Wall–chamber decompositions for generalised Monge–Ampère equations

Abstract

Generalised Monge–Ampère (gMA) equations form a large class of PDE including Donaldson's J-equation, inverse Hessian equations, some supercritical deformed Hermitian–Yang–Mills (dHYM) equations and some Z-critical equations. Solvability of these equations is characterised by numerical criteria involving intersection numbers over all subvarieties, and in this paper, we aim to characterise algebraically what happens when these non-linear 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-Moishezon type criterion, and such subvarieties are rigid in a suitable sense. This gives the first effective solvability criteria for these families of PDE, thus improving on work of Chen, Datar–Pingali, Song and Fang–Ma, and provides the first existence results in higher dimensions of a natural PDE analogue of Bridgeland's locally finite wall–chamber decomposition.