ScalingStacks

In this section we will give some examples where Theorem 1.1 applies. The constructions are well-known and come from algebraic geometry.

Let’s look at the case $n=2$ first, the case $n=1$ being trivial. The only projective Calabi-Yau surfaces are tori, bi-elliptic, Enriques and $K3$ surfaces (recall that the Calabi Conjecture has been successfully applied to the study of $K3$ surfaces by Todorov [To] and Siu [Si]). If $X$ is a torus and $L$ is a nef and big line bundle on $X$, then $L$ is ample, and so Theorem 1.1 is vacuous in this case. Similarly if $X$ is bi-elliptic, then $X$ is a finite unramified quotient of a torus, so a nef and big line bundle on $X$ pulls back to a nef and big line bundle on a torus. But this must be ample, and so the original line bundle is ample too (Corollary 1.2.28 in [L]) and Theorem 1.1 is again empty. If $X$ is an Enriques surface, then $X$ is an unramified $2:1$ quotient of a $K3$ surface, so the study of Ricci-flat metrics on $X$ is reduced to the case of a $K3$ surface. Finally let’s see that there exist projective $K3$s that admit a nef and big line bundle that is not ample, to which Theorem 1.1 applies. For example let $Y$ be the quotient surface $T/i$ where $T$ is the standard torus $\mathbb{C}^{2}/\mathbb{Z}^{4}$ and $i$ is induced by the involution $i(z,w)=(-z,-w)$ of $\mathbb{C}^{2}$. The surface $Y$ has 16 singular points, that are rational double points, and is a Calabi-Yau model. Blowing up these 16 points gives a smooth projective $K3$ surface $X$ (called a Kummer surface), and we can take $L$ to be the pullback of any ample divisor on $Y$. The set $E$, being equal to the null locus of $L$, is readily seen to be the union of the $16$ exceptional divisors, that are $(-2)$-curves. Then Theorem 1.1 applies, and the limit of smooth Ricci-flat metrics on $X$ with classes approaching $c_{1}(L)$ is the pullback of the unique Ricci-flat (actually flat) orbifold Kähler metric on $Y$ in the given class. This originally appeared as Theorem 8 in [KT]. Now we show that conversely all examples of Theorem 1.1 on $K3$ surfaces with $\alpha=c_{1}(L)$ are of the form $f:X\to Y$ where $Y$ is an orbifold $K3$ surface, $kL=f^{*}A$, for some $k\geq 1$ and some $A$ ample divisor on $Y$. Let $X$ be a projective $K3$ surface and $L$ a nef and big line bundle on $X$. By Theorem 2.1 we know that some power $kL$ is globally generated, and we might as well assume that $k=1$. Then the contraction map $f$ of $L$ contracts an irreducible curve $C$ to a point if and only if $C\cdot L=0$. But since $L\cdot L>0$, the Hodge Index theorem implies that $C\cdot C<0$. The long exact sequence in cohomology associated to the sequence

$$0\to\mathcal{O}_{X}(-C)\to\mathcal{O}_{X}\to\mathcal{O}_{C}\to 0,$$

gives that $H^{1}(X,\mathcal{O}(-C))=0$. Serre duality on the other hand gives $H^{2}(X,\mathcal{O}(C))=H^{0}(X,\mathcal{O}(-C))=0$, and $H^{1}(X,\mathcal{O}(C))=H^{1}(X,\mathcal{O}(-C))=0$. Riemann-Roch then gives

$$\dim H^{0}(X,\mathcal{O}(C))=2+\frac{1}{2}C\cdot C,$$

which implies that $C\cdot C$ must be even. But since $\pi(C)=\frac{C\cdot C}{2}+1$, the virtual genus of $C$, is nonnegative, we see that $C\cdot C=-2$. This implies that $\pi(C)=0$ and so $C$ is a smooth rational curve with self-intersection $-2$. Then the point $f(C)$ is a rational double point, and so $Y=f(X)$ is an orbifold $K3$ surface. Notice that Ricci-flat orbifold metrics on $Y$ exist by [Y2], [Kob].

Now we turn to examples in dimension $3$. The first one is known as conifold in the physics literature [GMS], and is described in detail in section 1.2 of [Ro], for example. Roughly speaking, it is a $3$ dimensional Calabi-Yau model $Y$ that sits in $\mathbb{P}^{4}$ as a nodal quintic. It has $16$ singular points, that are nodes and not of orbifold type. Moreover there exists a small resolution $f:X\to Y$, that is a birational morphism with $X$ a smooth Calabi-Yau threefold, that is an isomorphism outside the preimages of the nodes, which are $16$ rational curves. If $L$ is the pullback of any ample divisor on $Y$, then $L$ is nef and big on $X$, and the limit of smooth Ricci-flat metrics on $X$ with classes approaching $c_{1}(L)$ is the pullback of the unique singular Ricci-flat metric on $Y$, which exists by [EGZ]. The convergence is smooth on compact sets outside the union of the $16$ exceptional curves (which is clearly equal to the null locus of $L$). There are also other $3$ dimensional examples where the singularities of $Y$ are not isolated: one of these is described in Example 4.6 in [W1], and $Y$ has a curve $C$ of singularities. Blowing up $C$ gives a Calabi-Yau threefold $X$; if $L$ is the pullback of any ample divisor on $Y$, then the null locus of $L$ is the exceptional divisor $S$ which is a smooth surface ruled over $C$. Again our Theorem 1.1 applies, and the convergence is smooth off $S$.