Problem 24.4.337E8975FF8CBB

Problem 24.4.33.7E8975FF8CBB Let $X$ be an Oeljeklaus–Toma threefold of type $(2,1)$, with the Hermitian form $\omega $ of proposition 10.29.10. Does $X$ have the bounded mass property, that is, is $v_+(\omega )<+\infty $? Is the lower mass $v_-(\omega )$ positive? Equivalently, writing $F(u)$ for the fluctuation term in (10.29.12), is $F$ bounded above on smooth $\omega $-plurisubharmonic functions $u$, and is $\inf _ u\bigl (\mathrm M_\omega (Au)+F(u)\bigr )>0$? The same questions can be asked for Oeljeklaus–Toma manifolds of any type $(s,t)$ with $s\geq 2$.