On the Lower Boundedness of Modified K-energy

Abstract

In this paper we prove that the modified K-energy on a Fano manifold X is bounded from below if X admits a special degeneration whose central fiber is a Kähler—Ricci soliton and the soliton vector field can be lifted to X. We will use this result to derive some equivalent criteria of the modified K-semistability. We will also use this result to study the limit of Kähler—Ricci flow.