The Preservation of Convexity by Geodesics in the Space of Kähler Potentials on Complex Affine Manifolds

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.