Metrics with minimal singularities and the Abundance conjecture

The Abundance conjecture predicts that on a minimal projective klt pair (X,Δ), the adjoint divisor KX+Δ is semiample. When χ(X, OX)≠0, we give a necessary and sufficient condition for the conjecture to hold in terms of the asymptotic behaviour of multiplier ideals of currents with minimal singularities of small twists of KX+Δ. Furthermore, we prove fundamental structural properties as well as regularity and weak convergence behaviour of an important class of currents with minimal singularities: the supercanonical currents. The results of the paper indicate strongly that supercanonical currents are central to the completion of the proof of the Abundance conjecture for minimal klt pairs (X,Δ) with χ(X, OX)≠0.