Pome810CC86832BFCorollary 15.27.5.810CC86832BF The following conditions are equivalent:
$v_-(h)>0$.
For every nonzero finite non-pluripolar measure $\mu $ there are $c>0$ and $u\in \mathcal E_{\mathrm{env}}(X,h,0)$ such that $\mathrm{MA}^{\mathrm{ext}}_ h(u)=c\mu $.
For every probability density $f\in L^1(\mathrm{d}V)$ bounded below by a positive constant, there are $u\in \mathrm{PSH}(X,h)$ and $a>0$ such that $\mathrm{MA}^{\mathrm{ext}}_ h(u)\geq af\, \mathrm{d}V$.
Proof. The Hermitian form $h$ is semipositive and big, and the reference $0$ satisfies (16.28.67), since $P_ h[0]=0$ and $P_{\varepsilon h}(0)=0$ for every $0<\varepsilon <1$. Hence, if $v_-(h)>0$, theorem 15.37.4, applied with $\theta =h$ and $\varphi =0$, gives (2). Condition (2) implies (3), because a finite measure that is absolutely continuous with respect to $\mathrm{d}V$ is non-pluripolar. Finally, if $v_-(h)=0$, the density of theorem 15.27.3 violates (3); thus (3) implies (1). $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.