Least negative intersections of positive closed currents on compact Kähler manifolds

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Let X be a compact Kähler manifold of dimension k. Let R be a positive closed (p,p) current on X, and T₁,,Tₖ₋ₚ be positive closed (1,1) currents on X. We define a so-called least negative intersection of the currents T₁,T₂,,Tₖ₋ₚ and R, as a sublinear bounded operator eqnarray* (T₁,,Tₖ₋ₚ,R): C⁰(X)→ R. eqnarray* This operator is symmetric in T₁,,Tₖ₋ₚ. It is independent of the choice of a quasi-potential uᵢ of Tᵢ, of the choice of a smooth closed (1,1) form θᵢ in the cohomology class of Tᵢ, and of the choice of a Kähler form on X. Its total mass < (T₁,,Tₖ₋ₚ,R),1> is the intersection in cohomology {T₁}{T₂} {Tₖ₋ₚ}.{R}. It has a semi-continuous property concerning approximating Tᵢ by appropriate smooth closed (1,1) forms, plus some other good properties. If p=0 and T₁= =Tₖ=T, we have a least negative Monge-Ampere operator MA(T)= (T,,T). If the set where T has positive Lelong numbers does not contain any curve, then MA(T) is positive. Several examples are given.