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Minimal model theory for relatively trivial log canonical pairs. (La théorie des modèles minimaux pour des paires log-canoniques relativement triviaux.) (English. French summary) Zbl 1423.14111
Summary: We study relative log canonical pairs with relatively trivial log canonical divisors. We fix such a pair $$(X, \Delta) / Z$$ and establish the minimal model theory for the pair $$(X, \Delta)$$ assuming the minimal model theory for all Kawamata log terminal pairs whose dimension is not greater than $$\dim Z$$. We also show the finite generation of log canonical rings for log canonical pairs of dimension five which are not of log general type.

##### MSC:
 1.4e+31 Minimal model program (Mori theory, extremal rays)
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##### References:
 [1] Dan Abramovich & Kalle Karu, “Weak semistable reduction in characteristic $$0$$”, Invent. Math.139 (2000) no. 2, p. 241-273 · Zbl 0958.14006 [2] Florin Ambro, “The moduli $$b$$-divisor of an lc-trivial fibration”, Compos. Math.141 (2005) no. 2, p. 385-403 · Zbl 1094.14025 [3] Caucher Birkar, “On existence of log minimal models II”, J. Reine Angew. Math.658 (2011), p. 99-113 · Zbl 1226.14021 [4] Caucher Birkar, “Existence of log canonical flips and a special LMMP”, Publ. Math., Inst. Hautes Étud. Sci.115 (2012), p. 325-368 · Zbl 1256.14012 [5] Caucher Birkar, Paolo Cascini, Christopher D. Hacon & James McKernan, “Existence of minimal models for varieties of log general type”, J. Am. Math. Soc.23 (2010) no. 2, p. 405-468 · Zbl 1210.14019 [6] Caucher Birkar & Zhengyu Hu, “Log canonical pairs with good augmented base loci”, Compos. Math.150 (2014) no. 4, p. 579-592 · Zbl 1314.14027 [7] Jean-Pierre Demailly, Christopher D. Hacon & Mihai Păun, “Extension theorems, non-vanishing and the existence of good minimal models”, Acta Math.210 (2013) no. 2, p. 203-259 · Zbl 1278.14022 [8] Osamu Fujino, Special termination and reduction to pl flips, Flips for $$3$$-folds and $$4$$-folds, Oxford Lecture Series in Mathematics and its Applications 35, Oxford University Press, 2007, p. 63-75 · Zbl 1286.14025 [9] Osamu Fujino, “Finite generation of the log canonical ring in dimension four”, Kyoto J. Math.50 (2010) no. 4, p. 671-684 · Zbl 1210.14020 [10] Osamu Fujino, “Fundamental theorems for the log minimal model program”, Publ. Res. Inst. Math. Sci.47 (2011) no. 3, p. 727-789 · Zbl 1234.14013 [11] Osamu Fujino, “Some remarks on the minimal model program for log canonical pairs”, J. Math. Sci., Tokyo22 (2015) no. 1, p. 149-192 · Zbl 1435.14017 [12] Osamu Fujino, Foundations of the minimal model program, MSJ Memoirs 35, Mathematical Society of Japan, 2017 [13] Osamu Fujino & Yoshinori Gongyo, “On canonical bundle formulas and subadjunctions”, Mich. Math. J.61 (2012) no. 2, p. 255-264 · Zbl 1260.14010 [14] Osamu Fujino & Yoshinori Gongyo, “Log pluricanonical representations and the abundance conjecture”, Compos. Math.150 (2014) no. 4, p. 593-620 · Zbl 1314.14029 [15] Osamu Fujino & Yoshinori Gongyo, “On the moduli b-divisors of lc-trivial fibrations”, Ann. Inst. Fourier64 (2014) no. 4, p. 1721-1735 · Zbl 1314.14030 [16] Osamu Fujino & Yoshinori Gongyo, On log canonical rings, Higher dimensional algebraic geometry, Advanced Studies in Pure Mathematics 74, Mathematical Society of Japan, 2017, p. 159-169 · Zbl 1388.14058 [17] Osamu Fujino & Shigefumi Mori, “A canonical bundle formula”, J. Differ. Geom.56 (2000) no. 1, p. 167-188 · Zbl 1032.14014 [18] Yoshinori Gongyo, Remarks on the non-vanishing conjecture, Algebraic geometry in east Asia—Taipei 2011, Advanced Studies in Pure Mathematics 65, Mathematical Society of Japan, 2015, p. 107-116 · Zbl 1360.14054 [19] Yoshinori Gongyo & Brian Lehmann, “Reduction maps and minimal model theory”, Compos. Math.149 (2013) no. 2, p. 295-308 · Zbl 1264.14025 [20] Christopher D. Hacon, James McKernan & Chenyang Xu, “ACC for log canonical thresholds”, Ann. Math.180 (2014) no. 2, p. 523-571 · Zbl 1320.14023 [21] Christopher D. Hacon & Chenyang Xu, “Existence of log canonical closures”, Invent. Math.192 (2013) no. 1, p. 161-195 · Zbl 1282.14027 [22] Kenta Hashizume, “Remarks on the abundance conjecture”, Proc. Japan Acad. Ser. A Math. Sci.92 (2016) no. 9, p. 101-106 · Zbl 1365.14019 [23] Yujiro Kawamata, Variation of mixed Hodge structures and the positivity for algebraic fiber spaces, Algebraic geometry in east Asia—Taipei 2011, Advanced Studies in Pure Mathematics 65, Mathematical Society of Japan, 2015, p. 27-57 · Zbl 1360.14034 [24] János Kollár & Sándor J. Kovács, “Log canonical singularities are Du Bois”, J. Am. Math. Soc.23 (2010) no. 3, p. 791-813 · Zbl 1202.14003 [25] János Kollár & Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge University Press, 1998 · Zbl 0926.14003 [26] Ching-Jui Lai, “Varieties fibered by good minimal models”, Math. Ann.350 (2011) no. 3, p. 533-547 · Zbl 1221.14018 [27] Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs 14, Mathematical Society of Japan, 2004 © Annales de L’Institut Fourier - ISSN (électronique) : 1777-5310
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