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Towards the second main theorem on complements. (English) Zbl 1159.14020

The paper under review studies complements on log Fano varieties, this is a second chapter of the saga started in [Yu. G. Prokhorov, V. V. Shokurov, Izv. Math. 65, No. 6, 1169–1196 (2001); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 65, No. 6, 99–128 (2001; Zbl 1068.14018)]. Both the definition and the statement of theorems are too technical to be put in a review. I will try to rephrase it in a very simple situation. Assume that \(X\) is a Fano manifold. That is \(X\) is smooth and the anticanonical class is ample. The main theorem of the paper states, conditionally to some standard conjecture, that there is an integer \(m\) depending only on the dimension of \(X\) such that \(|-mK_X|\) is not empty. The true statement is directed to a much wider class of singular varieties called of Fano type.

MSC:

14J40 \(n\)-folds (\(n>4\))
14E30 Minimal model program (Mori theory, extremal rays)

Citations:

Zbl 1068.14018

References:

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