ScalingStacks

Proof. 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. ∎