SinGular Monge-Ampère equations – SiGMA

Motivated by M-theory, String theory in theoretical physics and the Minimal Model Problem in algebraic geometry we study singular Kähler spaces and in particular we focus on investigating their special structures (of a differential geometry nature) and their interaction with various parts of analysis.

More precisely, we search for special (singular) Kähler metrics having nice curvature properties. Examples of these are Kähler-Einstein (KE) or constant scalar curvature (cscK) metrics. The problem of the existence of these metrics can be reformulated in terms of a Monge-Ampère equation (which is a non-linear PDE). The KE case has been settled by Aubin, Yau (solving the Calabi conjecture), and Chen-Donaldson-Sun (solving the Yau-Tian-Donaldson conjecture); the cscK case has been very recently worked out by Chen-Cheng (solving a conjecture due to Tian). But these results only hold on a smooth Kähler manifold, and one still needs to deal with singular varieties.

This is where and why Pluripotential Theory steps into the picture: it has indeed been proved by Boucksom-Eyssidieux-Guedj-Zeriahi and by myself together with Darvas and Lu that pluripotential methods are very flexible and can be adapted to work with (singular) Monge-Ampère equations. Finding a solution to this type of equations which is smooth outside of the singular locus is in turn equivalent to the existence of singular KE or cscK metrics.

At this point a crucial ingredient is missing: the regularity of these (weak) solutions. The core of this project aims to solve that, using new techniques and ideas, which, in turn, might serve in attacking problems in complex analysis and algebraic geometry as well.