Destabilising subvarieties for partial differential equations in complex geometry

In this thesis, we consider the set of destabilising subvarieties associated to various geometric partial differential equations (PDEs) of Monge-Ampère type arising in complex geometry, including the $J$-equation, the deformed Hermitian Yang-Mills equation and certain generalised Monge-Ampère equations. Each of these PDEs has an associated Nakai-Moishezon type criterion characterising their solvability in terms of a certain stability condition involving subvarieties. We show that the set of subvarieties that violate this criterion is finite under certain mild hypotheses of positivity, which are always satisfied on compact Kähler surfaces. We use the results to show that the locus of stable (or solvable) PDEs is open in the locus of all the PDEs, and admits a locally finite wall-chamber structure whose walls are cut out by equations involving certain rigid subvarieties.