Mahler measures, Stable Pairs, and the Global coercive estimate for the Mabuchi Functional

2021/05/04 by Paul, Sean Timothy
#Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2105.01240

Abstract

We show that the Mabuchi energy of any polarized manifold (X,L) is (bounded below) proper on the full space of Kahler metrics in the first Chern class of L if and only if (X,L) is asymptotically (semi)stable. In particular it now follows from work of Xiuxiong Chen and Jinguri Cheng that there exists a cscK metric in the first Chern class of L if and only if (X,L) is asymptotically stable, provided the reduced automorphism group of (X,L) is finite.

Related

• Stable Pairs and Coercive Estimates for The Mabuchi Functional
• Uniform K-stability and asymptotics of energy functionals in Kähler geometry
• On the strict convexity of the K-energy
• Faltings Heights, Igusa Local Zeta Functions, and the Stability Conjectures in Kahler Geometry I
• Stability, energy functionals, and Kähler-Einstein metrics