Lower bounds on the modified K-energy and complex deformations

TL;DR

This paper proves that on certain deformations of polarized Kähler manifolds with extremal metrics, the modified K-energy remains bounded below, extending previous results and providing explicit examples where extremal metrics do not exist.

Contribution

It generalizes lower bounds on the modified K-energy to extremal metrics and extends convexity inequalities to this setting, with explicit examples.

Findings

Modified K-energy is bounded below on nearby deformations preserving symmetry.

Extension of convexity inequality to extremal metric setup.

Explicit examples of blow-ups with bounded below modified K-energy but no extremal metric.

Abstract

Let (X,L) be a polarized K\"ahler manifold that admits an extremal K\"ahler metric in c1(L). We show that on a nearby polarized deformation that preserves the symmetry induced by the extremal vector field of (X,L), the modified K-energy is bounded from below. This generalizes a result of Chen, Sz\'ekelyhidi and Tosatti to extremal metrics. Our proof also extends a convexity inequality on the space of K\"ahler potentials due to X.X. Chen to the extremal metric setup. As an application, we compute explicit polarized 4-points blow-ups of CP1\times CP1 that carry no extremal metric but with modified K-energy bounded from below.