ScalingStacks

In this section we will review some definitions and results from algebraic geometry, mainly from Mori’s Program, that will be used in the proof.

Let $X$ be a compact Calabi-Yau $n-$fold, that is a compact Kähler manifold of dimension $n$ and such that $c_{1}(X)=0$ in $H^{2}(X,\mathbb{R})$. We don’t insist that $X$ is simply connected. Notice that it follows that $aK_{X}\cong\mathcal{O}_{X}$ for some integer $a>0$: in fact by Theorem 1 in [Be] a finite unramified $a:1$ cover of $X$, $p:\tilde{X}\to X$, has trivial canonical bundle. But we have that $p^{*}K_{X}\cong K_{\tilde{X}}\cong\mathcal{O}_{\tilde{X}}$ and then Lemma 16.2 in [BHPV] implies that $aK_{X}\cong\mathcal{O}_{X}$. This can be rewritten as $K_{X}\sim_{\mathbb{Q}}0$ where $\sim_{\mathbb{Q}}$ indicates $\mathbb{Q}$-linear equivalence of Cartier $\mathbb{Q}$-divisors. For the rest of this section we will assume that $X$ is projective.

Let $L$ be a nef line bundle on $X$, and let $\kappa(X,L)$ be its Iitaka dimension, that is

and $\kappa(X,L)=-\infty$ if $kL$ has no sections for all $k\geq 0$. We call $\nu(X,L)$ its numerical dimension, that is the largest nonnegative integer $m$ such that there exists an $m-$cycle $V$ such that $(L^{m}\cdot V)>0$. It is always true that

When $|kL|$ is base-point-free, we get a morphism $\Phi_{|kL|}:X\to\mathbb{P}H^{0}(X,kL)^{*}$. Notice that if $L$ is big, that is $\kappa(X,L)=n$, then it is automatically good. The following is an immediate consequence of the base-point-free Theorem (Theorem 6.1.11 in [KMM]).

The next theorem is classical (see Theorem 2.1.33 in [L]).

We’ll call $f$ the contraction map of $L$. We also have the following theorem (Theorem 5.7 in [Ka1] or Theorem 1.9 in [Ka2]).

If $L$ is a line bundle, its stable base locus is the intersection of the base loci of $|mL|$ for all $m\geq 1$. It is equal to the base locus of $|mL|$ for some $m$ (see Prop. 2.1.21 in [L]). If $L$ now is nef and big, we define the augmented base locus of $L$, $\mathbf{B}_{+}(L)$, to be the stable base locus of $L-\varepsilon H$ for any $H$ ample divisor and any $\varepsilon>0$ small enough rational number. This definition is well-posed (see Lemma 10.3.1 in [L]) and a theorem of Nakamaye ([N], [L]) says that $\mathbf{B}_{+}(L)$ is equal to the null locus of $L$, that is the union of all positive-dimensional subvarieties $V\subset X$ such that $(L^{\dim V}\cdot V)=0$.

Finally let us state a well-known conjecture (see 10.3 of Peternell’s lectures in [MP]).

If $L$ is effective, this conjecture follows from the log abundance conjecture. Indeed for any small rational $\varepsilon>0$, the pair $(X,\varepsilon L)$ is klt, and the log abundance conjecture would imply that $K_{X}+\varepsilon L\sim_{\mathbb{Q}}\varepsilon L$ is semiample.

Notice that when $X$ is a surface, Conjecture 2.1 holds: in fact if $L$ is nef and non trivial, then $H^{2}(X,L)=H^{0}(X,K_{X}-L)=0$ and by Riemann-Roch

thus $L$ is effective. Then we can apply the log abundance theorem for surfaces (see e.g. [FM]) and get the result.