to $L$. It has canonical singularities, so its singular set is a subvariety of complex codimension at least $2$, and when $n=2$ canonical singularities are precisely rational double points, that are of orbifold type. We will discuss the case $n=2$ with more details in section 5. Let us also mention the results of Ruan [Ru]. He studies the Gromov-Hausdorff limits of sequences of Kähler metrics on a fixed compact manifold $X$, with uniformly bounded sectional curvature. Roughly speaking, he proves that there exists an analytic subvariety $E\subset X$ such that a subsequence of the metrics converges in the Gromov-Hausdorff topology on $X\backslash E$ to $\omega$ a smooth Hermitian form, which is either Kähler (non-collapsing) or pointwise nonnegative with determinant zero (collapsing). Moreover in the collapsing case, the kernel of $\omega$ gives a holomorphic foliation with singularities on $X$. Unfortunately in our setting the curvature is not bounded in general, so Ruan’s results don’t apply, but our Theorem 1.1 gives in particular Ruan’s conclusion in the non-collapsing case. We’ll discuss the collapsing case in section 5.

Of course, the above-mentioned results apply in more general situations than ours. Also, all the results in this section work in the case when $X$ is not projective, and the ample cone is replaced by the (bigger) Kähler cone. Then the above theorems still apply, but for technical reasons our Theorem 1.1 doesn’t (see section 6 for more discussions).