ScalingStacks

Denote by $\Omega$ the smooth volume form on $X$ given by

$$\Omega=\frac{\omega_{0}^{n}}{\int_{X}\omega_{0}^{n}},$$

which satisfies $\int_{X}\Omega=1$. We can write $\Omega=F\omega^{n},$ where $F\in L^{1}(\omega^{n})$, $F>0$. The following argument to show that actually $F\in L^{p}(\omega^{n})$ for some $p>1$ is similar to Lemma 3.2 in [EGZ]. First of all $1/F$ is smooth, nonnegative, and vanishes precisely on the exceptional set of $f$. Fixing local coordinates $(z^{i})$ on a polydisc $D\subset X$ and a local embedding $G:f(D)\to\mathbb{C}^{m}$, we see that $1/F$ is comparable to

$$\left|\frac{\partial G}{\partial z^{1}}\wedge\dots\wedge\frac{\partial G}{\partial z^{n}}\right|^{2}$$

on $D$. But this is in turn comparable to

$$\sum_{i=1}^{r}|g_{i}|^{2},$$

where the $g_{i}$ are holomorphic functions on $D$, and so $F^{\varepsilon}\in L^{1}(D,\Omega)$ for some small $\varepsilon>0$ that depends on the vanishing orders of the $g_{i}$. Then

(4.2)

$$\int_{D}F^{1+\varepsilon}\omega^{n}=\int_{D}F^{\varepsilon}\Omega<\infty.$$

The compactness of $X$ gives $F\in L^{1+\varepsilon}(\omega^{n})$, and so we can apply Theorem 2.1 and Proposition 3.1 of [EGZ] (which rely on the seminal work of Kołodziej [Koł]) to get a unique continuous $\varphi\in PSH(X,\omega)$ such that

(4.3)

$$(\omega+\sqrt{-1}\partial\overline{\partial}\varphi)^{n}=\alpha^{n}\Omega,$$

and $\sup_{X}\varphi=0$. Moreover we can see that $\varphi$ descends to a function on $Y$: if $V$ is a fiber of $f$, the restriction of $\varphi$ to $V$ is a plurisubharmonic function, because $\omega|_{V}=0$. Desingularizing $V$ and applying the maximum principle we see that $\varphi|_{V}$ has to be constant, and so $\varphi$ descends to $Y$. Since $\omega$ by construction is the pullback of a (singular) Kähler form on $Y$, we see that $\omega+\sqrt{-1}\partial\overline{\partial}\varphi$ is a singular Ricci-flat metric on $Y$, in the terminology of [EGZ]. On $X$, the closed positive current $\omega_{1}=\omega+\sqrt{-1}\partial\overline{\partial}\varphi$ clearly lies in the class $\alpha$ and has continuous potentials. Intuitively, our goal is to get estimates in the open set where $\omega$ is positive. This can be done rigorously in the following way, which was first used by H.Tsuji [Ts] (see also [TZ], [CL] for a recent revisiting of his approach). Since $L$ is nef and big, by Kodaira’s lemma (Example 2.2.19 in [L]) there exists $E$ effective Cartier divisor such that for all $\varepsilon>0$ small enough, $\alpha-\varepsilon E=\kappa_{\varepsilon}$ is Kähler. We’ll show that $\varphi$ is smooth on $X\backslash E$, and so $\omega_{1}$ is a smooth Ricci-flat metric there, and that the Ricci-flat metrics $\omega_{t}$ converge to $\omega_{1}$ in the $C^{\infty}$ topology on compact sets of $X\backslash E$. Notice that the metric $\omega_{1}$ on $X\backslash E$ cannot be complete, since its diameter is finite by the result in section 3. Our argument is very similar to the proof of Theorem 3.5 in [EGZ] (see also [Y2]). Once this is proved, we can repeat the argument for any other $E$ given by Kodaira’s lemma, and by uniqueness we see that $\omega_{1}$ is smooth off $E^{\prime}$, the intersection of the supports of all such $E$. We claim that $E^{\prime}$ is equal to the null locus of $L$, and by Nakamaye’s Theorem all we need to show is that it is equal to the augmented base locus of $L$. If $x\in X$ is a point outside the augmented base locus, then there exist $H$ an ample divisor and $k,m$ large enough so that $x$ is not in the base locus of $mL-\frac{m}{k}H$. But this means that $mL-\frac{m}{k}H\sim N$ where $N$ is an effective divisor that doesn’t pass through $x$, and moreover the cohomology class of $L-\frac{1}{m}N$ is Kähler. So we can take $\varepsilon=\frac{1}{m}$ and $E=N$, and we see that $E^{\prime}$ is contained in the null locus of $L$. Conversely, if $x$ belongs to the null locus, then there exists a subvariety $V$ through $x$ with $\dim V=k$ and $(L^{k}\cdot V)=0$. Since the potentials for the current $\omega_{1}$ are continuous, the self-intersection $\omega_{1}^{k}$ is a well-defined closed positive current [BT], which restricts to a nonnegative Borel measure on $V$. The integral $\int_{V}\omega_{1}^{k}$ is then equal to the cohomological intersection number $(L^{k}\cdot V)$ (see e.g. Corollary 9.3 in [De2]) which is zero. But if $x$ is not in $E^{\prime}$ then $\omega_{1}$ is smooth and Kähler near $x$ and the volume of $V$ with respect to $\omega_{1}$ would be positive, which is a contradiction.