Uniform non-collapsing for Hermitian metrics

Determine whether every Hermitian metric ω on a compact complex manifold X is uniformly non-collapsing; concretely, ascertain the validity of a uniform positive lower bound for Monge–Ampère volumes of bounded ω-plurisubharmonic functions, i.e., prove or refute that there exists a constant c(X,ω,M)>0 depending only on X, ω, and M such that inf_{v∈PSH(X,ω), 0≤v≤M} ∫_X (ω+dd^c v)^n ≥ c(X,ω,M).

Background

This remark appears in the discussion of the main difficulty in extending capacity comparisons from the Kähler to the Hermitian setting. The inequality comparing the supremum of the extremal function with the global Bedford–Taylor capacity on X hinges on lower bounds for Monge–Ampère volumes of bounded ω-plurisubharmonic functions.

In the Kähler case, standard comparison principles facilitate such estimates. For general Hermitian metrics, however, the existence of uniform non-collapsing lower bounds is not yet settled. Establishing this property would directly address obstacles in deriving capacity-based estimates needed for regularity results on compact Hermitian manifolds.