Let’s assume first that that $\alpha=c_{1}(L)$ for some line bundle $L$, which is equivalent to requiring that $\alpha\in N^{1}(X)_{\mathbb{Z}}$. Now $L$ is nef and big and so Theorem 2.1 implies that $L$ is semiample, so there exists some $k\geq 1$ such that $kL$ is globally generated. This gives a morphism $f:X\to\mathbb{P}^{N}$ such that $f^{*}\mathcal{O}(1)=kL$. If we let $\omega_{FS}$ be the Fubini-Study metric on $\mathbb{P}^{N}$, then $\omega=\frac{f^{*}\omega_{FS}}{k}$ is a pointwise nonnegative smooth real $(1,1)$ form in the class $\alpha$. If $\alpha\in N^{1}(X)_{\mathbb{Q}}$, then $k\alpha\in N^{1}(X)_{\mathbb{Z}}$ for some integer $k\geq 1$, and we can proceed as above. If finally $\alpha\in N^{1}(X)_{\mathbb{R}}$ then by Theorem 2.3 we know that the subcone of nef and big classes is locally rational polyhedral. Hence $\alpha$ lies on a face of this cone which is cut out by linear equations with rational coefficients. It follows that rational points on this face are dense, and it is then possible to write $\alpha$ as a linear combination of classes in $N^{1}(X)_{\mathbb{Q}}$ which are nef and big, with nonnegative coefficients. It is now clear that we can represent $\alpha$ by a smooth nonnegative form $\omega$.

Now fix a ball $\mathcal{U}$ in $N^{1}(X)_{\mathbb{R}}$ centered at $\alpha$, such that $\mathcal{K}_{NS}\cap\mathcal{U}$ is defined by $\{\Phi_{\beta}>0\}_{1\leq\beta\leq k}$ where the $\Phi_{\beta}$ are linear forms with rational coefficients. Since the big cone is open, up to shrinking $\mathcal{U}$ we may also assume that all the classes in $\partial\mathcal{K}_{NS}\cap\mathcal{U}$ are big. We may add some more linear forms to the $\Phi_{\beta}$, until they define a strongly convex rational polyhedral cone $C$ which is contained in $\overline{\mathcal{K}}_{NS}\cap\mathcal{U}$. We can then write

$$C=\left\{\sum_{i=1}^{\ell}a_{i}\gamma_{i}\ \bigg|\ a_{i}\geq 0\right\},$$

where the $\gamma_{i}$ are nef and big classes in $\mathcal{U}$. We claim that, when $t$ is bigger than some $t_{0}<1$, it is possible to write the path $\alpha_{t}$ as $\sum_{i}a_{i}(t)\gamma_{i}$ where the functions $a_{i}(t)$ are continuous and nonnegative. Assume first that the cone $C$ is simplicial, which means that the $\gamma_{i}$ are linearly independent. Then the path $\alpha_{t}$ enters and eventually stays in $C$, and so it can be expressed uniquely as

(4.1)

$$\alpha_{t}=\sum_{i=1}^{\ell}a_{i}(t)\gamma_{i},$$

where the $a_{i}(t)$ are smooth and nonnegative, $t_{0}\leq t\leq 1$. If on the other hand $C$ is not simplicial, it can be written as a finite union of simplicial subcones that intersect only along faces, and that are spanned by some linearly independent subsets of the $\gamma_{i}$. On any time interval when $\alpha_{t}$ belongs to the interior of a simplicial cone, the coefficients $a_{i}(t)$ in (4.1) vary smoothly, and on a common face of two simplicial cones the coefficients agree, hence the $a_{i}(t)$ vary continuously when $t_{0}\leq t<1$. Moreover since we only have finitely many simplicial subcones, we see that as $t\to 1$ the $a_{i}(t)$ converge to the coefficients of $\alpha_{1}$ in any of the simplicial cones that contains it, and so the $a_{i}(t)$ are continuous on the whole interval $t_{0}\leq t\leq 1$.

By the first part of the proof we know that we can choose $\delta_{i}\in\gamma_{i}$ a smooth nonnegative representative, for all $i$. Choose a smooth function $\varepsilon(t):[t_{0},1]\to\mathbb{R}$ that is positive on $[t_{0},1)$ and $\varepsilon(1)=0$, and that is small enough so that the classes $\tilde{\alpha}_{t}=\alpha_{t}-\varepsilon(t)\alpha_{t_{0}}$ are ample for all $t_{0}\leq t<1$. Then the new path $\tilde{\alpha}_{t}$ is also converging to $\alpha$ as $t\to 1$, and by the previous claim we can write

$$\tilde{\alpha}_{t}=\sum_{i=1}^{\ell}\tilde{a}_{i}(t)\gamma_{i},$$

where $\tilde{a}_{i}(t)$ is a continuous nonnegative function, for all $i$. Then the smooth $(1,1)$ forms

$$\tilde{\beta}_{t}:=\sum_{i=1}^{\ell}\tilde{a}_{i}(t)\delta_{i}$$

are nonnegative representatives of $\tilde{\alpha}_{t}$ that vary continuously in $t$. When $t$ approaches $1$, the forms $\tilde{\beta}_{t}$ converge in the $C^{\infty}$ topology to a smooth nonnegative form $\tilde{\omega}$ representing $\alpha$. If $\chi$ is a Kähler form in $\alpha_{t_{0}}$, then the forms $\beta_{t}=\tilde{\beta}_{t}+\varepsilon(t)\chi$ defined on $[t_{0},1)$ are Kähler, represent $\alpha_{t}$ and converge to $\tilde{\omega}$ as $t\to 1$. Up to replacing $\omega$ by $\tilde{\omega}$, this gives the desired family of forms on $[t_{0},1)$. It is very easy to extend the family $\beta_{t}$ on the whole $[0,1)$, and since we’re not going to use this, we leave the proof to the reader. ∎

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

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

∎