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The weak ordinarity conjecture and $$F$$-singularities. (English) Zbl 1390.14060
Oguiso, Keiji (ed.) et al., Higher dimensional algebraic geometry. In honour of Professor Yujiro Kawamata’s sixtieth birthday. Proceedings of the conference, Tokyo, Japan, January 7–11, 2013. Tokyo: Mathematical Society of Japan (MSJ) (ISBN 978-4-86497-046-4/hbk). Advanced Studies in Pure Mathematics 74, 11-39 (2017).
Summary: Recently M. Mustaţă and V. Srinivas [Nagoya Math. J. 204, 125–157 (2011; Zbl 1239.14011)] related a natural conjecture about the Frobenius action on the cohomology of the structure sheaf after reduction to characteristic $$p>0$$ with another conjecture connecting multiplier ideals and test ideals. We generalize this relation to the case of singular ambient varieties.
Additionally, we connect these results to a conjecture relating $$F$$-injective and Du Bois singularities. Finally, using an unpublished result of O. Gabber [Private communication (2012)], we also show that $$F$$-injective and Du Bois singularities have a common definition in terms of smooth hypercovers.
For the entire collection see [Zbl 1388.14012].

##### MSC:
 14F18 Multiplier ideals 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure 14J17 Singularities of surfaces or higher-dimensional varieties 14B05 Singularities in algebraic geometry 14F20 Étale and other Grothendieck topologies and (co)homologies
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