## Bounded Deformations of $\large\boldsymbol{(\epsilon,\delta)}$-Log Canonical Singularities

J. Math. Sci. Univ. Tokyo
Vol. 27 (2020), No. 1, Page 1-28.

Han, Jingjun; Liu, Jihao; Moraga, Joaquín
Bounded Deformations of $\large\boldsymbol{(\epsilon,\delta)}$-Log Canonical Singularities
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.