We prove a uniform diameter estimate and a uniform local non-collapsing of volumes for a large family of Kaehler metrics generalizing those obtained recently by Guo-Phong-Song-Sturm to the extent that no assumption on the uniform (pointwise) non-vanishing of volume forms is required. We treat also similar questions in the singular setting where the cohomology classes are allowed to degenerate to non-big classes. This again extends results by the latter authors where only the non-collapsing case was considered.
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Duc-Viet Vu (2024) studied this question.
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