Since the proof of the Calabi conjecture given by Yau, complex Monge-Ampère equations on compact Kähler manifolds have been intensively studied.
In this talk we consider complex Monge-Ampère equations with prescribed singularities. More precisely, we fix a potential and we show existence and uniqueness of solutions of complex Monge-Ampère equations which have the same singularity type of the model potential we chose. This result can be interpreted as a generalisation of Yau's theorem (in this case the model potential is smooth).
As a corollary we obtain the existence of singular Kähler-Einstein metrics with prescribed singularities on general type and Calabi-Yau manifolds.
This is a joint work with Tamas Darvas and Chinh Lu.[-]

Given a domain D in $C^n$ and a compact subset K of D, the set $A^D_K$ of all restrictions of functions holomorphic on D the modulus of which is bounded by 1 is a compact subset of the Banach space $C(K)$ of continuous functions on K. The sequence $d_m(A^D_K)$ of Kolmogorov m-widths of $A^D_K$ provides a measure of the degree of compactness of the set $A^D_K$ in $C(K)$ and the study of its asymptotics has a long history, essentially going back to Kolmogorov's work on epsilon-entropy of compact sets in the 1950s. In the 1980s Zakharyuta gave, for suitable D and K, the exact asymptotics of these diameters (1), and showed that is implied by a conjecture, now known as Zakharyuta's Conjecture, concerning the approximability of the regularised relative extremal function of K and D by certain pluricomplex Green functions. Zakharyuta's Conjecture was proved by Nivoche in 2004 thus settling (1) at the same time. We shall give a new proof of the asymptotics (1) for D strictly hyperconvex and K nonpluripolar which does not rely on Zakharyuta's Conjecture. Instead we proceed more directly by a two-pronged approach establishing sharp upper and lower bounds for the Kolmogorov widths. The lower bounds follow from concentration results of independent interest for the eigenvalues of a certain family of Toeplitz operators, while the upper bounds follow from an application of the Bergman–Weil formula together with an exhaustion procedure by special holomorphic polyhedral.[-]