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Du Bois pairs and vanishing theorems. (English) Zbl 1218.14021
Du Bois (or simply DB) singularities were first introduced by Steenbrink, based upon the work of Deligne and Du Bois, see [P. Du Bois, Bull. Soc. Math. Fr. 109, 41–81 (1981; Zbl 0465.14009); J. H. M. Steenbrink, Compos. Math. 42, 315–320 (1981; Zbl 0428.32017)].
In this paper, the author generalizes DB singularities to the context of pairs. He works in particular in the context $$(X, \Sigma)$$, where $$\Sigma$$ is a subscheme of a variety $$X$$. Suppose that $D^{.} \to \underline{\Omega}_X^{0} \to \underline{\Sigma}_X^{0} \to D^{.}[+1]$ is a triangle in $$D^b_{\text{coh}}(X)$$ where the map $$\underline{\Omega}_X^{0} \to \underline{\Sigma}_X^{0}$$ is the natural one. In this setting, $$(X, \Sigma)$$ is called Du Bois if $$D^{.}$$ is quasi-isomorphic to the ideal sheaf defining $$\Sigma$$.
This definition differs from the more common definitions of pairs appearing throughout the minimal model program in that it is a relative notion. In particular, if the pair $$(X, \Sigma)$$ is DB, then $$X$$ has Du Bois singularities if and only if $$\Sigma$$ does (see Proposition 5.1 in the paper). However, it is not clear whether $$(X, \Sigma)$$ being DB implies that $$X$$ is DB.
In Section 6 of this paper, the author proves several vanishing theorems for DB pairs. As an application, he proves that if $$(X, \Delta)$$ is a log canonical pair and $$\pi : \widetilde{X} \to X$$ is a log resolution, and $$\widetilde{\Delta} = (\pi_*^{-1} \lfloor \Delta \rfloor + E)_{\text{red}}$$, where $$E$$ denotes the union of the exceptional non-klt places, then $R^i \pi_* \mathcal{O}_{\widetilde{X}}(-\widetilde{\Delta}) = 0$ for $$i > 0$$.

##### MSC:
 14J17 Singularities of surfaces or higher-dimensional varieties 14F17 Vanishing theorems in algebraic geometry
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##### References:
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