Scalar Curvature Behavior for Finite Time Singularity of Kähler-ricci Flow

Ricci flow, since the debut in the famous original work [4] by R. Hamilton, has been one of the major driving forces for the development of Geometric Analysis in the past decades. Its astonishing power is best demonstrated by the breakthrough in solving Poincaré Conjecture and Geometrization Program. For this amazing story, we refer to [1], [7], [10] and the references therein. Meanwhile, KählerRicci flow, which is Ricci flow with initial metric being Kähler, has shown some of its own characters coming from the natural relation with complex Monge-Ampère equation and many interesting Algebraic Geometric objects. G. Tian’s Program, as described in [14] or [15], has illustrated the direction to further improve people’s understanding in many classic topics of great importance by Kähler-Ricci flow, for example, the Minimal Model Program in Algebraic Geometry. In the current work, we give some very general discussion on Kähler-Ricci flows over closed manifolds. The closed manifold under consideration is denoted by X with dimCX = n. The computation would be done for the following version of Kähler-Ricci flow,