On the complex Monge–Ampère operator for quasi‐plurisubharmonic functions with analytic singularities

We give a modified, very natural definition for the complex Monge–Ampère operator for an ω‐plurisubharmonic (psh) function φ with analytic singularities on a Kähler manifold (X,ω) of dimension n which has the property ∫X(ω+ddcφ)n=∫Xωn if X is compact. This means that, unlike in the previous definition, no mass is lost here. In fact, the definition works for any smooth (1,1)‐form ω (we need neither closedness nor positivity) and quasi‐psh φ with analytic singularities.