Fujita-type freeness for quasilog canonical curves and surfaces. (English) Zbl 1465.14009

Fujino, Osamu; Liu, Haidong

Kyoto J. Math. 60, No. 4, 1453-1467 (2020).

We work over the complex number field \(\mathbb{C}\). One of the most important open problems in algebraic geometry is Fujita’s freeness conjecture, which predicts that if \(X\) is a smooth projective variety and \(N\) is an ample Cartier divisor on \(X\), then \(|K_X +(\dim X+1)N|\) is base point free. It is widely open when \(\dim X \geq 6\). Reider’s theorem [I. Reider, Ann. Math. (2) 127, No. 2, 309–316 (1988; Zbl 0663.14010)] motivates the following stronger conjecture:
Let \(X\) be a smooth projective variety, and \(N\) be an ample Cartier divisor on \(X\). Assume that \(N^{\dim X} > (\dim X)^{\dim X}\) and \(N^{\dim Z}.Z \geq (\dim X)^{\dim Z}\) for every proper irreducible subvariety \(Z \subseteq X\). Then \(|K_X+N|\) is base point free.
The purpose of the paper under review is to study the stronger version of Fujita’s freeness conjecture for quasilog canonical (qlc) pairs, which would be useful for some inductive approach to the original conjecture. The main conjecture is the following (Conjecture 1.2):
Let \([X, \omega]\) be a projective qlc pair, and \(N:=M-\omega\), where \(M\) is a Cartier divisor on \(X\). Assume that \(N^{\dim X_i}.X_i > (\dim X_i)^{\dim X_i}\) for every positive dimensional irreducible component \(X_i\) of \(X\) and \(N^{\dim Z}.Z \geq n_Z^{\dim Z}\) for every proper irreducible subvariety \(Z \subseteq X\) which is not an irreducible component of \(X\), where \(n_Z=\min_i \{\dim X_i \mid Z \subseteq X_i\}\).
The purpose of the paper is to verify this conjecture when \(\dim X \leq 2\) (Theorem 1.3 and Corollary 1.5). The strategy to prove the conjecture is outlined in the introduction of the paper, and the authors carry out this strategy for the surface case. There are two main ingredients; (1) the first author’s result in [O. Fujino, “Fundamental properties of basic slc-trivial fibrations”, Preprint, arXiv:1804.1113], which comes from the theory of variations of mixed Hodge structure on cohomology with compact support, and (2) the authors’ previous result on normalization of qlc pairs [O. Fujino and H. Liu, Proc. Japan Acad., Ser. A 94, No. 10, 97–101 (2018; Zbl 1415.14009)].

Reviewer: Jinhyung Park (Seoul)

MSC:

14C20	Divisors, linear systems, invertible sheaves
14E30	Minimal model program (Mori theory, extremal rays)

Keywords:

quasilog canonical surfaces; semilog canonical surfaces; Fujita-type freeness; minimal model program

Citations:

Zbl 0663.14010; Zbl 1415.14009

References:

[1]	F. Ambro, Quasi-log varieties (in Russian), Tr. Mat. Inst. Steklova 240 (2003), 220-239; English translation in Proc. Steklov Inst. Math. 240, (2003), no. 1, 214-233. Zentralblatt MATH: 1081.14021 · Zbl 1081.14021
[2]	U. Angehrn and Y. T. Siu, Effective freeness and point separation for adjoint bundles, Invent. Math. 122 (1995), no. 2, 291-308. Zentralblatt MATH: 0847.32035 Digital Object Identifier: doi:10.1007/BF01231446 · Zbl 0847.32035 · doi:10.1007/BF01231446
[3]	F. Campana and T. Peternell, Algebraicity of the ample cone of projective varieties, J. Reine Angew. Math. 407 (1990), 160-166. Zentralblatt MATH: 0728.14004 · Zbl 0728.14004
[4]	O. Fujino, Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, 727-789. Zentralblatt MATH: 1234.14013 Digital Object Identifier: doi:10.2977/PRIMS/50 · Zbl 1234.14013 · doi:10.2977/PRIMS/50
[5]	O. Fujino, Fundamental theorems for semi log canonical pairs, Algebr. Geom. 1 (2014), no. 2, 194-228. Zentralblatt MATH: 1296.14014 Digital Object Identifier: doi:10.14231/AG-2014-011 · Zbl 1296.14014 · doi:10.14231/AG-2014-011
[6]	O. Fujino, Effective basepoint-free theorem for semi-log canonical surfaces, Publ. Res. Inst. Math. Sci. 53 (2017), no. 3, 349-370. Zentralblatt MATH: 1386.14039 Digital Object Identifier: doi:10.4171/PRIMS/53-3-1 · Zbl 1386.14039 · doi:10.4171/PRIMS/53-3-1
[7]	O. Fujino, Foundations of the minimal model program, MSJ Memoirs 35, Math. Soc. Japan, Tokyo, 2017. Zentralblatt MATH: 1386.14072 · Zbl 1386.14072
[8]	O. Fujino, Pull-back of quasi-log structures, Publ. Res. Inst. Math. Sci. 53 (2017), no. 2, 241-259. Zentralblatt MATH: 1388.14057 Digital Object Identifier: doi:10.4171/PRIMS/53-2-1 · Zbl 1388.14057 · doi:10.4171/PRIMS/53-2-1
[9]	O. Fujino, Fundamental properties of basic slc-trivial fibrations, to appear in Publ. Res. Inst. Math. Sci., preprint, arXiv:1804.11134v3 [math.AG]. arXiv: 1804.11134v3
[10]	O. Fujino and H. Liu,On normalization of quasi-log canonical pairs, Proc. Japan Acad. Ser. A Math. Sci. 94 (2018), no. 10, 97-101. Zentralblatt MATH: 1415.14009 Digital Object Identifier: doi:10.3792/pjaa.94.97 Project Euclid: euclid.pja/1543201232 · Zbl 1415.14009 · doi:10.3792/pjaa.94.97
[11]	H. Liu, Angehrn-Siu type effective basepoint freeness for quasi-log canonical pairs, Kyoto J. Math. 59 (2019), no. 2, 455-470. Zentralblatt MATH: 07080113 Digital Object Identifier: doi:10.1215/21562261-2019-0014 Project Euclid: euclid.kjm/1557216017 · Zbl 1472.14016 · doi:10.1215/21562261-2019-0014
[12]	F. · Zbl 1454.14019 · doi:10.1016/j.aim.2020.107210