Proposition 12.12.4.FBFFC53EC40F Set $S=\sum _{j=0}^{n-2}\gamma _0^ j\wedge \kappa _0^{n-2-j}$. Then
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12.12.5
\begin{equation} \Delta _\mu (B)=V_0^{-(n-1)/n} \left(I_ B+\int _ Xu_0\, \mathrm{dd}^{\mathrm{c}}B\wedge S\right). \end{equation}
If $-C_ B\kappa _0^2\leq \mathrm{dd}^{\mathrm{c}}B\leq C_ B\kappa _0^2$ in the sense of weak positivity of $(2,2)$-forms, then
\[ \Delta _\mu (B)\leq V_0^{-(n-1)/n}I_ B +(n-1)C_ B\operatorname {osc}_ X(u_0)V_0^{1/n}. \]
Such a finite constant $C_ B$ exists for every fixed smooth $B$.
Proof.
The factors of $S$ are closed, and $\gamma _0^{n-1}-\kappa _0^{n-1}=\mathrm{dd}^{\mathrm{c}}u_0\wedge S$. Integration by parts therefore gives (12.12.5). Since $\int _ X\mathrm{dd}^{\mathrm{c}}B\wedge S=0$, the right side does not change when a constant is added to $u_0$, so we may assume $\sup _ Xu_0=0$. Then $-\operatorname {osc}_ X(u_0)\leq u_0\leq 0$, and the lower bound for $\mathrm{dd}^{\mathrm{c}}B$ together with the positivity of the summands of $S$ bounds the correction term above by
\[ C_ B\operatorname {osc}_ X(u_0)\int _ X\kappa _0^2\wedge S =(n-1)C_ B\operatorname {osc}_ X(u_0)V_0. \]
The equality holds because the cohomologous Kähler forms $\gamma _0,\kappa _0$ have all mixed top intersections equal to $V_0$. Finally, restrict $\mathrm{dd}^{\mathrm{c}}B$ to the complex two-planes of $TX$ and normalize by $\kappa _0$; compactness of the bundle of these planes bounds the restrictions and gives $C_ B$.
$\square$
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