Bounded deformations of $(\hat{I}\mu ,\hat{I}´)$-log canonical singularities

Abstract

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

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