Geometric Flow, Multiplier Ideal Sheaves and Optimal Destabilizer for a Fano Manifold

Abstract

In (Donaldson in J Differ Geom 70(3):453–472, 2005), it was asked whether the lower bound of the Calabi functional is achieved by a sequence of the normalized Donaldson–Futaki invariants. We answer to the question for the Ricci curvature formalism, in place of the scalar curvature. Our principle is that the stability indicator is optimized by the multiplier ideal sheaves of certain weak geodesic ray asymptotic to the geometric flow. We actually obtain the results for the two cases: the inverse Monge–Ampère flow and the Kähler–Ricci flow.