Complex Analytic and Differential Geometry 2025, 06 - Monge-Ampere Operators

A family of locally bounded Psh functions (u) and (v) a family of decreasing sequences of Psh functions, converging index-wise to (u), then ⩓u_i^kdd^cu_i+1^k ∧ T → ⩓u_idd^cu_i+1 ∧ T weakly, and ⩓^kdd^cu_i+1^k ∧ T → ⩓dd^cu_i+1 ∧ T weakly. The latter is symmetric wrt. (u). A decreasing sequence of upper semi-continuous functions converging to some f on a separable locally compact space X and a sequence (μ) of positive measures converging on X, then the weak limit ν of f_iμ_i has ν ≤ fμ. Compact subsets K, L of X so that L is in the open hull of K, with Psh functions V, (u) on X with locally bounded (u) have ||V ⩓dd^cu_i||_L ≤ C_{K, L}||V||_Lⁱ(K)||u_i||_L^∞(K)

T has non zero bidimension, and X is covered by a family of Stein open sets Ω⋐X with a boundary disjoint from L(u)∩Supp[T]. The current uT has locally finite mass in X. Psh-functions (u) on an X covered by Stein open sets Ω with ∂Ω∩L(u_j)∩Supp[T] = 0, then ⩓ dd^cu_i ∧ T = dd^c(u₁ ⩓_{i = 2} dd^c u_i ∧ T), and if (u) are decreasing sequences of Psh functions converging pointwise. A Stein open subset Ω, if V is a Psh function on X with (u), are Psh so that the boundary of Ω is disjoint from L(u), then V ⩓ dd^cu_i has locally finite mass in Ω.

Psh functions (u) on X haave wel defined currents and locally finite mass in X as soon as q ≤ p and H_{2p - 2m +1}(⋒L(u) ∩ Supp[T]) = 0. A closed complex set F with H_{2s + 1}(F) = 0 for 0 ≤ s < n, then for almost all unitary coordinates and almost all radii of balls, {0}×∂B(0, r) does not intersect F. A closed positive current of bidimension (p, p) with Psh function u on X with L(u) ∩ Supp[T] is within an analytic set of dim ≤ p - 1, then uT and dd^cu∧T are well defined have locally finite mass in X.