25.6 Bounded products and extended measures9E2E80C382FD

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The Bedford–Taylor product of bounded potentials is local even when the background is not closed: expanding against smooth local corrections, as Kołodziej and Nguyen [Section 1, KN15] and Boucksom, Guedj and Lu [Section 2.1, BGL25] do, gives proposition 5.28.2. Restricting the products of the truncations to the sets where the potential exceeds the truncation level then defines the extended measure of theorem 13.14.3, by the mechanism of [Proposition 2.1, BGL25]. Finiteness becomes decisive when these restrictions are removed in the category of currents, and [Theorem 2.2, BGL25] uses a uniform bound on the variation for this purpose. An infinite extended positive measure is still given by the local construction, but it is not a global current of order zero.

The independence of the bounded mass property from the metric, and the reduction to smooth potentials, in proposition 13.14.15 recover the comparison made after [Definition 1.16, BGL25]; decreasing regularization with exact curvature bounds gives the passage from smooth to bounded potentials. The Gauduchon trace bound and the cancellation of the exact top-degree wedge product used in proposition 13.15.2 also appear in Guedj and Lu's [Proposition 3.3(i), (vi), GL22], and the estimate in terms of the range of the potential keeps their polynomial dependence on the oscillation.

The local mixed Monge–Ampère inequality, theorem 5.27.5, is Dinew's [Theorem 1.2, Dinew09], for locally bounded plurisubharmonic functions whose Monge–Ampère measures dominate nonnegative measurable densities with respect to Lebesgue measure. Its extension to smooth backgrounds is proved in section 5.28 and used in subsection 15.18.2. Freezing the coefficients of the two smooth backgrounds gives lemma 5.28.7 and the trace inequalities (15.14.3) and lemma 13.14.12, the latter keeping the mass of the linear trace on the pole set. The same argument also recovers the inequality for smooth volume forms used by Nguyen [Lemma 1.9, Nguyen16].