Relative mmp without $ \mathbb{Q} $-factoriality

János Kollár; János Kollár

doi:10.3934/era.2021033

Electronic Research Archive

2021, Volume 29, Issue 5: 3193-3203. doi: 10.3934/era.2021033

Previous Article Next Article

Special Issues

Relative mmp without $ \mathbb{Q} $-factoriality

János Kollár

Princeton University, Princeton NJ 08544-1000, USA

Received: 01 December 2020 Revised: 01 February 2021 Published: 15 April 2021
Primary: 14E30, 14E05, 14J17; Secondary: 14F17, 14J40

We consider the minimal model program for varieties that are not $ \mathbb{Q}$-factorial. We show that, in many cases, its steps are simpler than expected. The main applications are to log terminal singularities, removing the earlier $ \mathbb{Q} $-factoriality assumption from several theorems of Hacon-Witaszek and de Fernex-Kollár-Xu.
- Minimal model,
- dlt singularity,
- vanishing theorems
Citation: János Kollár. Relative mmp without $ \mathbb{Q} $-factoriality[J]. Electronic Research Archive, 2021, 29(5): 3193-3203. doi: 10.3934/era.2021033

Related Papers:

Abstract

We consider the minimal model program for varieties that are not $ \mathbb{Q}$-factorial. We show that, in many cases, its steps are simpler than expected. The main applications are to log terminal singularities, removing the earlier $ \mathbb{Q} $-factoriality assumption from several theorems of Hacon-Witaszek and de Fernex-Kollár-Xu.

References

[1]	Flips and Abundance for Algebraic Threefolds, Société Mathématique de France, Paris, 1992, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211, (1992).
[2]	H. Ahmadinezhad, M. Fedorchuk and I. Krylov, Stability of fibrations over one-dimensional bases, 2019, arXiv: 1912.08779.
[3]	V. Alexeev, Moduli of Weighted Hyperplane Arrangements, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer, Basel, 2015, Edited by Gilberto Bini, Martí Lahoz, Emanuele Macrì and Paolo Stellari. doi: 10.1007/978-3-0348-0915-3
[4]	E. Arvidsson, F. Bernasconi and J. Lacini, On the Kawamata-Viehweg vanishing for log del Pezzo surfaces in positive characteristic, 2020, arXiv: 2006.03571.
[5]	F. Bernasconi and J. Kollár, Vanishing theorems for threefolds in characteristic $p>5$, 2020, arXiv: 2012.08343.
[6]	B. Bhatt, L. Ma, Z. Patakfalvi, K. Schwede, K. Tucker, J. Waldron and J. Witaszek, Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic, 2020, arXiv: 2012.15801.
[7]	Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. (2010) 23: 405-468.
[8]	J.-F. Boutot, Schéma de Picard Local, vol. 632 of Lecture Notes in Mathematics, Springer, Berlin, 1978.
[9]	J. Carvajal-Rojas and A. Stäbler, On the local étale fundamental group of KLT threefold singularities, 2020, arXiv: 2004.07628, With an appendix by János Kollár.
[10]	Resolution of singularities of arithmetical threefolds. J. Algebra (2019) 529: 268-535.
[11]	T. de Fernex, J. Kollár and C. Xu, The dual complex of singularities, in Higher Dimensional Algebraic Geometry--in Honour of Professor Yujiro Kawamata's Sixtieth Birthday, vol. 74 of Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, (2017), 103-129. doi: 10.2969/aspm/07410103
[12]	O. Fujino, Foundations of the Minimal Model Program, vol. 35 of MSJ Memoirs, Mathematical Society of Japan, Tokyo, 2017.
[13]	Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen. Invent. Math. (1970) 11: 263-292.
[14]	On the rationality of Kawamata log terminal singularities in positive characteristic. Algebr. Geom. (2019) 6: 516-529.
[15]	C. Hacon and J. Witaszek, On the relative minimal model program for threefolds in low characteristics, 2019, arXiv: 1909.12872.
[16]	C. Hacon and J. Witaszek, On the relative minimal model program for fourfolds in positive characteristic, 2020, arXiv: 2009.02631.
[17]	J. Han, J. Liu and V. V. Shokurov, Acc for minimal log discrepancies of exceptional singularities, 2019, arXiv: 1903.04338.
[18]	J. Kollár, Lectures on Resolution of Singularities, vol. 166 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 2007.
[19]	A local version of the Kawamata-Viehweg vanishing theorem. Pure Appl. Math. Q. (2011) 7: 1477-1494.
[20]	J. Kollár, Singularities of the Minimal Model Program, vol. 200 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2013, With a collaboration of Sándor Kovács. doi: 10.1017/CBO9781139547895
[21]	J. Kollár, Appendix to "On the local tale fundamental group of KLT threefold singularities", 2020.
[22]	J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, vol. 134 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1998, With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. doi: 10.1017/CBO9780511662560
[23]	J. Kollár and J. Witaszek, Resolution and alteration with ample exceptional divisor, 2021, arXiv: 2102.03162.
[24]	J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math., 36 (1969), 195-279, URL http://www.numdam.org/item?id=PMIHES_1969__36__195_0.
[25]	Log-canonical models of singular pairs and its applications. Math. Res. Lett. (2012) 19: 325-334.
[26]	A variant of the Noether-Lefschetz theorem: Some new examples of unique factorisation domains. J. Algebraic Geom. (1994) 3: 81-115.
[27]	Algebraizations with minimal class group. Internat. J. Math. (1993) 4: 989-996.
[28]	M. Reid, Young person's guide to canonical singularities, in Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), vol. 46 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1987,345-414.
[29]	Three-dimensional log perestroikas. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya (1992) 56: 105-203.
[30]	T. Stacks project authors, The stacks project, https://stacks.math.columbia.edu, 2021.
[31]	Minimal model program for excellent surfaces. Ann. Inst. Fourier (Grenoble) (2018) 68: 345-376.
[32]	Desingularization of quasi-excellent schemes in characteristic zero. Adv. Math. (2008) 219: 488-522.
[33]	D. Villalobos-Paz, 2021, (in preparation).
[34]	J. Witaszek, Non Q-factorial flips in dimension three, 2020, Available from: http://www-personal.umich.edu/ jakubw/Non-Q-factorial-flips-in-dimension-three.pdf.
[35]	Finiteness of algebraic fundamental groups. Compos. Math. (2014) 150: 409-414.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)