The Preservation of Convexity by Geodesics in the Space of K\"ahler Potentials on Complex Affine Manifolds

TL;DR

This paper introduces a new convexity concept in the space of K"ahler potentials on complex affine manifolds and proves its preservation along geodesics, ensuring non-degeneracy of metrics under certain conditions.

Contribution

It defines $(S, ext{omega}_0)$-convexity and proves its preservation by geodesics in the K"ahler potential space, a novel result in complex geometry.

Findings

$(S, ext{omega}_0)$-convexity is preserved by geodesics

Geodesics between strictly $(S, ext{omega}_0)$-convex potentials maintain non-degenerate metrics

Introduction of a new convexity concept in K"ahler geometry

Abstract

On a compact complex affine manifold with a constant coefficient K\"ahler metric $\omega_0$, we introduce a concept: $(S,\omega_0)$-convexity and show that $(S,\omega_0)$-convexity is preserved by geodesics in the space of K\"ahler potentials. This implies that if two potentials are both strictly $(S,\omega_0)$-convex, then the metrics along the geodesic connecting them are non-degenerate.