Multipoint Okounkov bodies, strong topology of ω-plurisubharmonic functions and Kähler-Einstein metrics with prescribed singularities

Doctoral thesis, 2020

The most classical topic in Kähler Geometry is the study of Kähler-Einstein metrics as solution of complex Monge-Ampère equations. This thesis principally regards the investigation of a strong topology for ω-plurisubharmonic functions on a fixed compact Kähler manifold (X,ω), its connection with complex Monge-Ampère equations with prescribed singularities and the consequent study of singular Kähler-Einstein metrics. However the first part of the thesis, Paper I, provides a generalization of Okounkov bodies starting from a big line bundle over a projective manifold and a bunch of distints points. These bodies encode renowned global and local invariants as the volume and the multipoint Seshadri constant.

In Paper II the set of all ω-psh functions slightly more singular than a fixed singularity type are endowed with a complete metric topology whose distance represents the analog of the L1 Finsler distance on the space of Kähler potentials. These spaces can be also glued together to form a bigger complete metric space when the singularity types are totally ordered. Then Paper III shows that the corresponding metric topology is actually a strong topology given as coarsest refinement of the usual topology for ω-psh functions such that the relative Monge-Ampère energy becomes continuous. Moreover the main result of Paper III proves that the extended Monge-Ampère operator produces homeomorphisms between these complete metric spaces and natural sets of singular volume forms endowed their strong topologies. Such homeomorphisms extend Yau's famous solution to the Calabi's conjecture and the strong topology becomes a significant tool to study the stability of solutions of complex Monge-Ampère equations with prescribed singularities. Indeed Paper IV introduces a new continuity method with movable singularities for classical families of complex Monge-Ampère equations typically attached to the search of log Kähler-Einstein metrics. The idea is to perturb the prescribed singularities together with the Lebesgue densities and asking for the strong continuity of the solutions. The results heavily depend on the sign of the so-called cosmological constant and the most difficult and interesting case is related to the search of Kähler-Einstein metrics on a Fano manifold. Thus Paper V contains a first analytic characterization of the existence of Kähler-Einstein metrics with prescribed singularities on a Fano manifold in terms of the relative Ding and Mabuchi functionals. Then extending the Tian's α-invariant into a function on the set of all singularity types, a first study of the relationships between the existence of singular Kähler-Einstein metrics and genuine Kähler-Einstein metrics is provided, giving a further motivation to study these singular special metrics since the existence of a genuine Kähler-Einstein metric is equivalent to an algebrico-geometric stability notion called K-stability which in the last decade turned out to be very important in Algebraic Geometry.