Proposition 4.1 gives us $\omega\in\alpha$ a smooth nonnegative representative, and $\beta_{t}\in\alpha_{t}$ continuously varying Kähler forms, when $t<1$, such that $\beta_{t}\to\omega$ as $t\to 1$. Let’s assume first that the class $\alpha=c_{1}(L)$ for some nef and big line bundle $L$. As before, Theorem 2.1 gives a morphism $f:X\to\mathbb{P}^{N}$ such that $f^{*}\mathcal{O}(1)=kL$. Also by Theorem 2.2 the image of $f$ is a normal irreducible projective variety $Y$, $f:X\to Y$ is birational and $f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}$. Then setting $D_{0}=0$ as Cartier divisors on $Y$, we have $aK_{X}=f^{*}D_{0}$ for some integer $a>0$, so

$$f_{*}(aK_{X})=D_{0}=0$$

holds as Weil divisors, but since $f$ is birational we also have $f_{*}(aK_{X})=aK_{Y}$ (as Weil divisors), hence $aK_{Y}$ is Cartier and is equal to zero. So we have $f^{*}K_{Y}=K_{X}$ as $\mathbb{Q}$-divisors, which implies that $Y$ has at most canonical singularities and is a Calabi-Yau model (see also Corollary 1.5 of [Ka1]).

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.

Fix once and for all an $\varepsilon>0$ small enough so that Kodaira’s lemma holds. First of all notice that the classes $\alpha_{t}-\varepsilon E=\kappa_{\varepsilon}^{t}$ are all Kähler when $t$ is close to $1$. Choose a Kähler form $\chi_{\varepsilon}\in\kappa_{\varepsilon}$, let $\sigma\in H^{0}(X,\mathcal{O}_{X}(E))$ be the canonical section, and fix a Hermitian metric $|\cdot|$ on $E$ such that the following Poicaré-Lelong equation holds

(4.4)

$$\omega-\varepsilon[E]=\chi_{\varepsilon}-\varepsilon\sqrt{-1}\partial\overline{\partial}\log|\sigma|,$$

where $[E]$ denotes the current of integration on $E$. Then we have

$$\beta_{t}-\varepsilon[E]=\chi_{\varepsilon}+(\beta_{t}-\omega)-\varepsilon\sqrt{-1}\partial\overline{\partial}\log|\sigma|,$$

and $\chi_{\varepsilon}^{t}=\chi_{\varepsilon}+(\beta_{t}-\omega)$ is Kähler for $t$ close to $1$. There are smooth functions $\varphi_{t}$ solutions of

(4.5)

$$\omega_{t}^{n}=(\beta_{t}+\sqrt{-1}\partial\overline{\partial}\varphi_{t})^{n}=\alpha_{t}^{n}\Omega,$$

where the positive constants $\alpha_{t}^{n}$ approach $\alpha^{n}$ as $t$ goes to $1$, and $\sup_{X}\varphi_{t}=0$. We apply Theorem 2.1 and Proposition 3.1 of [EGZ] again, and get uniform estimates $\|\varphi_{t}\|_{C^{0}}\leq C_{0}$ independent of $t$. Outside $E$ we have

$$\beta_{t}=\chi_{\varepsilon}^{t}-\varepsilon\sqrt{-1}\partial\overline{\partial}\log|\sigma|,$$

so that the functions $\psi_{t}=\varphi_{t}-\varepsilon\log|\sigma|$ solve

(4.6)

$$(\chi_{\varepsilon}^{t}+\sqrt{-1}\partial\overline{\partial}\psi_{t})^{n}=\alpha_{t}^{n}\Omega=e^{F_{\varepsilon}^{t}}(\chi_{\varepsilon}^{t})^{n}$$

there, for some appropriate smooth functions $F_{\varepsilon}^{t}$, defined on the whole of $X$. As $t$ approaches $1$, the Kähler forms $\chi_{\varepsilon}^{t}$ are uniformly bounded in the smooth topology (with eigenvalues bounded away from $0$ uniformly), and so are the functions $F_{\varepsilon}^{t}$. Yau’s second order estimates [Y2] for the Monge-Ampère equation (4.6) give

(4.7)

$$\triangle^{\prime}_{t}(e^{-A\psi_{t}}(n+\triangle_{t}\psi_{t}))\geq e^{-A\psi_{t}}\left(-C_{1}-C_{2}(n+\triangle_{t}\psi_{t})+(n+\triangle_{t}\psi_{t})^{\frac{n}{n-1}}\right),$$

where $A,C_{1}$ and $C_{2}$ are uniform positive constants, $\triangle_{t}$ is the Laplacian of $\chi_{\varepsilon}^{t}$ and $\triangle^{\prime}_{t}$ is the Laplacian of $\chi_{\varepsilon}^{t}+\sqrt{-1}\partial\overline{\partial}\psi_{t}$. Now notice that on $X\backslash E$ we have

$$e^{-A\psi_{t}}(n+\triangle_{t}\psi_{t})=|\sigma|^{A\varepsilon}e^{-A\varphi_{t}}(n+\triangle_{t}\varphi_{t}-\varepsilon\triangle_{t}\log|\sigma|),$$

and

$$\left|\triangle_{t}\log|\sigma|\right|\leq C,$$

for some uniform constant $C$. Hence the function $e^{-A\psi_{t}}(n+\triangle_{t}\psi_{t})$ goes to zero when we approach $E$, and so its maximum will be attained. The maximum principle applied to (4.7) then gives

$$n+\triangle_{t}\psi_{t}\leq Ce^{A(\psi_{t}-\inf_{X\backslash E}\psi_{t})},$$

on the whole of $X\backslash E$. But noticing that $\inf_{X\backslash E}\psi_{t}\geq\inf_{X}\varphi_{t}-C$ for a uniform constant $C$, and recalling that $|\varphi_{t}|\leq C_{0}$, we get

$$n+\triangle_{t}\varphi_{t}\leq C+n+\triangle_{t}\psi_{t}\leq C(1+|\sigma|^{-A\varepsilon}).$$

This gives uniform interior $C^{2}$ estimates of $\varphi_{t}$ and $\psi_{t}$ on compact sets of $X\backslash E$. Then the Harnack estimate of Evans-Krylov gives uniform $C^{2,\gamma}$ estimates, for some $0<\gamma<1$, and a standard bootstrapping argument gives uniform $C^{k,\gamma}$ estimates for all $k\geq 2$, on compact sets of $X\backslash E$, independent of $t<1$. Thus the family $(\varphi_{t})$ is precompact $C^{k,\gamma^{\prime}}(X\backslash E)$ for any $0<\gamma^{\prime}<\gamma$, and any limit point $\psi$ belongs to $PSH(X\backslash E,\omega)$, it satisfies

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

on $X\backslash E$, and is bounded near $E$. Hence $\psi$ extends to a bounded function in $PSH(X,\omega)$ and the above Monge-Ampère equation holds on $X$ because the Borel measure $(\omega+\sqrt{-1}\partial\overline{\partial}\psi)^{n}$ doesn’t charge the analytic set $E$. Then by the uniqueness part of Theorem 2.1 of [EGZ], we must have $\psi=\varphi$. This implies that $\varphi_{t}\to\varphi$ in $C^{\infty}$ on compact sets of $X\backslash E$, and that $\varphi$ is smooth there.

If now we only have that $\alpha\in N^{1}(X)_{\mathbb{Q}}$ is nef and big, then $k\alpha\in N^{1}(X)_{\mathbb{Z}}$ for some integer $k\geq 1$. Kodaira’s lemma still holds, the function $F$ is obviously still in $L^{p}(\omega^{n})$, $p>1$, and the reasoning proceeds as above.

If finally $\alpha\in N^{1}(X)_{\mathbb{R}}$ then we use Kodaira’s lemma for big and nef $\mathbb{R}$-divisors (Example 2.2.23 in [L]) to get an effective $\mathbb{R}$-divisor $E$ such that for all $\varepsilon>0$ small enough, $\alpha-\varepsilon E=\kappa_{\varepsilon}$ is Kähler. The proof proceeds as above, once we show that $F$ is again in $L^{p}(\omega^{n})$, $p>1$. Recall from the proof of Proposition 4.1 that we can write

$$\alpha=\sum_{i}a_{i}\gamma_{i},$$

where $a_{i}\in\mathbb{R}_{>0}$ and $\gamma_{i}\in N^{1}(X)_{\mathbb{Q}}$ are nef and big. Moreover

$$\omega=\sum_{i}a_{i}\delta_{i},$$

where $\delta_{i}\in\gamma_{i}$ are smooth pointwise nonnegative forms. Then, by the previous case when $\alpha\in N^{1}(X)_{\mathbb{Q}}$, we know that there exist (nonsmooth) positive functions $F_{i}$ such that $\Omega=F_{i}a_{i}^{n}\delta_{i}^{n}$ and $F_{i}\in L^{1+\varepsilon}(\delta_{i}^{n})$ for some $\varepsilon>0$. As in (4.2) we see that for every $i$

$$\int_{X}F_{i}^{\varepsilon}\Omega=a_{i}^{n}\int_{X}F_{i}^{1+\varepsilon}\delta_{i}^{n}<\infty.$$

Then

$$\Omega=F\omega^{n}=F\left(\sum_{i}a_{i}\delta_{i}\right)^{n},$$

and since for each $j$ the function

$$\frac{(\sum_{i}a_{i}\delta_{i})^{n}}{a_{j}^{n}\delta_{j}^{n}}$$

is bigger than or equal to $1$ a.e., it follows that $F\leq F_{j}$ a.e. for all $j$. So

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

∎