2021/07/05 by Chi Li, Li, Chi
Mathematics · #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2107.02088
openalex publication_date 2021/07/05 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
This is largely an exposition article that expands the author's talk at the Xiamen International Conference on Geometric Analysis in June 2021. We first survey the author's joint work with Jiyuan Han on the Yau-Tian-Donaldson (YTD) conjecture for g-weighted Kähler-Ricci solitons (g-solitons). We then review recent works of Apostolov-Canderbank-Jubert-Lahdili which establish a connection between a particular g-soliton equation with Ricci-flat Kähler cone metrics (or equivalently Sasaki-Einstein metrics). The main interest in this connection is the transformation of a possibly irregular Sasaki-Einstein metric to a particular g-soliton equation on any quasi-regular quotient. We will revisit this transformation by understanding how the corresponding transversal complex Monge-Ampère equations are transformed under the deformation of Reeb vector fields. Finally we explain how this PDE/pluripotential point of view allows one to prove the YTD conjecture for general Fano cones by combining the YTD conjecture for g-solitons on log Fano pairs and the above transformation.