Wall-chamber decompositions for generalized Monge-Amp\`ere equations

Generalized Monge-Amp\`ere equations form a large class of PDE including Donaldson's J-equation, inverse Hessian equations, some supercritical deformed Hermitian-Yang Mills equations, and some Z-critical equations. Solvability of these equations is characterized by numerical criteria involving intersection numbers over all subvarieties, and in this paper, we aim to characterize algebraically what happens when these nonlinear Nakai-Moishezon type criteria fail. As a main result, we show that under mild positivity assumptions, there is a finite number of subvarieties violating the Nakai type criterion, and such subvarieties are moreover rigid in a suitable sense. This gives first effective solvability criteria for these familes of PDE, thus improving on work of Gao Chen, Datar-Pingali, Song and Fang-Ma. As an application, we obtain first examples in higher dimension of polarized compact K\"ahler manifolds exhibiting a natural PDE analog of Bridgeland's locally finite wall-chamber decomposition.