On a generalized canonical bundle formula for generically finite morphisms
Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 2047-2077.

We prove a canonical bundle formula for generically finite morphisms in the setting of generalized pairs (with -coefficients). This complements Filipazzi’s canonical bundle formula for morphisms with connected fibres. It is then applied to obtain a subadjunction formula for log canonical centers of generalized pairs. As another application, we show that the image of an anti-nef log canonical generalized pair has the structure of a numerically trivial log canonical generalized pair. This readily implies a result of Chen–Zhang. Along the way we prove that the Shokurov type convex sets for anti-nef log canonical divisors are indeed rational polyhedral sets.

Nous prouvons une formule de fibré canonique pour des morphismes génériquement finis dans le cadre de paires généralisées (avec -coefficients). Cela complète la formule de fibré canonique de Filipazzi pour les morphismes à fibres connectées. Elle est ensuite appliquée pour obtenir une formule de sous-jonction pour les centres log canoniques de paires généralisées. Comme une autre application, nous montrons que l’image d’une paire généralisée canonique anti-nef log a la structure d’une paire généralisée canonique log numériquement triviale. Cela implique un résultat de Chen–Zhang. Au passage, nous prouvons que les ensembles convexes de type de Shokurov pour les diviseurs log canoniques anti-nef sont en effet des ensembles polyédriques rationnels.

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DOI: 10.5802/aif.3437
Classification: 14E30,  14N30
Keywords: generalized pair, canonical bundle formula, subadjunction
Han, Jingjun 1, 2, 3; Liu, Wenfei 4

1 Department of Mathematics, The University of Utah, Salt Lake City, UT 84112, (USA)
2 Mathematical Sciences Research Institute, Berkeley, CA 94720, (USA)
3 Shanghai Center for Mathematical Sciences, Fudan University, Jiangwan Campus, Shanghai, 200438, (China)
4 Xiamen University School of Mathematical Sciences Siming South Road 422 Xiamen, Fujian 361005 China
License: CC-BY-ND 4.0
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Han, Jingjun; Liu, Wenfei. On a generalized canonical bundle formula for generically finite morphisms. Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 2047-2077. doi : 10.5802/aif.3437. https://aif.centre-mersenne.org/articles/10.5802/aif.3437/

[1] Ambro, Florin Shokurov’s boundary property, J. Differ. Geom., Volume 67 (2004) no. 2, pp. 229-255 | MR | Zbl

[2] Ambro, Florin The moduli b-divisor of an lc-trivial fibration, Compos. Math., Volume 141 (2005) no. 2, pp. 385-403 | DOI | MR | Zbl

[3] Birkar, Caucher On existence of log minimal models. II, J. Reine Angew. Math., Volume 658 (2011), pp. 99-113 | DOI | MR | Zbl

[4] Birkar, Caucher Anti-pluricanonical systems on Fano varieties, Ann. Math., Volume 190 (2019) no. 2, pp. 345-463 | DOI | MR | Zbl

[5] Birkar, Caucher Geometry and moduli of polarised varieties (2020) (https://arxiv.org/abs/2006.11238)

[6] Birkar, Caucher On connectedness of non-klt loci of singularities of pairs (2020) (https://arxiv.org/abs/2010.08226)

[7] Birkar, Caucher Generalised pairs in birational geometry, EMS Surv. Math. Sci., Volume 8 (2021) no. 1-2, pp. 5-24 | DOI | MR | Zbl

[8] Birkar, Caucher; Chen, Yifei Singularities on toric fibrations (2020) (https://arxiv.org/abs/2010.07651)

[9] Birkar, Caucher; Zhang, De-Qi Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs, Publ. Math., Inst. Hautes Étud. Sci., Volume 123 (2016), pp. 283-331 | DOI | MR | Zbl

[10] Chen, Guodu Boundedness of n-complements for generalized pairs (2020) (https://arxiv.org/abs/2003.04237)

[11] Chen, Guodu; Xue, Qingyuan Boundedness of (ϵ,n)-Complements for projective generalized pairs of Fano type (2020) (https://arxiv.org/abs/2008.07121)

[12] Chen, Meng; Zhang, Qi On a question of Demailly–Peternell–Schneider, J. Eur. Math. Soc., Volume 15 (2013) no. 5, pp. 1853-1858 | DOI | MR | Zbl

[13] Corti, Alessio 3-fold flips after Shokurov, Flips for 3-folds and 4-folds (Oxford Lecture Series in Mathematics and its Applications), Volume 35, Oxford University Press, 2007, pp. 18-48 | DOI | MR | Zbl

[14] Cutkosky, Steven Dale Generically finite morphisms and simultaneous resolution of singularities, Commutative algebra. Interactions with algebraic geometry. Proceedings of the international conference, Grenoble, France, July 9–13, 2001 and the special session at the joint international meeting of the American Mathematical Society and the Société Mathématique de France, Lyon, France, July 17–20, 2001 (Contemporary Mathematics), Volume 331, American Mathematical Society, 2003, pp. 75-99 | DOI | MR | Zbl

[15] Filipazzi, Stefano Boundedness of log canonical surface generalized polarized pairs, Taiwanese J. Math., Volume 22 (2018) no. 4, pp. 813-850 | DOI | MR | Zbl

[16] Filipazzi, Stefano Generalized pairs in birational geometry (2019) (Ph. D. Thesis) | MR

[17] Filipazzi, Stefano On a generalized canonical bundle formula and generalized adjunction, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 21, Spec. Iss. (2020), pp. 1187-1221 | MR | Zbl

[18] Filipazzi, Stefano On the boundedness of n-folds with κ(X)=n-1 (2020) (https://arxiv.org/abs/2005.05508)

[19] Filipazzi, Stefano; Svaldi, Roberto Invariance of plurigenera and boundedness for generalized pairs, Mat. Contemp., Volume 47 (2020), pp. 114-150 | MR

[20] Filipazzi, Stefano; Svaldi, Roberto On the connectedness principle and dual complexes for generalized pairs (2020) (https://arxiv.org/abs/2010.08018)

[21] Filipazzi, Stefano; Waldron, Joe Connectedness principle in characteristic p>5 (2020) (https://arxiv.org/abs/2010.08414)

[22] Fujino, Osamu Applications of Kawamata’s positivity theorem, Proc. Japan Acad., Ser. A, Volume 75 (1999) no. 6, pp. 75-79 | MR | Zbl

[23] Fujino, Osamu Some remarks on the minimal model program for log canonical pairs, J. Math. Sci., Tokyo, Volume 22 (2015) no. 1, pp. 149-192 | MR | Zbl

[24] Fujino, Osamu Foundations of the minimal model program, MSJ Memoirs, 35, Mathematical Society of Japan, 2017 | MR | Zbl

[25] Fujino, Osamu Fundamental properties of basic slc-trivial fibrations (2018) (https://arxiv.org/abs/1804.11134, to appear in Publications of the Research Institute for Mathematical Sciences, Kyoto University)

[26] Fujino, Osamu; Gongyo, Yoshinori On canonical bundle formulas and subadjunctions, Mich. Math. J., Volume 61 (2012) no. 2, pp. 255-264 | DOI | MR | Zbl

[27] Fujino, Osamu; Gongyo, Yoshinori On the moduli b-divisors of lc-trivial fibrations, Ann. Inst. Fourier, Volume 64 (2014) no. 4, pp. 1721-1735 | DOI | Numdam | MR | Zbl

[28] Fujino, Osamu; Mori, Shigefumi A canonical bundle formula, J. Differ. Geom., Volume 56 (2000) no. 1, pp. 167-188 | MR | Zbl

[29] Hacon, Christopher D.; Moraga, Joaquín On weak Zariski decompositions and termination of flips, Math. Res. Lett., Volume 27 (2020) no. 5, pp. 1393-1421 | DOI | MR | Zbl

[30] Han, Jingjun; Li, Zhan Weak Zariski decompositions and log terminal models for generalized polarized pairs (2018) (https://arxiv.org/abs/1806.01234)

[31] Han, Jingjun; Li, Zhan On Fujita’s conjecture for pseudo-effective thresholds, Math. Res. Lett., Volume 27 (2020) no. 2, pp. 377-396 | DOI | MR | Zbl

[32] Han, Jingjun; Li, Zhan On accumulation points of pseudo-effective thresholds, Manuscr. Math., Volume 165 (2021) no. 3-4, pp. 537-558 | DOI | MR | Zbl

[33] Han, Jingjun; Liu, Jihao Effective birationality for sub-pairs with real coefficients (2020) (https://arxiv.org/abs/2007.01849)

[34] Han, Jingjun; Liu, Wenfei On numerical nonvanishing for generalized log canonical pairs, Doc. Math., Volume 25 (2020), pp. 93-123 | MR | Zbl

[35] Kawamata, Yujiro Subadjunction of log canonical divisors for a subvariety of codimension 2, Birational algebraic geometry. A conference on algebraic geometry in memory of Wei-Liang Chow (1911–1995), Baltimore, MD, USA, April 11–14, 1996 (Contemporary Mathematics), Volume 207, American Mathematical Society, 1997, pp. 79-88 | DOI | MR | Zbl

[36] Kawamata, Yujiro Subadjunction of log canonical divisors. II, Am. J. Math., Volume 120 (1998) no. 5, pp. 893-899 | DOI | MR | Zbl

[37] Kodaira, Kunihiko On the structure of compact complex analytic surfaces. I, Am. J. Math., Volume 86 (1964), pp. 751-798 | DOI | MR | Zbl

[38] Kollár, János Flips and abundance for algebraic threefolds, Astérisque, 211, Société Mathématique de France, 1992, pp. 1-258 (Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991) | Numdam | MR | Zbl

[39] Kollár, János Singularities of the minimal model program, Cambridge Tracts in Mathematics, 200, Cambridge University Press, 2013, x+370 pages (with a collaboration of Sándor Kovács) | DOI | MR | Zbl

[40] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, 1998 (With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original) | DOI | MR | Zbl

[41] Lazić, Vladimir; Moraga, Joaquín; Tsakanikas, Nikolaos Special termination for log canonical pairs (2020) (https://arxiv.org/abs/2007.06458)

[42] Lazić, Vladimir; Peternell, Thomas On generalised abundance. I, Publ. Res. Inst. Math. Sci., Volume 56 (2020) no. 2, pp. 353-389 | DOI | MR | Zbl

[43] Lazić, Vladimir; Tsakanikas, Nikolaos On the existence of minimal models for log canonical pairs (2019) (https://arxiv.org/abs/1905.05576, to appear in Publications of the Research Institute for Mathematical Sciences, Kyoto University)

[44] Li, Zhan Boundedness of the base varieties of certain fibrations (2020) (https://arxiv.org/abs/2002.06565)

[45] Liu, Jihao Sarkisov program for generalized pairs, Osaka J. Math, Volume 58 (2021) no. 4, pp. 899-920 | MR | Zbl

[46] Mori, Shigefumi Classification of higher-dimensional varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) (Proceedings of Symposia in Pure Mathematics), Volume 46, American Mathematical Society, 1987, pp. 269-331 | MR | Zbl

[47] Prokhorov, Yu. G.; Shokurov, Vyacheslav V. Towards the second main theorem on complements, J. Algebr. Geom., Volume 18 (2009) no. 1, pp. 151-199 | DOI | MR | Zbl

[48] Stacks Project Authors Stacks Project, 2018 (https://stacks.math.columbia.edu)

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