Pluricomplex Green's functions and Fano manifolds

Abstract

We show that if a Fano manifold does not admit Kahler-Einstein metrics then the Kahler potentials along the continuity method subconverge to a function with analytic singularities along a subvariety which solves the homogeneous complex Monge-Ampere equation on its complement, confirming an expectation of Tian-Yau. Comment: EpiGA Volume 3 (2019), Article Nr. 9

Rooftop envelopes and residual plurisubharmonic functions

“…where sup * stands for the upper semicontinuous regularization of the envelope as in (1.1); later on, we will use the similar notation lim * . This was continued in [18]- [21], [32], [33], [46], and other recent papers. We would like especially to refer to [19] where the following was proved: for θ-psh functions u and v such that θ n v := (θ + dd c v) n has positive nonpluripolar mass and u ≤ v + C (in other words, u has a stronger singularity than v), the condition P θ [u](v) = v is equivalent to the equality P θ [u](0) = P θ [v](0) (the envelopes of the singularity types of u and v coincide), as well as to the equality of their total non-pluripolar Monge-Ampère masses: θ n u (X) = θ n v (X).…”

Plurisubharmonic Interpolation and Plurisubharmonic Geodesics

“…We would especially like to refer to [12,22] where the connectivity problem for quasiplurisubharmonic functions on compact Kähler manifolds was for the first time treated in terms of rooftop envelopes; the approach was developed then in [15][16][17][18][19]24,25] and other recent papers. A nice overview of this activity can be found in [26].…”