This paper by Gang Tian addresses the existence of Kähler-Einstein metrics on compact Kähler manifolds with positive first Chern class and no nontrivial holomorphic vector fields. The author disproves a long-standing conjecture that such manifolds necessarily admit Kähler-Einstein metrics. Instead, Tian establishes an analytic criterion for the existence of these metrics, which is satisfied on stable Kähler manifolds under certain conditions. The paper also reviews the historical context, including the work of Calabi, Yau, Aubin, and others, and discusses the challenges and new phenomena that arise when the first Chern class is positive. Tian's results provide new insights into the geometric aspects of the Calabi problem and highlight the importance of the Futaki invariant in understanding the existence of Kähler-Einstein metrics.This paper by Gang Tian addresses the existence of Kähler-Einstein metrics on compact Kähler manifolds with positive first Chern class and no nontrivial holomorphic vector fields. The author disproves a long-standing conjecture that such manifolds necessarily admit Kähler-Einstein metrics. Instead, Tian establishes an analytic criterion for the existence of these metrics, which is satisfied on stable Kähler manifolds under certain conditions. The paper also reviews the historical context, including the work of Calabi, Yau, Aubin, and others, and discusses the challenges and new phenomena that arise when the first Chern class is positive. Tian's results provide new insights into the geometric aspects of the Calabi problem and highlight the importance of the Futaki invariant in understanding the existence of Kähler-Einstein metrics.