It is well known that solvability of the complex Monge- Ampere (CMA) equation on compact Kaehler manifolds is related to the positivity of certain intersection numbers. In fact, this follows from combining Yau’s resolution of the Calabi conjecture, with Demailly and Paun’s generalization of the classical Nakai-Mozhesoin criteria. This correspondence was recently extended to a broad class of complex non-linear PDEs including the J-equation and the deformed Hermitian-Yang-Mills (dHYM) equations by the work of Gao Chen and others. A natural question to ask is whether solutions (necessarily singular) exist in any reasonable sense if the Nakai criteria fails. A motivating example is that of CMA equations in big classes. In this talk, I will provide an overview of a program for constructing such singular solutions for the J equation and the dHYM equations, and state some conjectures and problems. Analogous to the characterization of volumes of big classes, a key point is to define a minimal slope using birational models. Finally, I will outline how to resolve some of these questions on Kahler surfaces and some manifolds with large symmetry groups. This is joint work with Ramesh Mete and Jian Song.
