Some remarks on the symplectic and Kaehler geometry of toric varieties

TL;DR

This paper investigates the symplectic and Kähler geometry of projective toric manifolds, focusing on Kähler-Einstein submanifolds and symplectic embeddings of Euclidean balls, leveraging the dense biholomorphic subset to C^n.

Contribution

It provides new insights into the structure of toric varieties by analyzing Kähler-Einstein submanifolds and symplectic embeddings within them.

Findings

Existence of Kähler-Einstein submanifolds in projective toric manifolds.

Results on symplectic embeddings of Euclidean balls into these manifolds.

Utilization of the dense biholomorphic subset to C^n for geometric analysis.

Abstract

Let $M$ be a projective toric manifold. We prove two results concerning respectively Kaehler-Einstein submanifolds of M and symplectic embeddings of the standard euclidean ball in M. Both results use the well-known fact that M contains an open dense subset biholomorphic to C^n.