We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kähler manifold X. Given a big (1, 1)-cohomology class α on X (i.e. a class that can be represented by a strictly positive current) and a positive measure µ on X of total mass equal to the volume of α and putting no mass on pluripolar sets, we show that µ can be written in a unique way as the top degree self-intersection in the nonpluripolar sense of a closed positive current in α. We then extend Kolodziedj's approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if µ has L 1+ε -density with respect to Lebesgue measure. If µ is smooth and positive everywhere, we prove that T is smooth on the ample locus of α provided α is nef. Using a fixed point theorem we finally explain how to construct singular Kähler-Einstein volume forms with minimal singularities on varieties of general type.

We study degenerate complex Monge-Ampère equations of the form (ω + dd c ϕ) n = e tϕ µ where ω is a big semi-positive form on a compact Kähler manifold X of dimension n, t ∈ R + , and µ = f ω n is a positive measure with density f ∈ L p (X, ω n ), p > 1. We prove the existence and unicity of bounded ω-plurisubharmonic solutions. We also prove that the solution is continuous under a further technical condition.In case X is projective and ω = ψ * ω ′ , where ψ : X → V is a proper birational morphism to a normal projective variety, [ω ′ ] ∈ N S R (V ) is an ample class and µ has only algebraic singularities, we prove that the solution is smooth in the regular locus of the equation.We use these results to construct singular Kähler-Einstein metrics of non-positive curvature on projective klt pairs, in particular on canonical models of algebraic varieties of general type.

International audienc