# Boundedness of $\mathbb{Q}$-Fano varieties with degrees and alpha-invariants bounded from below

## TL;DR

This paper proves that $Q$-Fano varieties with fixed dimension, bounded anti-canonical degrees, and alpha-invariants form a bounded family, with implications for K-semistable varieties, advancing understanding of their geometric properties.

## Contribution

It establishes boundedness results for $Q$-Fano varieties under specific numerical invariants, including a corollary for K-semistable cases, which was previously unknown.

## Key findings

- Boundedness of $Q$-Fano varieties with fixed dimension and invariants.
- Boundedness of K-semistable $Q$-Fano varieties under similar conditions.
- Extension of boundedness results to broader classes of Fano varieties.

## Abstract

We show that $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family.