# Calabi-Yau structure and special Lagrangian submanifold of the complexified symmeric space

## TL;DR

This paper explores Calabi-Yau structures on complexified symmetric spaces, detailing their description via Schwarz's theorem and constructing invariant special Lagrangian submanifolds using the Stenzel metric.

## Contribution

It provides a detailed description of Calabi-Yau structures on complexified symmetric spaces and constructs invariant special Lagrangian submanifolds within this framework.

## Key findings

- Calabi-Yau structures are described in terms of Schwarz's theorem.
- Constructed invariant special Lagrangian submanifolds of any phase.
- Applied the Stenzel metric to these structures.

## Abstract

It is known that there exist Calabi-Yau structures on the complexifications of symmetric spaces of compact type. In this paper, we describe the Calabi-Yau structures of the complexified symmetric spaces in terms of the Schwarz's theorem in detail. We consider the case where the Calabi-Yau structure arises from the Riemannian metric corresponding to the Stenzel metric. In the complexified symmetric spaces equipped with such a Calabi-Yau structure, we give constructions of special Lagrangian submanifolds of any phase which are invariant under the actions of symmetric subgroups of the isometry group of the original symmetric space of compact type.