Proposition 10.16.5.BA31DCC7CCB7 Let $Z$ be a compact Kähler manifold of dimension $k>0$ and $b\in H^{1,1}_{\mathrm{BC}}(Z,\mathbb R)$. Then
\[ \mathcal R_ Z(b)>0\quad \Longleftrightarrow \quad b\text{ is big}. \]
If $b$ is nef, then
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10.16.6
\begin{equation} \mathcal R_ Z(b)=\int _ Zb^ k. \end{equation}
Suppose that $b$ is nef with $I=\int _ Zb^ k>0$. Fix a Kähler form $\kappa $, let $p_1,p_2:Z\times Z\to Z$ be the projections and $\Delta \subset Z\times Z$ the diagonal, and put
\[ M=k\int _ Zb^{k-1}\wedge \{ \kappa \} . \]
Then there are $d>0$ and a closed positive $(k,k)$-current $\Theta $ in $(p_1^*b+p_2^*b)^ k$ with $\Theta \geq d[\Delta ]$ such that
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10.16.7
\begin{equation} K_ b=M^{-1}(p_1)_*(\Theta \wedge p_2^*\kappa )\in b, \qquad K_ b\geq (d/M)\kappa . \end{equation}
Proof.
If $\mathcal R_ Z(b)>0$, some positive representative has positive non-pluripolar top mass, so $b$ is big by the first fact above. Conversely, if $S\in b$ satisfies $S\geq \delta \kappa $ with $\delta >0$, then $S-\delta \kappa $ is positive and closed, and expanding the non-pluripolar product gives $\langle S^ k\rangle \geq \delta ^ k\kappa ^ k$. Hence $\mathcal R_ Z(b)>0$.
Suppose now that $b$ is nef. For $t>0$, the class $b+t\{ \kappa \} $ is Kähler. Adding $t\kappa $ to an arbitrary positive representative of $b$ and using the Kähler mass bound of Step 1 in the proof of proposition 10.16.3 yields
\[ \mathcal R_ Z(b)\leq \int _ Z(b+t\{ \kappa \} )^ k. \]
Letting $t$ decrease to zero gives the upper bound in (10.16.6). In particular, equality holds if $I=0$.
Let $I>0$, and choose $L>0$ such that $L\{ \kappa \} -b$ is Kähler. Since intersection numbers of nef classes are nonnegative,
\[ 0\leq \int _ Zb^{k-1}\wedge (L\{ \kappa \} -b) =LM/k-I, \]
so $M>0$. The class $B=p_1^*b+p_2^*b$ is nef and satisfies
\[ \int _{Z\times Z}B^{2k}={2k\choose k}I^2>0. \]
The classical mass concentration theorem along the diagonal gives $\Theta \in B^ k$ with $\Theta \geq d[\Delta ]$ for some $d>0$. When $B^ k\wedge p_2^*\{ \kappa \} $ is pushed forward by $p_1$, only the term containing exactly one factor $p_1^*b$ survives. Thus
\[ (p_1)_*(B^ k\wedge p_2^*\{ \kappa \} )=Mb, \qquad (p_1)_*([\Delta ]\wedge p_2^*\kappa )=\kappa . \]
Since $\Theta -d[\Delta ]$ is positive, these identities prove (10.16.7); the $\partial \bar\partial $-lemma for currents on the compact Kähler manifold $Z$ identifies the cohomology class computed in this way with the Bott–Chern class $b$. Therefore $b$ is big, and by the second fact above a current with minimal singularities in $b$ has non-pluripolar mass $\int _ Zb^ k$. This gives the remaining lower bound.
$\square$
Pome
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