Describe limit metrics induced by symmetric-power Hermitian metrics

Describe the limit metric on \(\mathcal{O}_{\mathbb{P}^{n}}(1)\), or characterize its singularities, when the metric is obtained as a limit of the normalized metrics \(h_m^{1/m}\) induced by arbitrary smooth Hermitian metrics \(H_m\) on \(\operatorname{Sym}^{m}(\mathbb{C}^{n+1})\).

Background

The paper studies singular metrics on projective bundles arising from Hermitian metrics on vector spaces and uses a precise description of limits induced by metrics on $\mathbb{C}^{n+1}$ in Proposition \ref{addlemma}. That proposition handles a special family: after unitary changes of coordinates and rescaling, the limiting metric on $\mathcal{O}_{\mathbb{P}^{n}}(1)$ has a weight of the form $\log\sum_i\lambda_i|\zeta_i|^2$.

The authors note that pseudo-effectivity involves metrics on all symmetric powers, so a broader problem is to understand limits of the normalized induced metrics $h_m^{1/m}$ when the underlying Hermitian metrics on $\operatorname{Sym}^{m}(\mathbb{C}^{n+1})$ vary arbitrarily. They explicitly state this as a question and do not resolve it; the problem is intended to clarify the possible limiting singularities relevant to pseudo-effective sheaves.