Explicit lower bounds for Monge–Ampère volume infima on Hermitian manifolds

Derive an explicit uniform lower bound, in terms of the data (X, ω, M), for the quantity inf_{v∈PSH(X,ω), 0≤v≤M} ∫_X (ω+dd^c v)^n, where PSH(X,ω) denotes the ω-plurisubharmonic functions on X and M>0 is a fixed bound.

Background

The authors need a quantitative lower bound on Monge–Ampère volumes of bounded ω-plurisubharmonic functions to compare extremal function suprema with global Bedford–Taylor capacity on Hermitian manifolds. While positivity of the infimum has been established (for each fixed M) by Guedj and Lu, no explicit bound depending on X, ω, and M is known.

Such an explicit estimate would strengthen capacity comparisons and could remove technical barriers that necessitate detours via relative capacities in Cn\mathbb{C}^n in the present work.

References

Although Guedj and Lu Proposition~3.4 proved that $$0<\inf{\int_X(\omega+ddcv)n:v\in PSH(X,\omega),\;0\leq v\leq M}$$ for each M\in(0,\infty), but an explicit uniform lower bound for the infimum in terms of X,\omega and M is not known.

— Hölder regularity of Siciak-Zaharjuta extremal functions on compact Hermitian manifolds  (2604.01887 - Ahn, 2 Apr 2026) in Remark (Introduction; “Difficulty compared to Kähler case”)