Pluricomplex Green's functions and Fano manifolds

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

Existence of coupled Kähler–Einstein metrics using the continuity method

In this paper, we prove the existence of coupled Kähler–Einstein metrics on complex manifolds whose canonical bundle is ample. These metrics were introduced and their existence in the said case was proven by Hultgren and Nyström using calculus of variations. We prove the result using the method of continuity. In the process of proving estimates, akin to the usual Kähler–Einstein metrics, we reduce existence in the Fano case to a [Formula: see text] estimate.