PomeProblem 24.10.2.4813DF6B9029 Let $\pi :X\to B$ be a surjective holomorphic submersion of connected compact complex manifolds with connected fibers, where $B$ is Kähler of dimension $q\geq 1$. Fix a Hermitian form $\omega $ on $X$ and $\varphi \in \mathrm{PSH}(X,\omega )$, and a holomorphic line bundle $A$ on $B$ with a smooth Hermitian metric $h$. Set $\theta =c_1(A,h)$. For positive integers $m$, identify
\[ V_ m=H^0\bigl (X,\pi ^*A^ m\otimes \mathcal I(m\varphi )\bigr ) \subseteq H^0(B,A^ m) \]
by holomorphic descent. Find geometric or analytic conditions that produce $\psi \in \mathrm{PSH}(B,\theta )$ of positive non-pluripolar mass such that, for all sufficiently large $m$,
\[ \begin{gathered} W_ m=H^0\bigl (B,A^ m\otimes \mathcal I(m\psi )\bigr )\subseteq V_ m,\\ \dim V_ m-\dim W_ m=o(m^ q). \end{gathered} \]
Determine which features of the singularities of $\varphi $ along the fibers suffice for this approximation.
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