A K-energy functional for complexified K\"ahler classes

TL;DR

This paper extends the K-energy functional to complexified K"ahler classes, enabling a variational approach to scalar curvature equations with B-field and establishing convexity and uniqueness results.

Contribution

It introduces a new K-energy functional for complexified K"ahler classes and demonstrates its convexity along geodesics, leading to uniqueness results for scalar curvature solutions.

Findings

Extended K-energy functional is convex along geodesics.

Solutions to scalar curvature equations with B-field are unique up to automorphisms.

Provides a variational framework for complexified K"ahler classes.

Abstract

The K-energy functional is extended to complexified K\"ahler classes, providing a variational approach to study the scalar curvature equation with B-field introduced by Schlitzer and Stoppa. The extended K-energy is convex along geodesics in the space of almost calibrated representatives of the complexified K\"ahler class. This fact is used to show that, in some situations, solutions of the scalar curvature equation with B-field are unique in their class, up to pullbacks by reduced automorphisms of the manifold.