On the log abundance for compact Kähler threefolds.(English)Zbl 1536.14012
The Abundance Conjecture in algebraic geometry asserts that if the caonical divisor \(K_X\) of a projective manifold \(X\) is nef, then \(K_X\) is semi-ample; i.e., \(|-mK_X|\) is base point free for some \(m>0\). This is one of the most important conjectures in algebraic geometry and it has many applications.
This conjecture is classically known for surfaces and it is confirmed for threefold byY. Miyaoka [Compos. Math. 68, No. 2, 203–220 (1988;Zbl 0681.14019)],Y. Miyaoka [Math. Ann. 281, No. 2, 325–332 (1988;Zbl 0625.14023)] andY. Kawamata [Invent. Math. 108, No. 2, 229–246 (1992;Zbl 0777.14011)], and it is still wide open in dimension at least four. Moreover, inspired by the minimal model program, it is also expected that the abundance conjecture also holds for log canonical pairs. This is again confirmed in dimension at most three [S. Keel et al., Duke Math. J. 75, No. 1, 99–119 (1994;Zbl 0818.14007)].
On the other hand, in the last decade, the minimal model program was established for Kähler manifolds byA. Höring andT. Peternell [Invent. Math. 203, No. 1, 217–264 (2016;Zbl 1337.32031)], which are generalisations of projective manifolds. Also the Abundance Conjecture is also confirmed for Kähler threefolds with terminal singualrities and numerical dimension not equal to 2 byF. Campana et al. [Ann. Sci. Éc. Norm. Supér. (4) 49, No. 4, 971–1025 (2016;Zbl 1386.32020)].
In the paper under review, the authors proves the Abundance Conjecutre for log canonocal Kähler pair \((X,\Delta)\) in the case where \(\nu(K_X+\Delta)\not=2\), and hence genralises the result of Campana, Höring and Peternell in the terminal case. We also remark that the remaining case is solved by the same authors in a recent preprint [O. Das andW. Ou, “On the Log Abundance for Compact {Kähler} threefolds II”, Preprint,arXiv:2306.00671].
This conjecture is classically known for surfaces and it is confirmed for threefold byY. Miyaoka [Compos. Math. 68, No. 2, 203–220 (1988;Zbl 0681.14019)],Y. Miyaoka [Math. Ann. 281, No. 2, 325–332 (1988;Zbl 0625.14023)] andY. Kawamata [Invent. Math. 108, No. 2, 229–246 (1992;Zbl 0777.14011)], and it is still wide open in dimension at least four. Moreover, inspired by the minimal model program, it is also expected that the abundance conjecture also holds for log canonical pairs. This is again confirmed in dimension at most three [S. Keel et al., Duke Math. J. 75, No. 1, 99–119 (1994;Zbl 0818.14007)].
On the other hand, in the last decade, the minimal model program was established for Kähler manifolds byA. Höring andT. Peternell [Invent. Math. 203, No. 1, 217–264 (2016;Zbl 1337.32031)], which are generalisations of projective manifolds. Also the Abundance Conjecture is also confirmed for Kähler threefolds with terminal singualrities and numerical dimension not equal to 2 byF. Campana et al. [Ann. Sci. Éc. Norm. Supér. (4) 49, No. 4, 971–1025 (2016;Zbl 1386.32020)].
In the paper under review, the authors proves the Abundance Conjecutre for log canonocal Kähler pair \((X,\Delta)\) in the case where \(\nu(K_X+\Delta)\not=2\), and hence genralises the result of Campana, Höring and Peternell in the terminal case. We also remark that the remaining case is solved by the same authors in a recent preprint [O. Das andW. Ou, “On the Log Abundance for Compact {Kähler} threefolds II”, Preprint,arXiv:2306.00671].
Reviewer: Jie Liu (Beijing)
MSC:
| 14E30 | Minimal model program (Mori theory, extremal rays) |
| 32J17 | Compact complex \(3\)-folds |
| 32J27 | Compact Kähler manifolds: generalizations, classification |
Cite
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