Pome

Example 11.41.8121EF09A28A5

Example 11.41.8.121EF09A28A5 Let $(X_0,h_0)$ be a connected compact Hermitian $m$-fold, let $k\ge 2$, and let $\kappa $ be the mass-one Fubini–Study form on $\mathbb P^ k$. On $X_0\times \mathbb P^ k$, choose a smooth positive volume form $\mathrm dV$. With pullbacks implicit, put

\[ \begin{aligned} N& =m+k,& C& =\binom Nm,& V_0& =\int _{X_0}h_0^ m,\\ H& =h_0+\kappa ,& \mu & =(h_0^ m/V_0)\wedge \kappa ^ k,& R& =\mu /\mathrm dV. \end{aligned} \]

For $D=\{ Z_0=0\} $ pulled back to the product, take $E=\lambda D$, $0<\lambda <1$, with

\[ \ell _ E=\lambda \log \frac{|Z_0|^2}{\sum _{i=0}^ k|Z_ i|^2}, \qquad H_ E=h_0+(1-\lambda )\kappa . \]

The global threshold of $R$ is $CV_0$, while its ordinary, relative envelope and one-sided thresholds prescribed by $E$ are all $CV_0(1-\lambda )^ k$, attained by $\ell _ E$. If $v_-(h_0)=0$, both $H$ and $H_ E$ are nonclosed and have zero lower mass.

Proof. The Poincaré–Lelong formula gives $[D]=\kappa +\mathrm{dd}^{\mathrm{c}}(\ell _ E/\lambda )$. Dimensions and the mass-one projective factor give

\[ H^ N=CV_0R\, \mathrm dV,\qquad H_ E^ N=CV_0(1-\lambda )^ kR\, \mathrm dV. \]

Thus the zero residual solves each smooth positive equation and fixes its unique scalar. The theorem identifies all three restricted coefficients, and the correspondence gives $\mathrm{MA}^{\mathrm{ext}}_ H(\ell _ E)=H_ E^ N$. Under the factor hypothesis, pullbacks of bounded $h_0$-psh potentials with mass tending to zero have masses multiplied respectively by $C$ and $C(1-\lambda )^ k$. Bounded submersion pullback and the product expansion therefore give zero lower mass for both backgrounds. A closed positive $h_0$ would instead have every bounded mass equal to $V_0$ by Stokes, so both backgrounds are nonclosed. $\square$

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