On the 4-dimensional minimal model program for Kähler varieties.(English)Zbl 1540.14029
As one of the most significant tools to classify algebraic or analytic varieties, the minimal model program (MMP for short) has been substantially developed for complex projective varieties during the last two decades (see [C. Birkar et al., J. Am. Math. Soc. 23, No. 2, 405–468 (2010;Zbl 1210.14019)]). It has also been well-understood for compact Kähler threefolds (see [A. Höring andT. Peternell, Invent. Math. 203, No. 1, 217–264 (2016;Zbl 1337.32031);F. Campana et al., Ann. Sci. Éc. Norm. Supér. (4) 49, No. 4, 971–1025 (2016;Zbl 1386.32020);O. Das andW. Ou, Manuscr. Math. 173, No. 1–2, 341–404 (2024;Zbl 1536.14012);O. Das andC. Hacon, “The log minimal model program for Kähler \(3\)-folds”, Preprint,arXiv:2009.05924]) while much less is known about Kähler varieties in higher dimension.
In the paper under review, the authors prove that the MMP holds for effective dlt Kähler 4-fold pairs and for (strongly) semistable families of 3-folds over curves (see Theorems 1.1 and 1.2). Besides, by studying the finite generation conjecture, the authors show the existence of flips for analytic varieties in all dimensions (see Theorem 1.3). The relative MMP for projective morphisms between analytic varieties has also been established (see Theorem 1.4). In [“Minimal model program for projective morphisms between complex analytic spaces”, Preprint,arXiv:2201.11315],O. Fujino obtains a similar result to Theorems 1.3 and 1.4 by using different approaches.
In the paper under review, the authors prove that the MMP holds for effective dlt Kähler 4-fold pairs and for (strongly) semistable families of 3-folds over curves (see Theorems 1.1 and 1.2). Besides, by studying the finite generation conjecture, the authors show the existence of flips for analytic varieties in all dimensions (see Theorem 1.3). The relative MMP for projective morphisms between analytic varieties has also been established (see Theorem 1.4). In [“Minimal model program for projective morphisms between complex analytic spaces”, Preprint,arXiv:2201.11315],O. Fujino obtains a similar result to Theorems 1.3 and 1.4 by using different approaches.
Reviewer: Guolei Zhong (Daejeon)
MSC:
| 14E05 | Rational and birational maps |
| 14E30 | Minimal model program (Mori theory, extremal rays) |
| 32Q57 | Classification theorems for complex manifolds |
Cite
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