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Finite generation of the log canonical ring in dimension four. (English) Zbl 1210.14020
Let \(X\) be a smooth projective variety and \(B=\sum b_iB_i\) an effective simple normal crossings \(\mathbb Q\)-divisor on \(X\). One of the fundamental results of the minimal model program states that if \(b_i<1\), then the log canonical ring \(R(X,K_X+B)=\oplus _{m\geq 0}\mathcal O _X(m(K_X+B))\) is finitely generated [C. Birkar et al., J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019)]. In the article under review, the author shows that if \(\dim X =4\) and \(b_i\leq 1\), then the log canonical ring \(R(X,K_X+B)\) is finitely generated. The case when \(\dim X=3\) is known by S. Keel, K. Matsuki, J. McKernan [Duke Math. J. 75, No. 1, 99-119 (1994; Zbl 0818.14007)], and [J. Kollár, Flips and abundance for algebraic threefolds. A summer seminar at the University of Utah, Salt Lake City, 1991. Astérisque. 211. Paris: Société Mathématique de France, (1992; Zbl 0782.00075)]. The author also proves abundance for \(n+1\) irregular canonical varieties under the assumption that the minimal model conjecture and the abundance conjecture hold in dimension \(\leq n\).

MSC:
14E30 Minimal model program (Mori theory, extremal rays)
14J35 \(4\)-folds
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