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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 \(i > 0\).

14J17 Singularities of surfaces or higher-dimensional varieties
14F17 Vanishing theorems in algebraic geometry
Full Text: DOI arXiv
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