Proof. The first statement follows from Poincaré duality. Pick any linear form $\ell$ on $\operatorname {c-N}^{d-k}({ \mathcal {X} })$. For each model $X'$, there exists a unique class $\alpha _{X'}\in \operatorname {N}^{k}(X')$ such that $\ell ([\beta ]) = (\alpha _{X'}\cdot \beta )$ for all $\beta \in \operatorname {N}^{d-k}(X')$. For any proper birational morphism $\pi \colon X'' \to X'$ and any class $\beta \in \operatorname {N}^{d-k}(X')$, we have $(\alpha _{X'}\cdot \beta ) = \ell ([\beta ]) = (\alpha _{X''}\cdot \pi ^{*}\beta )$ hence $\pi _*\alpha _{X''}=\alpha _{X'}$. The collection of numerical classes $(\alpha _{X'})$ thus defines a Weil class $\alpha$ which satisfies $\ell (\gamma ) = (\alpha \cdot \gamma )$ for any Cartier class $\gamma$.

Before we prove the second statement, note that a basis of neighborhoods of $0$ for the weak topology on $\operatorname {w-N}^{d-k}({ \mathcal {X} })$ is given by sets of the form $[U,X']:=\{\alpha \in \operatorname {w-N}^{d-k}({ \mathcal {X} }) \mid \alpha _{X'} \in U\}$ where $X'$ is a smooth model and $U$ is an open neighborhood of $0$ in $\operatorname {N}^{d-k}(X')$. As the latter space is finite dimensional, it is isomorphic to the dual of $\operatorname {N}^{k}(X')$. It follows that the family of open sets

\[ V(\{\beta_i\}, \{r_i\}):=\{\alpha \in \operatorname{w-N}^{d-k}({ \mathcal{X} }) \mid |(\alpha_{X'}\cdot \beta_i)| < r_i \} \]

also forms a basis of neighborhood of $0$, where $\beta _i$ are classes in $\operatorname {N}^{k}(X')$ for some model $X'$ and $r_i$ are positive real numbers.

Now take any weakly continuous linear form $\ell$ on $\operatorname {w-N}^{d-k}({ \mathcal {X} })$. By what precedes, we may find a model $X'$, classes $\beta _i\in \operatorname {N}^{k}(X')$ and $r_i>0$ such that $|\ell (\alpha )|<1$ if $|(\alpha \cdot [\beta _i])| < r_i$ for all $i$. Write $\ell _i(\alpha ) := (\alpha \cdot [\beta _i])$. Then we first infer $| \ell | \leqslant C \max |\ell _i|$ with $C= \max r^{-1}_i$, hence $\ell =0$ on $\cap _i \{ \ell _i =0\}$. This implies $\ell$ is a linear combination of the $\ell _i$'s (see, e.g., [Reference RudinRud91, Lemma 3.9]). We obtain a family of real numbers $a_i$ such that

\[ \ell(\alpha) = \bigg(\alpha \cdot \sum a_i [\beta_i]\bigg) \]

for all $\alpha \in \operatorname {w-N}^{d-k}({ \mathcal {X} })$ as required.

Proof. Pick any class $\alpha \in \operatorname {w-N}^{k}({ \mathcal {X} })$. For any model $X'$, and for any open subset $U$ of $\operatorname {N}^{k}(X')$, let $[U,X'] = \{ \beta \in \operatorname {w-N}^{k} ({ \mathcal {X} }), \beta _{X'} \in U\}$ as previously. Then the collection $I_\alpha$ of all sets containing $\alpha$ and of the form $[U,X']$ forms a basis of neighborhood of $\alpha$. It is also an inductive set for the order relation $[U_0,X_0] \leqslant [U_1,X_1]$ iff $X_0 \leqslant X_1$ and the preimage of $U_0$ by the pullback morphism $\operatorname {N}^{k}(X_0)\to \operatorname {N}^{k}(X_1)$ contains $U_1$.

Suppose that $\alpha$ is BPF. Then for each $i=[U,X']\in I_\alpha$ we may choose a Cartier BPF $b$-class $\alpha _i$ in $[U,X']$. By construction, we have $(\alpha _i)_{X'} \to \alpha _{X'}$ for all model $X'$. The converse statement is clear.

Proof. This is a rephrasing of the negativity lemma alluded to previously.

Proof. Fix a class $\alpha \in \operatorname {Nef^{1}}({ \mathcal {X} })$, and note that it follows from the definition that

\[ \lVert {\alpha} \rVert_{\operatorname{BPF}} \leqslant (\alpha \cdot {{\omega}}^{d-1}). \]

Conversely, write $\alpha = \alpha _+ - \alpha _-$ where $\alpha _\pm \in \operatorname {Nef^{1}}({ \mathcal {X} })$ are chosen so that $(\alpha _+ \cdot {{\omega }}^{d-1}) + (\alpha _- \cdot {{\omega }}^{d-1}) \leqslant \lVert {\alpha } \rVert _{\operatorname {BPF}} + \epsilon$ with $\epsilon >0$. Then $(\alpha \cdot {{\omega }}^{d-1}) \leqslant (\alpha _+ \cdot {{\omega }}^{d-1}) + (\alpha _- \cdot {{\omega }}^{d-1})\leqslant \lVert {\alpha } \rVert _{\operatorname {BPF}} +\epsilon$, and this implies the reverse inequality.

Proof. Suppose $\alpha$ is BPF. By Siu's inequalities (Proposition 1.1), we obtain $0\leqslant \alpha \leqslant C_d (\alpha \cdot {{\omega }}^{d-k}) {{\omega }}^{k}$ so that $\lVert {\alpha } \rVert _{\omega } \leqslant C_d (\alpha \cdot {{\omega }}^{d-k})$. Conversely, $\alpha \leqslant C {{\omega }}^{k}$ implies $(\alpha \cdot {{\omega }}^{d-k})\leqslant C ({{\omega }}^{d})$ hence (8) holds.

Pick any Cartier $b$-class $\alpha$, and write $\alpha = \alpha _+ - \alpha _-$ with $\alpha _\pm \in \operatorname {BPF}^{k}({ \mathcal {X} })$. The previous arguments show

\[ \alpha = \alpha_+ - \alpha_- \leqslant \alpha_+ \leqslant C_d (\alpha_+\cdot{{\omega}}^{d-k}) \leqslant C_d \lVert {\alpha} \rVert_{\operatorname{BPF}} \]

and, similarly, $\alpha \geqslant - C_d \lVert {\alpha } \rVert _{\operatorname {BPF}}$, hence $\lVert {\alpha } \rVert _{\omega } \leqslant C_d \lVert {\alpha } \rVert _{\operatorname {BPF}}$.

Suppose that $- C {{\omega }} \leqslant \alpha \leqslant C {{\omega }}$ for some positive constant $C>0$, and pick any class $\gamma \in \operatorname {BPF}^{d-1}({ \mathcal {X} })$. We obtain $|(\alpha \cdot \gamma )|\leqslant C ({{\omega }}\cdot \gamma )$, hence $\lVert {\alpha } \rVert _{\operatorname {BPF}}^{\vee }\leqslant C$.

Proof. Suppose $\alpha$ is a Cartier class determined in some smooth model $X'$. We only need to show that $\lVert {\alpha } \rVert _{\omega }\leqslant \lVert {\alpha } \rVert _{\operatorname {BPF}}^{\vee }$.

Let us first treat the case $\alpha \in \operatorname {c-N}^{d-1}({ \mathcal {X} })$. For any nef class $\gamma \in \operatorname {Nef}(X')$ such that ${(\gamma \cdot {{\omega }}^{d-1})=1}$ we have

\[ ((\alpha + \lVert {\alpha} \rVert_{\operatorname{BPF}}^{\vee} {{\omega}}^{d-1})\cdot[\gamma]) \geqslant 0, \]

hence the class $(\alpha + \lVert {\alpha } \rVert _{\operatorname {BPF}}^{\vee } {{\omega }}^{d-1})_{X'}$ is psef by duality. Similarly, we obtain $(-\alpha + \lVert {\alpha } \rVert _{\operatorname {BPF}}^{\vee } {{\omega }}^{d-1})_{X'}\geqslant 0$, hence $\lVert {\alpha } \rVert _{\omega }\leqslant \lVert {\alpha } \rVert _{\operatorname {BPF}}^{\vee }$ as required.

Let us now treat the case where $\alpha$ is a $b$-divisor. Recall from [Reference Boucksom, Demailly, Păun and PeternellBDPP13] that a class $\alpha _0\in \operatorname {N}^{1}(X)$ is psef if and only if $(\alpha _0\cdot \gamma _0)\geqslant 0$ for any strongly movable curve class $\gamma _0\in \operatorname {N}^{d-1}(X)$.

Suppose first that $\alpha$ is a Cartier $b$-divisor determined in some smooth model $X'$. For any strongly movable class $\gamma _0\in \operatorname {N}^{d-1}(X')$, there exists a class $\gamma \in \operatorname {c-BPF}^{d-1}({ \mathcal {X} })$ such that $\gamma _{X'} = \gamma _0$. It follows that

\[ ((\alpha + \lVert {\alpha} \rVert_{\operatorname{BPF}}^{\vee} {{\omega}})\cdot\gamma_0) = ((\alpha + \lVert {\alpha} \rVert_{\operatorname{BPF}}^{\vee} {{\omega}})\cdot\gamma) \geqslant 0, \]

hence $\alpha + \lVert {\alpha } \rVert _{\operatorname {BPF}}^{\vee } {{\omega }}\geqslant 0$ as required.

Proof. Suppose $\beta \geqslant \gamma \in \operatorname {Nef^{1}}({ \mathcal {X} })$, and $\beta$ is determined in $X'$. Then $\beta _{X'}\geqslant \gamma _{X'}$ and by Lemma 1.4 there exists a net $\gamma _i\in \operatorname {c-Nef^{1}}({ \mathcal {X} })$ such that $\gamma _{i,X'} \to \gamma _{X'}$. As $\beta _{X'}$ is big, we have $\gamma _{i,X'}\leqslant (1+\epsilon )\beta _{X'}$ for any $\epsilon >0$ and for $i \in I_\epsilon$ in a tail set. We infer from Lemma 1.5 $\gamma _i\leqslant [\gamma _{i,X'}] \leqslant (1+\epsilon )\beta$, hence $\gamma _i\leqslant (1+\epsilon ) P(\beta )$ for $i \in I_\epsilon$. We conclude that $\gamma \leqslant P(\beta )$ by taking the limit of the net and then $\epsilon \to 0$.

Suppose that $\beta '$ is a nef $b$-divisor class such that $\gamma \leqslant \beta '$ for any $\gamma \in \operatorname {Nef^{1}}({ \mathcal {X} })$ satisfying $\gamma \leqslant \beta$. Applying these inequalities to Cartier $b$-divisor classes, we get $P(\beta )_{X'} \leqslant \beta '_{X'}$ in each model, which concludes the proof.

Proof. Pick any $\alpha \in \operatorname {c-N}^{1}({ \mathcal {X} }) \cap \operatorname {Nef^{1}}({ \mathcal {X} })$. Without loss of generality, we may suppose that $\alpha$ is determined in $X$, and we need to show that $\alpha _X$ is nef. By adding a small real ample class of $X$, we may suppose that $\alpha _X = c_1(L)$ for some big line bundle $L\to X$. As $\alpha$ is nef, we have $P(\alpha _X)=\alpha$. Redoing the proof of Proposition 2.7, we find a sequence of elements

\[ \beta_m=\frac 1{2^{m}} \beta'_{2^{m}} = \frac 1{2^{m}}[K_X+ (d+1) A]+ \alpha - \frac 1{2^{m}} [\mathcal{J}(\mathfrak{b}_\bullet^{2^{m}})]\in \operatorname{c-Nef^{1}}({ \mathcal{X} }), \]

which converges weakly to $\alpha$. This implies $- ({1}/{2^{m}}) [\mathcal {J}(\mathfrak {b}_\bullet ^{2^{m}})]$ to tend to $0$ weakly. As the sequence $- ({1}/{2^{m}}) [\mathcal {J}(\mathfrak {b}_\bullet ^{2^{m}})]\leqslant 0$ is decreasing, we obtain ${ \mathcal {J} }(\mathfrak {b}_\bullet ^{2^{m}})=0$ for all $m$, and $\beta _{m}$ is thus a nef class determined in $X$. This implies $\alpha _X$ to be nef as required.

Proof. Compatibility with the intersection of Cartier $b$-divisor classes is obvious from the definition. The symmetry and multilinearity follows from the corresponding properties for the intersection product of divisors. The fact that it is increasing is a consequence of the observation that $(\alpha '_1\cdot \alpha _2 \cdot \cdots \cdot \alpha _k \cdot \gamma ) \geqslant (\alpha _1\cdot \alpha _2 \cdot \cdots \cdot \alpha _k \cdot \gamma )$ as soon as $\alpha '_1\geqslant \alpha _1$. These arguments prove part (i).

We fix a class ${{\omega }}\in \operatorname {Nef^{1}}({ \mathcal {X} })$ determined by an ample class in $X$, and prove part (ii). Suppose $\alpha ^{(i)}_l\to \alpha _l$ weakly. Pick any class $\Delta$ in the accumulation locus of the net $(\alpha _1^{(j)}\cdot \cdots \cdot \alpha _k^{(j)} \cdot \gamma )$. We may replace the net $I$ by one indexed by the family of open neighborhoods of $\Delta$ (endowed with the partial ordering given by the inclusion) and assume that $(\alpha _1^{(j)}\cdot \cdots \cdot \alpha _k^{(j)} \cdot \gamma ) \to \Delta$.

Let $\beta ^{(i)}_l$ be nets of nef Cartier $b$-divisor classes decreasing to $\alpha _l$. Fix any large index $i$, and choose a model $X'$ in which $\beta ^{(i)}_l$, $l=1, \ldots, k$ are all determined. As $(\alpha _l^{(j)})_{X'} \to (\alpha _l)_{X'}$ in $\operatorname {N}^{1}(X')$ we have $(\alpha _l^{(j)})_{X'}\leqslant (\beta ^{(i)}_l)_{X'} + \epsilon {{\omega }}$ for an arbitrary small $\epsilon >0$ and all $j$ large enough, hence $\alpha _l^{(j)} \leqslant \beta ^{(i)}_l + \epsilon {{\omega }}$ by Lemma 1.5. We infer

\[ (\alpha_1^{(j)}\cdot \cdots \cdot \alpha_k^{(j)} \cdot \gamma ) \leqslant (\beta_1^{(l)}\cdot \cdots \cdot \beta_k^{(l)} \cdot \gamma ) + \epsilon kC^{k-1} ({{\omega}}^{k}\cdot \gamma), \]

where $C>0$ is a any constant such that $\beta _l^{(i_0)}\leqslant C{{\omega }}$, $l=1,\ldots,k$, and $i_0$ is a fixed index. We now let successively $j$ tend to infinity in the net $I$, then $\epsilon \to 0$, and finally let $l$ tend to infinity in $I$. We conclude that $\Delta \leqslant (\alpha _1\cdot \cdots \cdot \alpha _k \cdot \gamma )$ as required.

Now suppose that $\alpha ^{(j)}_i$ is a net of nef $b$-divisor classes (not necessarily Cartier) decreasing to $\alpha _i$. We claim that there exists a constant $C>0$ such that for any $\epsilon >0$ one has

\[ (\alpha_1\cdot \cdots \cdot \alpha_k \cdot \gamma ) \leqslant (\alpha_1^{(j)}\cdot \cdots \cdot \alpha_k^{(j)} \cdot \gamma ) \leqslant (\alpha_1\cdot \cdots \cdot \alpha_k \cdot \gamma ) + \epsilon kC^{k-1} ({{\omega}}^{k}\cdot \gamma) \]

for all $j$ large enough. Indeed, the first inequality is a consequence of the fact that the intersection product is increasing in each variable, and the second inequality follows from the previous proof. These estimates imply the convergence $(\alpha _1^{(j)}\cdot \cdots \cdot \alpha _k^{(j)} \cdot \gamma ) \to (\alpha _1\cdot \cdots \cdot \alpha _k \cdot \gamma )$ to hold in $\operatorname {N}_+^{k+l}({ \mathcal {X} })$ which proves part (iii).

Finally, we observe that if $\alpha$ and $\alpha '$ are two nef classes such that $\lVert {\alpha - \alpha '} \rVert _{\omega }\leqslant \epsilon$, then $- \epsilon {{\omega }} \leqslant (\alpha - \alpha ')\leqslant \epsilon {{\omega }}$ so that part (iv) follows from part (iii).

Proof. Pick any two nets $\alpha _j, \beta _j\in \operatorname {c-Nef^{1}}({ \mathcal {X} })$ parametrized by an inductive set $J$ and decreasing to $\alpha$ and $\beta$, respectively. Fix any $\epsilon >0$.

Observe that $(\alpha _j^{d}) \geqslant (\alpha ^{d})>0$ and $(\beta _j^{d}) \geqslant (\beta ^{d})>0$ so that $\alpha _j$ and $\beta _j$ are big. The largest non-negative number $s_j$ such that $\alpha _j - s_j \beta _j\geqslant 0$ is, thus, positive. By [Reference Boucksom, Favre and JonssonBFJ09, Theorem F], we have

\[ (\alpha_j^{d-1}\cdot\beta_j)^{{d}/({d-1})} - (\alpha_j^{d})(\beta_j^{d})^{{1}/({d-1})} \geqslant ( (\alpha_j^{d-1}\cdot\beta_j)^{{1}/({d-1})} - s_j (\beta_j^{d})^{ 1/({d-1})} )^{d}. \]

In other words, $s_j\geqslant \tau _j$ where

\[ \tau_j = \frac 1{(\beta_j^{d})^{{1}/({d-1})}} (\alpha_j^{d-1}\cdot\beta_j)^{{1}/({d-1})} - \frac{1}{(\beta_j^{d})^{{1}/({d-1})}} ((\alpha_j^{d-1}\cdot\beta_j)^{{d}/({d-1})} - (\alpha_j^{d})(\beta_j^{d})^{{1}/({d-1})})^{{1}/{d}}. \]

By Theorem 3.2(iii), we have

\[ \lim_J \tau_j = \tau := \frac{1}{(\beta^{d})^{ 1/({d-1})}} (\alpha^{d-1}\cdot\beta)^{{1}/({d-1})} - \frac 1{(\beta^{d})^{{1}/({d-1})}} ((\alpha^{d-1}\cdot\beta)^{{d}/({d-1})} - (\alpha^{d})(\beta^{d})^{{1}/({d-1})})^{{1}/{d}}, \]

and we obtain $\alpha - \tau \beta \geqslant 0$ as required.

Proof. When $\alpha _i$ are Cartier, the result is clear. By multilinearity and by replacing $\alpha _i$ by $\alpha _i + {{\omega }}$, we may suppose that $(\alpha _i^{d})>0$ for all $i$. Pick any $\epsilon >0$. By the previous corollary, one can find $\alpha '_i\in \operatorname {c-Nef^{1}}({ \mathcal {X} })$ such that $(1-\epsilon ) \alpha _i' \leqslant \alpha _i \leqslant \alpha _i'$. It follows that

\begin{align*} 0\leqslant (\alpha'_1 \cdot \cdots \cdot \alpha'_k\cdot\gamma) -(\alpha_1\cdot \cdots \cdot \alpha_k\cdot\gamma) &\leqslant ((1-\epsilon)^{-k} -1) (\alpha_1\cdot \cdots \cdot \alpha_k\cdot\gamma)\\ &\leqslant C \epsilon ({{\omega}}^{k}\cdot\gamma). \end{align*}

Suppose $\gamma _i \to \gamma$ is a net of classes converging weakly in $\operatorname {BPF}^{l}({ \mathcal {X} })$. Then

\begin{align*} \liminf_i (\alpha_1\cdot \cdots \cdot \alpha_k\cdot\gamma_i) &\geqslant \liminf_i (\alpha'_1 \cdot \cdots \cdot \alpha'_k\cdot\gamma_i) - C \epsilon ({{\omega}}^{k}\cdot\gamma_i)\\ &= (\alpha'_1 \cdot \cdots \cdot \alpha'_k\cdot\gamma) - C \epsilon ({{\omega}}^{k}\cdot\gamma) \geqslant (\alpha_1 \cdot \cdots \cdot \alpha_k\cdot\gamma) - C \epsilon ({{\omega}}^{k}\cdot\gamma), \end{align*}

which implies $\liminf _i (\alpha _1\cdot \cdots \cdot \alpha _k\cdot \gamma _i)\geqslant (\alpha _1\cdot \cdots \cdot \alpha _k\cdot \gamma )$.

We conclude using Theorem  3.2(i).

Proof. Let $\alpha$ be any nef class. As any Cartier class belongs to $\operatorname {N^{1}_{\Sigma }}({ \mathcal {X} })$, by adding ${{\omega }}$ if necessary, we may assume that $(\alpha ^{d})>0$. By Corollary 3.7, for each $n$, one can take a nef Cartier class $\alpha _n$ such that $(1+ {1}/{n})^{-1} \alpha _n\leqslant \alpha \leqslant \alpha _n$. Using Corollary 3.5, we infer for all integers $m\geqslant n$:

\begin{align*} \lVert {\alpha_n - \alpha_m} \rVert_{\Sigma}^{2} &=\sup_{ \substack{\gamma \in \operatorname{c-BPF}^{d-2}({ \mathcal{X} }) \\ (\gamma \cdot {{\omega}}^{2}) = 1}} 2 ((\alpha_n - \alpha_m)\cdot {{\omega}} \cdot \gamma )^{2} - ( (\alpha_n - \alpha_m)^{2} \cdot \gamma)\\ &\leqslant \sup_{ \substack{\gamma \in \operatorname{c-BPF}^{d-2}({ \mathcal{X} }) \\ (\gamma \cdot {{\omega}}^{2}) = 1}} \frac 2{n^{2}} (\alpha_m \cdot {{\omega}} \cdot \gamma )^{2} + \frac 2{n} (\alpha_m^{2}\cdot \gamma) \leqslant \frac{C}n \end{align*}

for some constant $C>0$. This shows that $(\alpha _n)$ is a Cauchy sequence in $\operatorname {N^{1}_{\Sigma }}({ \mathcal {X} })$. However, because it converges weakly to $\alpha$ we conclude that $\alpha$ belongs to $\operatorname {N^{1}_{\Sigma }}({ \mathcal {X} })$ and $\lVert {\alpha _n-\alpha } \rVert _{\Sigma } \to 0$.

By linearity, we obtain $\operatorname {Vect}(\operatorname {Nef^{1}}({ \mathcal {X} })) \subset \operatorname {N^{1}_{\Sigma }}({ \mathcal {X} })$ and the continuity follows from the fact that $\lVert {\alpha } \rVert _{\operatorname {BPF}} = (\alpha \cdot {{\omega }}^{d-1})$ for all $\alpha \in \operatorname {Nef^{1}}({ \mathcal {X} })$ by Lemma 1.7.

Proof. Take two sequences of nef Cartier $b$-divisor classes such that $\lVert {\alpha _n-\alpha } \rVert _{\Sigma }\to 0$ and $\lVert {\beta _n-\beta } \rVert _{\Sigma }\to 0$. By continuity of the intersection product in $\operatorname {N^{1}_{\Sigma }}({ \mathcal {X} })$, we have $\imath (\alpha )\cdot \imath (\beta ) = \lim _n \lim _m \alpha _n\cdot \beta _m$. By Corollary 3.8, we obtain $\lim _m \alpha _n\cdot \beta _m = \alpha _n\cdot \beta$; and then $\lim _n \alpha _n\cdot \beta = \alpha \cdot \beta$ which proves the result.

Proof. In view of Corollary 3.10, the statement is a direct consequence of the version of Hodge index theorem holding in $\operatorname {N^{1}_{\Sigma }}({ \mathcal {X} })$, see [Reference Dang and FavreDF21, Theorem 3.13].

Proof. Write $\alpha$ and $\beta$ as decreasing limits of sequences of nef Cartier classes $\alpha _n\downarrow \alpha$ and $\beta _n\downarrow \beta$. By the Khovanskii–Teissier inequalities, the sequence $(\alpha _n^{k}\cdot \beta _n^{d-k})$ is log-concave which implies by Theorem 3.2 the log-concavity of the sequence $e_k=(\alpha ^{k}\cdot \beta ^{d-k})$ by letting $n\to \infty$.

Suppose now that $\alpha$ and $\beta$ are big and nef, and that the sequence $e_k$ is linear. Scale both classes such that $(\alpha ^{d})=(\beta ^{d}) =1$ so that $(\alpha ^{d-1}\cdot \beta )=1$ too. Then $\alpha =\beta$ by Diskant's inequalities.

Proof. Observe that the last statement is a consequence of Cauchy–Schwarz inequality. Let us prove that $q$ is semi-negative on $H_\alpha$.

Pick any decreasing sequence $\alpha _n\in \operatorname {c-Nef^{1}}({ \mathcal {X} })$ converging to $\alpha$. Pick any sequence of models $X_{n+1}\geqslant X_n$ such that for each $n$, the classes $\gamma _n$ and $\alpha _n$ are determined in $X_n$. By perturbing slightly $\gamma _n$, we may assume it is represented by a surface $\Sigma$ in $X_n$. We may also assume that $\gamma _n \leqslant \omega ^{d-2}$ for some fixed ample class $\omega$ in $X$. Note that

\[ 0< c:= (\omega\cdot\alpha \cdot \gamma ) \leqslant (\omega\cdot\alpha_n \cdot \gamma_n ) \leqslant C:= (\omega\cdot\alpha_1 \cdot \gamma_1 ) . \]

Pick any class $\theta \in \operatorname {N^{1}_{\Sigma }}({ \mathcal {X} })$ such that $(\theta \cdot \alpha \cdot \gamma )=0$. Take any sequence of Cartier classes $\theta ^{(j)}\to \theta$ in $\operatorname {N^{1}_{\Sigma }}({ \mathcal {X} })$. By Theorem 1.6, we have $(\theta ^{(j)} \cdot \beta )\to (\theta \cdot \beta )$ in $\operatorname {N}^{{2},\vee }_{\operatorname {BPF}}({ \mathcal {X} })$ for any class $\beta \in \operatorname {Nef^{1}} ({ \mathcal {X} })$.

Fix $\epsilon >0$, and take $n$ large enough such that $|(\theta \cdot \alpha _n\cdot \gamma _n)|\leqslant \epsilon$. For all $j\gg 1$, we have $|(\theta ^{(j)} \cdot \alpha _n\cdot \gamma _n)|\leqslant 2\epsilon$. Set $\tilde {\theta }_j= \theta ^{(j)} - \eta _j \omega$ so that $(\tilde {\theta }_j \cdot \alpha _n\cdot \gamma _n)=0$, and observe that

\[ 0\leqslant |\eta_j| := \frac{|(\theta^{(j)}\cdot \alpha_n\cdot \gamma_n)|}{(\omega\cdot \alpha_n\cdot \gamma_n)} \leqslant 2c^{-1} \epsilon. \]

By the Hodge index theorem, we obtain

\[ (\tilde{\theta}^{2}_j\cdot \gamma_n ) = (\tilde{\theta}^{2}_j|_\Sigma) \leqslant 0. \]

As the sequence $\eta _j$ is bounded, we may extract a subsequence converging to some $0<\eta \leqslant 2c^{-1}\epsilon$ and letting $j\to \infty$, we obtain

\[ ((\theta - \eta \omega)^{2} \cdot\gamma_n ) \leqslant 0 \]

We conclude by letting first $n\to \infty$ and then $\epsilon \to 0$.

Proof. We may suppose that $\theta \cdot \gamma \neq 0$, $\theta '\cdot \gamma \neq 0$.

We claim that

\[ \theta^{2}\cdot \gamma = (\theta')^{2}\cdot \gamma = 0. \]

To see this pick any class $\beta \in \operatorname {c-BPF}^{d-k-2}({ \mathcal {X} })$ such that the curve classes $\theta \cdot \gamma \cdot \beta$ and $\theta '\cdot \gamma \cdot \beta$ are both non-zero. Observe that

\[ q_\gamma(a \theta+ a' \theta'):= ((a \theta+ a' \theta')^{2} \cdot \gamma\cdot \beta) =a^{2} (\theta^{2}\cdot \gamma\cdot\beta) + (a')^{2} ((\theta')^{2}\cdot \gamma\cdot \beta) \geqslant 0. \]

Pick any big and nef class $\alpha \in \operatorname {c-Nef^{1}}({ \mathcal {X} })$ such that $(\theta \cdot \gamma \cdot \beta \cdot \alpha ) \neq 0$, and $(\theta '\cdot \gamma \cdot \beta \cdot \alpha ) \neq 0$. By Proposition 3.12, we have $q_\gamma \leqslant 0$ on the hyperplane $H:= (\gamma \cdot \beta \cdot \alpha )^{\perp }$. One can thus find a pair $a,a' \neq 0$ such that $a\theta + a'\theta ' \in H$ is non-zero and such that $q_\gamma (a\theta + a'\theta ')=0$, and it follows that $(\theta ^{2}\cdot \gamma \cdot \beta ) = ((\theta ')^{2}\cdot \gamma \cdot \beta ) = 0$. As $\beta$ is arbitrary, the claim is proved.

Pick any class $\beta \in \operatorname {c-BPF}^{d-k-2}({ \mathcal {X} })$ such that both curve classes $\theta \cdot \Omega$ and $\theta '\cdot \Omega$ are non-zero with $\Omega =\beta \cdot \gamma$. We shall prove that $\theta \cdot \gamma \cdot \beta = c(\beta ) \theta '\cdot \gamma \cdot \beta$ for some $c(\beta )>0$. By adding a small multiple of ${{\omega }}^{k}$ to $\gamma _n$ and ${{\omega }}^{d-k-2}$ to $\beta$, we may assume that $\Omega _n:= \gamma _n\cdot \beta$ is strongly BPF, and determined in some model by a surface $\Sigma _n$ which is the image under a flat map of a complete intersection of ample divisors. We may also impose that $\theta \cdot \Omega \cdot {{\omega }} \neq 0$.

Choose any $t_n$ such that $(\theta _n^{2}\cdot \Omega _n) =0$ with $\theta _n = \theta + t_n {{\omega }}$. Observe that

\begin{align*} 0 = (\theta_n^{2}\cdot \Omega_n) &= (\theta^{2}\cdot \Omega_n) + 2 t_n ({{\omega}}\cdot \theta\cdot \Omega_n) +t_n^{2} ({{\omega}}^{2}\cdot \Omega_n)\\ &=(\theta^{2}\cdot (\Omega_n-\Omega)) + 2 t_n ({{\omega}}\cdot \theta\cdot \Omega_n) +t_n^{2} ({{\omega}}^{2}\cdot \Omega_n). \end{align*}

As $({{\omega }}^{2}\cdot \Omega _n)$ is uniformly bounded from below by $({{\omega }}^{2} \cdot \gamma \cdot \beta )$, and $(\theta ^{2}\cdot \Omega _n)\sim (\theta ^{2}\cdot \Omega ) =0$, such a constant $t_n$ exists and we may choose $t_n\to 0$ as $n\to \infty$.

Now pick another big and nef class ${{\omega }}'$ such that ${{\omega }} \cdot \Omega$ and ${{\omega }}' \cdot \Omega$ are not proportional, normalized by $({{\omega }} \cdot \Omega \cdot \theta )= ({{\omega }}' \cdot \Omega \cdot \theta )\neq 0$. Choose $r_n, r'_n$ such that

(17)\begin{align} 0&=(\theta_n'\cdot \theta_n\cdot \Omega_n ), \end{align}

(18)\begin{align} 0&=((\theta_n')^{2}\cdot \Omega_n) \end{align}

with $\theta '_n = \theta ' + r_n {{\omega }}+ r'_n {{\omega }}'$.

Proof. Let us prove that $M(\alpha ) := \alpha ^{d-1}$ is injective on the cone of big and nef classes. Recall that by Corollary 3.4, a nef class $\alpha$ satisfies $\alpha \geqslant c{{\omega }}$ for some $c>0$ iff $(\alpha ^{d})>0$.

If $\alpha$ and $\beta$ are two nef classes such that $\alpha ^{d-1}= \beta ^{d-1}$, then $(\alpha ^{d}) = (\alpha \cdot \beta ^{d-1})$, and $(\alpha ^{d-1}\cdot \beta ) = (\beta ^{d})$. By the log-concavity of the sequence $k\mapsto \log (\alpha ^{k}\cdot \beta ^{d-k})$ we obtain $0<(\alpha ^{d})= (\beta ^{d}) = (\alpha ^{k}\cdot \beta ^{d-k})$ for all $k$. This implies the equality between $\alpha$ and $\beta$ by Proposition 3.11.

We now prove the surjectivity. Fix any class $\gamma \in \operatorname {BPF}^{d-1}({ \mathcal {X} })$ such that $\gamma \geqslant c {{\omega }}^{d-1}$ for some $c>0$. We follow the classical variational method to solve a Monge-Ampère equation. Let $K:= \{ \alpha \in \operatorname {Nef^{1}}({ \mathcal {X} }) \,|\,(\alpha \cdot \gamma ) = 1\}$. Note that $K$ is compact for the weak topology of $\operatorname {w-N}^{1}({ \mathcal {X} })$ by [Reference Dang and FavreDF21, Proposition 2.13], as $K\subset \{ \alpha \in \operatorname {Nef^{1}}({ \mathcal {X} }), (\alpha \cdot {{\omega }}^{d-1}) \leqslant c^{-1}\}$. As the function $\alpha \mapsto (\alpha ^{d})$ is upper-semicontinuous by Theorem 3.2(ii), there exists a class $\alpha _\star \in K$ such that $(\alpha _\star ^{d})= \sup _K (\alpha ^{d})$.

Observe that $(\alpha _\star ^{d})>0$. Pick any Cartier class $\beta \in \operatorname {c-N}^{1}({ \mathcal {X} })$. For $t$ small enough, the class $\alpha _\star + t\beta$ is psef by Siu's inequality. Consider the nef envelope $P(\alpha _\star + t\beta )$ as defined in § 2.2.

Pick any big and nef class ${{\omega }}'\in \operatorname {c-Nef^{1}}({ \mathcal {X} })$ such that $\alpha _\star \leqslant \omega '$, and $\omega ' \pm \beta$ are both nef (any sufficiently large ample class in a model in which $\beta$ is determined does the job). Using [Reference Boucksom, Favre and JonssonBFJ09, Corollary 3.4] and by taking limits of Cartier nef classes decreasing to $\alpha _\star$, we have

\[ P(\alpha_\star+ t\beta)^{d}\geqslant (\alpha_\star^{d}) + d t (\alpha_\star^{(d-1)}\cdot \beta) - C t^{2} \]

for some $C>0$ which depends only on $(\omega ^{d})$ and for any $0 \leqslant t \leqslant 1$. Observe now that

\[ \alpha_t := \frac{P(\alpha_\star+ t\beta)}{(P(\alpha_\star+ t\beta)\cdot \gamma)} \in K \]

which implies

\begin{align*} (\alpha_\star^{d}) \geqslant \frac{P(\alpha_\star+ t\beta)^{d}}{(P(\alpha_\star+ t\beta)\cdot \gamma)^{d}} &\geqslant \frac{(\alpha_\star^{d}) + d t (\alpha_\star^{(d-1)}\cdot \beta) - C t^{2}}{((\alpha_\star+ t\beta)\cdot \gamma)^{d}}\\ &\geqslant (\alpha_\star^{d}) + d t (\alpha_\star^{(d-1)}\cdot \beta) - d t (\beta\cdot \gamma) (\alpha_\star^{d}) +O(t^{2}), \end{align*}

hence $(\alpha _\star ^{(d-1)}\cdot \beta ) \leqslant (\beta \cdot \gamma ) (\alpha _\star ^{d})$. Since this is true for $\pm \beta$ we infer $(\alpha _\star ^{(d-1)}\cdot \beta ) = (\beta \cdot \gamma ) (\alpha _\star ^{d})$. Rescaling $\alpha _\star$, we find a nef class such that $(\alpha ^{d-1}) = \gamma$ as required.

Proof. Let $\pi \colon X'\to X$ be any composition of blow-ups of smooth centers, and let $E$ be any $\pi$-exceptional prime divisor in $X'$. It follows from [Reference Boucksom, Favre and JonssonBFJ09, Theorem B] that

\[ P([\pi^{*}c_1(L)])^{d-1}\cdot E = \lim_{n\to\infty} \frac{(d-1)!}{n^{d-1}} \dim_\mathbb{C} (V_n), \]

where

\[ V_n:= \mathrm{Im} ( H^{0}(X',n\pi^{*}L) \to H^{0}(E,n\pi^{*}L|_E) ). \]

Observe that

\[ V_n\subset H^{0}(E, n\pi^{*}L|_E) = H^{0}(\pi(E), nL|_{\pi(E)}) \]

and because $\dim (\pi _*(E)) < d-1$, the asymptotic Riemann–Roch theorem implies $\dim _\mathbb {C} (V_n)= o(n^{d-1})$, hence $P([\pi ^{*}c_1(L)])^{d-1}\cdot E =0$.

Define the class $\alpha := P([c_1(L)])^{d-1}$. The previous computation shows

\[ (\alpha_{X'}\cdot E)=(\alpha\cdot [E]) = ( P([c_1(L)])^{d-1}\cdot [E]) = ( P([\pi^{*}c_1(L)])^{d-1}\cdot [E])= 0 \]

for any $\pi$-exceptional divisor $E$. As $\pi$ is a composition of blow-ups with smooth centers, there is a decomposition $\operatorname {N}^{1}(X') = \pi ^{*} \operatorname {N}^{1}(X) \oplus F$ where $F$ is the subspace spanned by classes of exceptional divisors. As the intersection of $\alpha _{X'}$ with any element in $F$ is zero, we have $\alpha _{X'} = \pi ^{*} \pi _*(\alpha _{X'})= \pi ^{*}\alpha _{X}$, hence $\alpha$ is a Cartier class determined in $X$.

Proof. Pick any $\gamma \in \operatorname {BPF}^{d-1}({ \mathcal {X} })$. By Theorem C, for any $t>0$ we may find a class $\delta _t\in \operatorname {Nef^{1}}({ \mathcal {X} })$ such that $\gamma + t{{\omega }}^{d-1} = (\delta _t^{d-1})$. Let $s = (\delta _t^{d-2}\cdot {{\omega }}^{2})$. We now estimate using [Reference Dang and FavreDF21, Theorem 3.11]:

\begin{align*} | (\alpha\cdot (\gamma+t{{\omega}}^{d-1})) |^{2} &= | (\alpha\cdot \delta_t^{d-1}) |^{2}\\ &\leqslant 9 q_{{{\omega}}, ({1}/{s}) \delta_t^{d-2}}(\alpha) q_{{{\omega}}, ({1}/{s}) \delta_t^{d-2}}(s \delta_t)\\ &\leqslant 18 ( (\gamma+t{{\omega}}^{d-1}) \cdot{{\omega}})^{2} \lVert {\alpha} \rVert_{\Sigma}^{2}, \end{align*}

and we conclude by letting $t\to 0$.

Proof. Pick $\alpha \in \operatorname {N^{1}_{\Sigma }}({ \mathcal {X} })$ and choose a sequence $\alpha _n\in \operatorname {c-N}^{1}({ \mathcal {X} })$ such that $\lVert {\alpha -\alpha _n} \rVert _{\Sigma } \to 0$. Suppose $\gamma _i\in \operatorname {BPF}^{k}({ \mathcal {X} })$ is a net of BPF classes converging weakly to $\gamma$. We may suppose that $C:= \sup _i(\gamma _i\cdot {{\omega }}^{d-k})< \infty$. Pick any BPF Cartier $b$-numerical class $\beta \in \operatorname {c-BPF}^{d-k-1}({ \mathcal {X} })$. By (19) and Corollary 3.4, we have

\[ |(\alpha\cdot\gamma_i\cdot\beta) -(\alpha_n\cdot\gamma_i\cdot\beta) | \leqslant 5 (\gamma_i\cdot\beta\cdot {{\omega}}^{d-k})\lVert {\alpha-\alpha_n} \rVert_{\Sigma} \leqslant CC_d (\beta\cdot {{\omega}}^{k+1}) \lVert {\alpha-\alpha_n} \rVert_{\Sigma} . \]

Pick any $\epsilon >0$. Choose $n_0$ large enough such that $CC_d (\beta \cdot {{\omega }}^{k+1}) \lVert {\alpha -\alpha _{n_0}} \rVert _{\Sigma } \leqslant \epsilon$. As the class $\alpha _{n_0}$ is Cartier, the map $\eta \mapsto \alpha _{n_0}\cdot \eta$ is weakly continuous, hence for $i$ large enough we have

\[ |(\alpha\cdot\gamma_i\cdot\beta) -(\alpha\cdot\gamma\cdot\beta) | \leqslant 2\epsilon+ |(\alpha_{n_0}\cdot\gamma_i\cdot\beta) -(\alpha_{n_0}\cdot\gamma\cdot\beta) | \leqslant 3\epsilon. \]

This proves $\gamma \in \operatorname {BPF}^{k}({ \mathcal {X} })\mapsto \alpha \cdot \gamma$ is weakly continuous.

The fact that $\gamma \mapsto \alpha \cdot \beta \cdot \gamma$ is continuous follows by the same token and [Reference Dang and FavreDF21, Theorem 3.16].

Proof. We may suppose that $\alpha \leqslant {{\omega }}^{d-1}$. We claim that for each integer $n$, there exists a class $\alpha _n\in \operatorname {c-BPF}^{d-1}({ \mathcal {X} })$ such that

(20)\begin{equation} \alpha+ \frac 1n {{\omega}}^{d-1} \leqslant \alpha_n \leqslant \bigg(1+\frac{1}{2^{n}}\bigg) \bigg(\alpha+ \frac{1}{n}{{\omega}}^{d-1}\bigg). \end{equation}

As for all $n\geqslant 2$

\begin{align*} \alpha_{n+1} &\leqslant \bigg(1+\frac 1{2^{n+1}}\bigg) \bigg(\alpha + \frac 1{n+1} {{\omega}}^{d-1}\bigg)\\ &\leqslant \alpha_n - \frac 1{n(n+1)} {{\omega}}^{d-1} +\frac 1{2^{n+1}} \bigg(\alpha + \frac 1{n+1} {{\omega}}^{d-1}\bigg)\leqslant \alpha_n \end{align*}

the sequence $\alpha _n$ is decreasing to $\alpha$ as required. When $\alpha$ is big, then for any $\epsilon >0$, we have ${{\omega }}^{d-1}/n \leqslant \epsilon \alpha$ for all $n$ large enough, hence it remains to prove (20).

By Theorem C, one can find a nef and big class $\beta ' \in \operatorname {Nef^{1}}({ \mathcal {X} })$ such that $(\beta ')^{d-1} = \alpha + ({1}/{n}){{\omega }}^{d-1}$. By Corollary 3.7, one can find a nef Cartier class $\beta (\epsilon )$ such that $\beta ' \leqslant \beta (\epsilon ) \leqslant (1+\epsilon ) \beta '$ and we conclude by taking $\alpha _n := \beta (\epsilon )^{d-1}$ with $(1+\epsilon )^{d-1} \leqslant 1+ {1}/{n}$.