• Previous Article
    Local well-posedness of perturbed Navier-Stokes system around Landau solutions
  • ERA Home
  • This Issue
  • Next Article
    Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity
doi: 10.3934/era.2021039

On Nonvanishing for uniruled log canonical pairs

1. 

Fachrichtung Mathematik, Campus, Gebäude E2.4, Universität des Saarlandes, 66123 Saarbrücken, Germany

2. 

Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA

* Corresponding author: Vladimir Lazić

Received  November 2020 Revised  April 2021 Published  May 2021

We prove the Nonvanishing conjecture for uniruled projective log canonical pairs of dimension $ n $, assuming the Nonvanishing conjecture for smooth projective varieties in dimension $ n-1 $. We also show that the existence of good minimal models for non-uniruled projective klt pairs in dimension $ n $ implies the existence of good minimal models for projective log canonical pairs in dimension $ n $.

Citation: Vladimir Lazić, Fanjun Meng. On Nonvanishing for uniruled log canonical pairs. Electronic Research Archive, doi: 10.3934/era.2021039
References:
[1]

F. Ambro, The moduli $b$-divisor of an lc-trivial fibration, Compos. Math., 141 (2005), 385-403.  doi: 10.1112/S0010437X04001071.  Google Scholar

[2]

C. Birkar, Ascending chain condition for log canonical thresholds and termination of log flips, Duke Math. J., 136 (2007), 173-180.  doi: 10.1215/S0012-7094-07-13615-9.  Google Scholar

[3]

C. Birkar, On existence of log minimal models II, J. Reine Angew. Math., 658 (2011), 99-113.  doi: 10.1515/CRELLE.2011.062.  Google Scholar

[4]

C. BirkarP. CasciniC. D. Hacon and J. M$^{\rm{c}}$Kernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., 23 (2010), 405-468.  doi: 10.1090/S0894-0347-09-00649-3.  Google Scholar

[5]

S. BoucksomJ.-P. DemaillyM. Păun and Th. Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom., 22 (2013), 201-248.  doi: 10.1090/S1056-3911-2012-00574-8.  Google Scholar

[6]

S. R. Choi, The Geography of Log Models and its Applications, PhD Thesis, Johns Hopkins University, 2008.  Google Scholar

[7]

O. Debarre, Higher-Dimensional Algebraic Geometry, Universitext, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-5406-3.  Google Scholar

[8]

J.-P. DemaillyC. D. Hacon and M. Păun, Extension theorems, non-vanishing and the existence of good minimal models, Acta Math., 210 (2013), 203-259.  doi: 10.1007/s11511-013-0094-x.  Google Scholar

[9]

T. Dorsch and V. Lazić, A note on the abundance conjecture, Algebraic Geometry, 2 (2015), 476-488.  doi: 10.14231/AG-2015-020.  Google Scholar

[10]

O. Fujino, Special termination and reduction to pl flips, in Flips for 3-folds and 4-folds (ed. A. Corti), vol. 35 of Oxford Lecture Ser. Math. Appl., Oxford Univ. Press, 2007, 63-75. doi: 10.1093/acprof:oso/9780198570615.003.0004.  Google Scholar

[11]

O. Fujino, What is log terminal?, in Flips for 3-folds and 4-folds (ed. A. Corti), vol. 35 of Oxford Lecture Ser. Math. Appl., Oxford Univ. Press, 2007, 49-62. doi: 10.1093/acprof:oso/9780198570615.003.0003.  Google Scholar

[12]

O. Fujino, Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci., 47 (2011), 727-789.  doi: 10.2977/PRIMS/50.  Google Scholar

[13]

O. Fujino and Y. Gongyo, On canonical bundle formulas and subadjunctions, Michigan Math. J., 61 (2012), 255-264.  doi: 10.1307/mmj/1339011526.  Google Scholar

[14]

Y. Gongyo, On the minimal model theory for dlt pairs of numerical log Kodaira dimension zero, Math. Res. Lett., 18 (2011), 991-1000.  doi: 10.4310/MRL.2011.v18.n5.a16.  Google Scholar

[15]

Y. Gongyo and B. Lehmann, Reduction maps and minimal model theory, Compos. Math., 149 (2013), 295-308.  doi: 10.1112/S0010437X12000553.  Google Scholar

[16]

C. D. Hacon and C. Xu, On finiteness of B-representations and semi-log canonical abundance, in Minimal Models and Extremal Rays (Kyoto, 2011), vol. 70 of Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 2016,361–377.  Google Scholar

[17]

J. Han and Z. Li, Weak Zariski decompositions and log terminal models for generalized polarized pairs, arXiv: 1806.01234. Google Scholar

[18]

K. Hashizume, On the non-vanishing conjecture and existence of log minimal models, Publ. Res. Inst. Math. Sci., 54 (2018), 89-104.  doi: 10.4171/PRIMS/54-1-3.  Google Scholar

[19]

K. Hashizume and Z.-Y. Hu, On minimal model theory for log abundant lc pairs, J. Reine Angew. Math., 767 (2020), 109-159.  doi: 10.1515/crelle-2019-0032.  Google Scholar

[20]

Z. Hu, Log canonical pairs over varieties with maximal Albanese dimension, Pure Appl. Math. Q., 12 (2016), 543-571.  doi: 10.4310/PAMQ.2016.v12.n4.a5.  Google Scholar

[21]

Y. Kawamata, Pluricanonical systems on minimal algebraic varieties, Invent. Math., 79 (1985), 567-588.  doi: 10.1007/BF01388524.  Google Scholar

[22]

S. KeelK. Matsuki and J. M$^{\rm{c}}$Kernan, Log abundance theorem for threefolds, Duke Math. J., 75 (1994), 99-119.  doi: 10.1215/S0012-7094-94-07504-2.  Google Scholar

[23]

J. Kollár and S. J. Kovács, Log canonical singularities are Du Bois, J. Amer. Math. Soc., 23 (2010), 791-813.  doi: 10.1090/S0894-0347-10-00663-6.  Google Scholar

[24] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, vol. 134 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9780511662560.  Google Scholar
[25]

V. Lazić and Th. Peternell, Abundance for varieties with many differential forms, Épijournal Geom. Algébrique, 2 (2018), Article 1. doi: 10.46298/epiga.2018.volume2.3867.  Google Scholar

[26]

V. Lazić and Th. Peternell, On Generalised Abundance, II, Peking Math. J., 3 (2020), 1-46.  doi: 10.1007/s42543-019-00022-1.  Google Scholar

[27]

V. Lazić and N. Tsakanikas, On the existence of minimal models for log canonical pairs, arXiv: 1905.05576, to appear in Publ. Res. Inst. Math. Sci. Google Scholar

[28]

Y. Miyaoka, The Chern classes and Kodaira dimension of a minimal variety, in Algebraic Geometry, Sendai, 1985, vol. 10 of Adv. Stud. Pure Math., North-Holland, Amsterdam, 1987,449–476. doi: 10.2969/aspm/01010449.  Google Scholar

[29]

Y. Miyaoka, On the Kodaira dimension of minimal threefolds, Math. Ann., 281 (1988), 325-332.  doi: 10.1007/BF01458437.  Google Scholar

[30]

N. Nakayama, Zariski-Decomposition and Abundance, vol. 14 of MSJ Memoirs, Mathematical Society of Japan, Tokyo, 2004.  Google Scholar

[31]

V. V. Shokurov, $3$-fold log models, J. Math. Sci., 81 (1996), 2667-2699.  doi: 10.1007/BF02362335.  Google Scholar

show all references

References:
[1]

F. Ambro, The moduli $b$-divisor of an lc-trivial fibration, Compos. Math., 141 (2005), 385-403.  doi: 10.1112/S0010437X04001071.  Google Scholar

[2]

C. Birkar, Ascending chain condition for log canonical thresholds and termination of log flips, Duke Math. J., 136 (2007), 173-180.  doi: 10.1215/S0012-7094-07-13615-9.  Google Scholar

[3]

C. Birkar, On existence of log minimal models II, J. Reine Angew. Math., 658 (2011), 99-113.  doi: 10.1515/CRELLE.2011.062.  Google Scholar

[4]

C. BirkarP. CasciniC. D. Hacon and J. M$^{\rm{c}}$Kernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., 23 (2010), 405-468.  doi: 10.1090/S0894-0347-09-00649-3.  Google Scholar

[5]

S. BoucksomJ.-P. DemaillyM. Păun and Th. Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom., 22 (2013), 201-248.  doi: 10.1090/S1056-3911-2012-00574-8.  Google Scholar

[6]

S. R. Choi, The Geography of Log Models and its Applications, PhD Thesis, Johns Hopkins University, 2008.  Google Scholar

[7]

O. Debarre, Higher-Dimensional Algebraic Geometry, Universitext, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-5406-3.  Google Scholar

[8]

J.-P. DemaillyC. D. Hacon and M. Păun, Extension theorems, non-vanishing and the existence of good minimal models, Acta Math., 210 (2013), 203-259.  doi: 10.1007/s11511-013-0094-x.  Google Scholar

[9]

T. Dorsch and V. Lazić, A note on the abundance conjecture, Algebraic Geometry, 2 (2015), 476-488.  doi: 10.14231/AG-2015-020.  Google Scholar

[10]

O. Fujino, Special termination and reduction to pl flips, in Flips for 3-folds and 4-folds (ed. A. Corti), vol. 35 of Oxford Lecture Ser. Math. Appl., Oxford Univ. Press, 2007, 63-75. doi: 10.1093/acprof:oso/9780198570615.003.0004.  Google Scholar

[11]

O. Fujino, What is log terminal?, in Flips for 3-folds and 4-folds (ed. A. Corti), vol. 35 of Oxford Lecture Ser. Math. Appl., Oxford Univ. Press, 2007, 49-62. doi: 10.1093/acprof:oso/9780198570615.003.0003.  Google Scholar

[12]

O. Fujino, Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci., 47 (2011), 727-789.  doi: 10.2977/PRIMS/50.  Google Scholar

[13]

O. Fujino and Y. Gongyo, On canonical bundle formulas and subadjunctions, Michigan Math. J., 61 (2012), 255-264.  doi: 10.1307/mmj/1339011526.  Google Scholar

[14]

Y. Gongyo, On the minimal model theory for dlt pairs of numerical log Kodaira dimension zero, Math. Res. Lett., 18 (2011), 991-1000.  doi: 10.4310/MRL.2011.v18.n5.a16.  Google Scholar

[15]

Y. Gongyo and B. Lehmann, Reduction maps and minimal model theory, Compos. Math., 149 (2013), 295-308.  doi: 10.1112/S0010437X12000553.  Google Scholar

[16]

C. D. Hacon and C. Xu, On finiteness of B-representations and semi-log canonical abundance, in Minimal Models and Extremal Rays (Kyoto, 2011), vol. 70 of Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 2016,361–377.  Google Scholar

[17]

J. Han and Z. Li, Weak Zariski decompositions and log terminal models for generalized polarized pairs, arXiv: 1806.01234. Google Scholar

[18]

K. Hashizume, On the non-vanishing conjecture and existence of log minimal models, Publ. Res. Inst. Math. Sci., 54 (2018), 89-104.  doi: 10.4171/PRIMS/54-1-3.  Google Scholar

[19]

K. Hashizume and Z.-Y. Hu, On minimal model theory for log abundant lc pairs, J. Reine Angew. Math., 767 (2020), 109-159.  doi: 10.1515/crelle-2019-0032.  Google Scholar

[20]

Z. Hu, Log canonical pairs over varieties with maximal Albanese dimension, Pure Appl. Math. Q., 12 (2016), 543-571.  doi: 10.4310/PAMQ.2016.v12.n4.a5.  Google Scholar

[21]

Y. Kawamata, Pluricanonical systems on minimal algebraic varieties, Invent. Math., 79 (1985), 567-588.  doi: 10.1007/BF01388524.  Google Scholar

[22]

S. KeelK. Matsuki and J. M$^{\rm{c}}$Kernan, Log abundance theorem for threefolds, Duke Math. J., 75 (1994), 99-119.  doi: 10.1215/S0012-7094-94-07504-2.  Google Scholar

[23]

J. Kollár and S. J. Kovács, Log canonical singularities are Du Bois, J. Amer. Math. Soc., 23 (2010), 791-813.  doi: 10.1090/S0894-0347-10-00663-6.  Google Scholar

[24] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, vol. 134 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9780511662560.  Google Scholar
[25]

V. Lazić and Th. Peternell, Abundance for varieties with many differential forms, Épijournal Geom. Algébrique, 2 (2018), Article 1. doi: 10.46298/epiga.2018.volume2.3867.  Google Scholar

[26]

V. Lazić and Th. Peternell, On Generalised Abundance, II, Peking Math. J., 3 (2020), 1-46.  doi: 10.1007/s42543-019-00022-1.  Google Scholar

[27]

V. Lazić and N. Tsakanikas, On the existence of minimal models for log canonical pairs, arXiv: 1905.05576, to appear in Publ. Res. Inst. Math. Sci. Google Scholar

[28]

Y. Miyaoka, The Chern classes and Kodaira dimension of a minimal variety, in Algebraic Geometry, Sendai, 1985, vol. 10 of Adv. Stud. Pure Math., North-Holland, Amsterdam, 1987,449–476. doi: 10.2969/aspm/01010449.  Google Scholar

[29]

Y. Miyaoka, On the Kodaira dimension of minimal threefolds, Math. Ann., 281 (1988), 325-332.  doi: 10.1007/BF01458437.  Google Scholar

[30]

N. Nakayama, Zariski-Decomposition and Abundance, vol. 14 of MSJ Memoirs, Mathematical Society of Japan, Tokyo, 2004.  Google Scholar

[31]

V. V. Shokurov, $3$-fold log models, J. Math. Sci., 81 (1996), 2667-2699.  doi: 10.1007/BF02362335.  Google Scholar

[1]

Joaquín Delgado, Eymard Hernández–López, Lucía Ivonne Hernández–Martínez. Bautin bifurcation in a minimal model of immunoediting. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1397-1414. doi: 10.3934/dcdsb.2019233

[2]

Marc-Auréle Lagache, Ulysse Serres, Vincent Andrieu. Minimal time synthesis for a kinematic drone model. Mathematical Control & Related Fields, 2017, 7 (2) : 259-288. doi: 10.3934/mcrf.2017009

[3]

Virginia González-Vélez, Amparo Gil, Iván Quesada. Minimal state models for ionic channels involved in glucagon secretion. Mathematical Biosciences & Engineering, 2010, 7 (4) : 793-807. doi: 10.3934/mbe.2010.7.793

[4]

Christian Winkel, Simon Neumann, Christina Surulescu, Peter Scheurich. A minimal mathematical model for the initial molecular interactions of death receptor signalling. Mathematical Biosciences & Engineering, 2012, 9 (3) : 663-683. doi: 10.3934/mbe.2012.9.663

[5]

Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1769-1781. doi: 10.3934/dcdsb.2014.19.1769

[6]

Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046

[7]

Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511

[8]

Gabriel Ponce, Ali Tahzibi, Régis Varão. Minimal yet measurable foliations. Journal of Modern Dynamics, 2014, 8 (1) : 93-107. doi: 10.3934/jmd.2014.8.93

[9]

Virginie Bonnaillie-Noël, Corentin Léna. Spectral minimal partitions of a sector. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 27-53. doi: 10.3934/dcdsb.2014.19.27

[10]

Kristian Bjerklöv, Russell Johnson. Minimal subsets of projective flows. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 493-516. doi: 10.3934/dcdsb.2008.9.493

[11]

Romar dela Cruz, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. On the minimum number of minimal codewords. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020130

[12]

Nancy Guelman, Jorge Iglesias, Aldo Portela. Examples of minimal set for IFSs. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5253-5269. doi: 10.3934/dcds.2017227

[13]

Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. A minimal approach to the theory of global attractors. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2079-2088. doi: 10.3934/dcds.2012.32.2079

[14]

A. Giambruno and M. Zaicev. Minimal varieties of algebras of exponential growth. Electronic Research Announcements, 2000, 6: 40-44.

[15]

Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems & Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036

[16]

José Ginés Espín Buendía, Daniel Peralta-salas, Gabriel Soler López. Existence of minimal flows on nonorientable surfaces. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4191-4211. doi: 10.3934/dcds.2017178

[17]

Fernando Alcalde Cuesta, Ana Rechtman. Minimal Følner foliations are amenable. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 685-707. doi: 10.3934/dcds.2011.31.685

[18]

Boju Jiang, Jaume Llibre. Minimal sets of periods for torus maps. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 301-320. doi: 10.3934/dcds.1998.4.301

[19]

Bernard Helffer, Thomas Hoffmann-Ostenhof, Susanna Terracini. Nodal minimal partitions in dimension $3$. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 617-635. doi: 10.3934/dcds.2010.28.617

[20]

K. Tintarev. Critical values and minimal periods for autonomous Hamiltonian systems. Discrete & Continuous Dynamical Systems, 1995, 1 (3) : 389-400. doi: 10.3934/dcds.1995.1.389

 Impact Factor: 0.263

Article outline

[Back to Top]