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Potentially Du Bois spaces. (English) Zbl 1323.14002
Du Bois singularities have the property that the natural map $$H^i(X^{\text{an}},\mathbb{C}) \to H^i(X^{\text{an}},\mathcal{O}_{X^{\text{an}}})$$ is surjective; together with the requirement that the general hyperplane section is also Du Bois this essentially characterises them. Log canonical singularities are Du Bois. In [S. J. Kovács, Kyoto J. Math. 51, No. 1, 47-69 (2011; Zbl 1218.14021)]. the notion was generalised to pairs $$(X,\Sigma)$$. The Authors call a singular point potentially Du Bois, if it has a Zariski open neighbourhood $$U$$ containing a subvariety $$\Sigma_U$$, such that $$(U,\Sigma_U)$$ is a Du Bois pair. They prove that the non Du Bois locus has codimension at least three, and that a normal variety $$X$$ with $$K_X$$ Cartier and potentially Du Bois singularities is log canonical and Du Bois. On the other hand, they develop in detail an example of a normal isolated 3-dimensional potentially Bu Bois singularity with $$K_X$$ $$\mathbb{Q }$$-Cartier, that is not Bu Bois.

##### MSC:
 14B05 Singularities in algebraic geometry 32S05 Local complex singularities
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