Pluricomplex Green's functions and Fano manifolds

TL;DR

This paper investigates the behavior of Kähler potentials on Fano manifolds lacking Kähler-Einstein metrics, showing they converge to singular solutions of a complex Monge-Ampère equation, confirming a conjecture by Tian-Yau.

Contribution

It demonstrates the convergence of Kähler potentials to singular solutions on Fano manifolds without Kähler-Einstein metrics, extending understanding of complex Monge-Ampère equations.

Findings

Kähler potentials subconverge to functions with analytic singularities

The singularities occur along a subvariety of the manifold

The limit functions solve the homogeneous complex Monge-Ampère equation

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.