On complex Legendre duality

TL;DR

This paper explores complex Legendre duality on Kähler manifolds, showing that fixed points of this transformation correspond to real analytic Kähler metrics, extending the understanding of geometric dualities.

Contribution

It demonstrates that fixed points of complex Legendre duality are necessarily real analytic Kähler metrics, providing new insights into the structure of these dualities.

Findings

Fixed points correspond to real analytic Kähler metrics.

Complex Legendre duality acts as a local isometry on Kähler potentials.

The result extends the understanding of dualities in Kähler geometry.

Abstract

Complex Legendre duality is a generalization of Legendre transformation from Euclidean spaces to Kahler manifolds, that Berndtsson and collaborators have recently constructed. It is a local isometry of the space of Kahler potentials. We show that the fixed point of such a transformation must correspond to a real analytic Kahler metric.