Invariant Calabi-Yau structures on punctured complexified symmetric spaces

TL;DR

This paper constructs G-invariant Calabi-Yau structures on complexified symmetric spaces using solutions to a Monge-Ampère type equation, providing explicit descriptions and proving the existence of solutions in certain cases.

Contribution

It introduces a method to build invariant Calabi-Yau structures via a Monge-Ampère equation and explicitly describes this equation for low-rank symmetric spaces.

Findings

Explicit Monge-Ampère equations for rank one and two cases

Existence of solutions to the Monge-Ampère equation in these cases

Construction of invariant Calabi-Yau structures on complexified symmetric spaces

Abstract

In this paper, we show that $G$-invariant Calabi-Yau structures on the complexification $G^{\mathbb C}/K^{\mathbb C}$ of a symmetric space $G/K$ of compact type are constructed from solutions of a Monge-Amp$\grave{\rm e}$re type equation. Also, we give an explicit descriptions of the Monge-Amp$\grave{\rm e}$re type equation in the case where the rank of $G/K$ is equal to one or two. Furthermore, we prove the existence of solutions of the Monge-Amp$\grave{\rm e}$re type equation.