# Bounded deformations of $(\epsilon,\delta)$-log canonical singularities

## TL;DR

This paper investigates the boundedness properties of $(,)$-log canonical singularities, establishing bounds in low dimensions and providing counterexamples in higher dimensions, thus advancing understanding of their deformation and boundedness behavior.

## Contribution

It proves boundedness of $(,)$-lc singularities in dimensions 2 and up to deformation in higher dimensions, and presents counterexamples showing unboundedness in higher dimensions.

## Key findings

- $n$-dimensional $(,)$-lc singularities are bounded up to deformation.
- 2-dimensional $(,)$-lc singularities form a bounded family.
- Counterexamples show unboundedness of $(,)$-lc singularities in higher dimensions.

## Abstract

In this paper we study $(\epsilon,\delta)$-lc singularites, i.e. $\epsilon$-lc singularities admitting a $\delta$-plt blow-up. We prove that $n$-dimensional $(\epsilon,\delta)$-lc singularities are bounded up to a deformation, and $2$-dimensional $(\epsilon,\delta)$-lc singularities form a bounded family. Furthermore, we give an example which shows that $(\epsilon,\delta)$-lc singularities are not bounded in higher dimensions, even in the analytic sense.