Appears in collection : Complex Analytic and Differential Geometry

Given a Kahler manifold (X, ω), finding smooth solutions to the equation (ø +i∂ ̄∂u)n=føn goes back to Yau’s solution of the Calabi conjecture in the seventies. In joint work with E. Di Nezza and C.H. Lu, we proposed to solve this same equation with the added constraint that u∈PSH(X, ω) has prescribed singularity type. As it turns out, this problem is well posed only for a certain class of (model) singularity types that we characterize, and we also solve the correspondingequation. Our results extend to the case of big cohomology classes as well.