# A gap theorem for minimal log discrepancies of non-canonical singularities in dimension three

## TL;DR

This paper proves a uniform gap in minimal log discrepancies for non-canonical singularities in three-dimensional varieties, with implications for the boundedness of certain Calabi-Yau threefolds.

## Contribution

It establishes a positive lower bound gap for minimal log discrepancies in threefolds with non-canonical singularities, advancing the understanding of their structure.

## Key findings

- Existence of a positive gap elta in minimal log discrepancies.
- Boundedness of non-canonical klt Calabi-Yau 3-folds modulo flops.
- Upper bounds on global indices of klt Calabi-Yau 3-folds.

## Abstract

We show that there exists a positive real number $\delta>0$ such that for any normal quasi-projective $\mathbb{Q}$-Gorenstein $3$-fold $X$, if $X$ has worse than canonical singularities, that is, the minimal log discrepancy of $X$ is less than $1$, then the minimal log discrepancy of $X$ is not greater than $1-\delta$. As applications, we show that the set of all non-canonical klt Calabi-Yau $3$-folds are bounded modulo flops, and the global indices of all klt Calabi-Yau $3$-folds are bounded from above.