Proposition 11.1.20682711871907

Proof. Let $u\in \mathcal H_\omega $. Since $C\eta +\mathrm{dd}^{\mathrm{c}}u\geq \omega +\mathrm{dd}^{\mathrm{c}}u>0$, the function $u/C$ belongs to $\mathcal H_\eta $. The top exterior power is monotone on positive $(1,1)$-forms: for positive Hermitian matrices $A\leq B$, all eigenvalues of $A^{-1/2}BA^{-1/2}$ are at least one, so $\det A\leq \det B$. Applying this pointwise gives

Taking the supremum over $u$ proves the inequality. Since any two Hermitian forms on the compact manifold $X$ are comparable in both directions, the last assertion follows. $\square$