# K\"ahler-Einstein metrics on Q-smoothable Fano varieties, their moduli and some applications

## TL;DR

This paper surveys recent advances in establishing K"ahler-Einstein metrics on smoothable Fano varieties and explores their role in constructing moduli spaces, along with applications and open problems.

## Contribution

It provides a comprehensive overview of the existence of K"ahler-Einstein metrics on Fano varieties and discusses their significance in moduli space construction and future research directions.

## Key findings

- Existence results for K"ahler-Einstein metrics on smoothable Fano varieties
- Construction of compact moduli spaces of K-polystable Fano varieties
- Identification of open problems and future research directions

## Abstract

We survey recent results on the existence of K\"ahler-Einstein metrics on certain smoothable Fano varieties, focusing on the importance of such metrics in the construction of compact algebraic moduli spaces of K-polystable Fano varieties. Moreover, we give some applications and we discuss some natural problems which deserve future investigations.