doi: 10.3934/era.2021040

On minimal 4-folds of general type with $ p_g \geq 2 $

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  January 2021 Revised  March 2021 Published  May 2021

We show that, for any nonsingular projective 4-fold $ V $ of general type with geometric genus $ p_g\geq 2 $, the pluricanonical map $ \varphi_{33} $ is birational onto the image and the canonical volume $ {\rm Vol}(V) $ has the lower bound $ \frac{1}{480} $, which improves a previous theorem by Chen and Chen.

Citation: Jianshi Yan. On minimal 4-folds of general type with $ p_g \geq 2 $. Electronic Research Archive, doi: 10.3934/era.2021040
References:
[1]

C. BirkarP. CasciniC. D. Hacon and J. McKernan, 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

[2]

E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math., 42 (1973), 171-219.   Google Scholar

[3]

G. Brown and A. Kasprzyk, Four-dimensional projective orbifold hypersurfaces, Exp. Math., 25 (2016), 176-193.  doi: 10.1080/10586458.2015.1054054.  Google Scholar

[4]

J. A. Chen and M. Chen, Explicit birational geometry of threefolds of general type, Ⅰ, Ann. Sci. Éc. Norm. Supér., 43 (2010), 365-394.  doi: 10.24033/asens.2124.  Google Scholar

[5]

J. A. Chen and M. Chen, Explicit birational geometry of threefolds of general type, Ⅱ, J. Differ. Geom., 86 (2010), 237-271.  doi: 10.4310/jdg/1299766788.  Google Scholar

[6]

J. A. Chen and M. Chen, Explicit birational geometry for 3-folds and 4-folds of general type, Ⅲ, Compos. Math., 151 (2015), 1041-1082.  doi: 10.1112/S0010437X14007817.  Google Scholar

[7]

M. Chen, Canonical stability in terms of singularity index for algebraic threefolds, Math. Proc. Cambridge Philos. Soc., 131 (2001), 241-264.  doi: 10.1017/S030500410100531X.  Google Scholar

[8]

M. Chen, A sharp lower bound for the canonical volume of 3-folds of general type, Math. Ann., 337 (2007), 887-908.  doi: 10.1007/s00208-006-0060-4.  Google Scholar

[9]

M. Chen, On pluricanonical systems of algebraic varieties of general type, in Algebraic Geometry in East Asia–Seoul 2008, Adv. Stud. Pure Math., 60 (Mathematical Society of Japan, Tokyo, 2010), 215–236. doi: 10.2969/aspm/06010215.  Google Scholar

[10]

M. Chen, Some birationality criteria on 3-folds with $p_g>1$, Sci. China Math., 57 (2014), 2215-2234.  doi: 10.1007/s11425-014-4890-3.  Google Scholar

[11]

M. Chen, On minimal 3-folds of general type with maximal pluricanonical section index, Asian J. Math., 22 (2018), 257-268.  doi: 10.4310/AJM.2018.v22.n2.a3.  Google Scholar

[12]

M. Chen, Y. Hu and M. Penegini, On projective threefolds of general type with small positive geometric genus, Electron. Res. Arch., 29 (2021), 2293–2323, arXiv: 1710.07799. doi: 10.3934/era.2020117.  Google Scholar

[13]

C. D. Hacon and J. McKernan, Boundedness of pluricanonical maps of varieties of general type, Invent. Math., 166 (2006), 1-25.  doi: 10.1007/s00222-006-0504-1.  Google Scholar

[14]

Y. Kawamata, A generalization of Kodaira-Ramanujam's vanishing theorem, Math. Ann., 261 (1982), 43-46.  doi: 10.1007/BF01456407.  Google Scholar

[15]

Y. Kawamata, On the extension problem of pluricanonical forms, in Algebraic Geometry: Hirzebruch 70 (Warsaw, 1998), Contemp. Math., 241 (American Mathematical Society, Providence, RI, 1999), 193–207. doi: 10.1090/conm/241/03636.  Google Scholar

[16]

Y. KawamataK. Matsuda and K. Matsuki, Introduction to the minimal model problem, Adv. Stud. Pure Math., 10 (1987), 283-360.  doi: 10.2969/aspm/01010283.  Google Scholar

[17] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9780511662560.  Google Scholar
[18]

Y.-T. Siu, Finite generation of canonical ring by analytic method, Sci. China Ser. A, 51 (2008), 481-502.  doi: 10.1007/s11425-008-0073-4.  Google Scholar

[19]

S. Takayama, Pluricanonical systems on algebraic varieties of general type, Invent. Math., 165 (2006), 551-587.  doi: 10.1007/s00222-006-0503-2.  Google Scholar

[20]

H. Tsuji, Pluricanonical systems of projective varieties of general type. I, Osaka J. Math., 43 (2006), 967-995.   Google Scholar

[21]

E. Viehweg, Vanishing theorems, J. Reine Angew. Math., 335 (1982), 1-8.  doi: 10.1515/crll.1982.335.1.  Google Scholar

show all references

References:
[1]

C. BirkarP. CasciniC. D. Hacon and J. McKernan, 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

[2]

E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math., 42 (1973), 171-219.   Google Scholar

[3]

G. Brown and A. Kasprzyk, Four-dimensional projective orbifold hypersurfaces, Exp. Math., 25 (2016), 176-193.  doi: 10.1080/10586458.2015.1054054.  Google Scholar

[4]

J. A. Chen and M. Chen, Explicit birational geometry of threefolds of general type, Ⅰ, Ann. Sci. Éc. Norm. Supér., 43 (2010), 365-394.  doi: 10.24033/asens.2124.  Google Scholar

[5]

J. A. Chen and M. Chen, Explicit birational geometry of threefolds of general type, Ⅱ, J. Differ. Geom., 86 (2010), 237-271.  doi: 10.4310/jdg/1299766788.  Google Scholar

[6]

J. A. Chen and M. Chen, Explicit birational geometry for 3-folds and 4-folds of general type, Ⅲ, Compos. Math., 151 (2015), 1041-1082.  doi: 10.1112/S0010437X14007817.  Google Scholar

[7]

M. Chen, Canonical stability in terms of singularity index for algebraic threefolds, Math. Proc. Cambridge Philos. Soc., 131 (2001), 241-264.  doi: 10.1017/S030500410100531X.  Google Scholar

[8]

M. Chen, A sharp lower bound for the canonical volume of 3-folds of general type, Math. Ann., 337 (2007), 887-908.  doi: 10.1007/s00208-006-0060-4.  Google Scholar

[9]

M. Chen, On pluricanonical systems of algebraic varieties of general type, in Algebraic Geometry in East Asia–Seoul 2008, Adv. Stud. Pure Math., 60 (Mathematical Society of Japan, Tokyo, 2010), 215–236. doi: 10.2969/aspm/06010215.  Google Scholar

[10]

M. Chen, Some birationality criteria on 3-folds with $p_g>1$, Sci. China Math., 57 (2014), 2215-2234.  doi: 10.1007/s11425-014-4890-3.  Google Scholar

[11]

M. Chen, On minimal 3-folds of general type with maximal pluricanonical section index, Asian J. Math., 22 (2018), 257-268.  doi: 10.4310/AJM.2018.v22.n2.a3.  Google Scholar

[12]

M. Chen, Y. Hu and M. Penegini, On projective threefolds of general type with small positive geometric genus, Electron. Res. Arch., 29 (2021), 2293–2323, arXiv: 1710.07799. doi: 10.3934/era.2020117.  Google Scholar

[13]

C. D. Hacon and J. McKernan, Boundedness of pluricanonical maps of varieties of general type, Invent. Math., 166 (2006), 1-25.  doi: 10.1007/s00222-006-0504-1.  Google Scholar

[14]

Y. Kawamata, A generalization of Kodaira-Ramanujam's vanishing theorem, Math. Ann., 261 (1982), 43-46.  doi: 10.1007/BF01456407.  Google Scholar

[15]

Y. Kawamata, On the extension problem of pluricanonical forms, in Algebraic Geometry: Hirzebruch 70 (Warsaw, 1998), Contemp. Math., 241 (American Mathematical Society, Providence, RI, 1999), 193–207. doi: 10.1090/conm/241/03636.  Google Scholar

[16]

Y. KawamataK. Matsuda and K. Matsuki, Introduction to the minimal model problem, Adv. Stud. Pure Math., 10 (1987), 283-360.  doi: 10.2969/aspm/01010283.  Google Scholar

[17] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9780511662560.  Google Scholar
[18]

Y.-T. Siu, Finite generation of canonical ring by analytic method, Sci. China Ser. A, 51 (2008), 481-502.  doi: 10.1007/s11425-008-0073-4.  Google Scholar

[19]

S. Takayama, Pluricanonical systems on algebraic varieties of general type, Invent. Math., 165 (2006), 551-587.  doi: 10.1007/s00222-006-0503-2.  Google Scholar

[20]

H. Tsuji, Pluricanonical systems of projective varieties of general type. I, Osaka J. Math., 43 (2006), 967-995.   Google Scholar

[21]

E. Viehweg, Vanishing theorems, J. Reine Angew. Math., 335 (1982), 1-8.  doi: 10.1515/crll.1982.335.1.  Google Scholar

[1]

Inês Cruz, Helena Mena-Matos, Esmeralda Sousa-Dias. The group of symplectic birational maps of the plane and the dynamics of a family of 4D maps. Journal of Geometric Mechanics, 2020, 12 (3) : 363-375. doi: 10.3934/jgm.2020010

[2]

Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 887-912. doi: 10.3934/dcdsb.2018047

[3]

Salim A. Messaoudi, Muhammad I. Mustafa. A general stability result in a memory-type Timoshenko system. Communications on Pure & Applied Analysis, 2013, 12 (2) : 957-972. doi: 10.3934/cpaa.2013.12.957

[4]

. Publisher Correction to: Probability, uncertainty and quantitative risk, volume 4. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 7-. doi: 10.1186/s41546-019-0041-7

[5]

Masoud Sabzevari, Joël Merker, Samuel Pocchiola. Canonical Cartan connections on maximally minimal generic submanifolds $\mathbf{M^5 \subset \mathbb{C}^4}$. Electronic Research Announcements, 2014, 21: 153-166. doi: 10.3934/era.2014.21.153

[6]

Stephen Baigent. Convex geometry of the carrying simplex for the May-Leonard map. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1697-1723. doi: 10.3934/dcdsb.2018288

[7]

Thomas Feulner. The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes. Advances in Mathematics of Communications, 2009, 3 (4) : 363-383. doi: 10.3934/amc.2009.3.363

[8]

Cuncai Hua, Guanrong Chen, Qunhong Li, Juhong Ge. Converting a general 3-D autonomous quadratic system to an extended Lorenz-type system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 475-488. doi: 10.3934/dcdsb.2011.16.475

[9]

Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789

[10]

Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021003

[11]

Jungkai A. Chen and Meng Chen. On projective threefolds of general type. Electronic Research Announcements, 2007, 14: 69-73. doi: 10.3934/era.2007.14.69

[12]

Thomas Feulner. Canonization of linear codes over $\mathbb Z$4. Advances in Mathematics of Communications, 2011, 5 (2) : 245-266. doi: 10.3934/amc.2011.5.245

[13]

Sami Baraket, Soumaya Sâanouni, Nihed Trabelsi. Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 1013-1063. doi: 10.3934/dcds.2020069

[14]

Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257

[15]

José R. Quintero, Alex M. Montes. Exact controllability and stabilization for a general internal wave system of Benjamin-Ono type. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021021

[16]

Xinlin Cao, Huaian Diao, Jinhong Li. Some recent progress on inverse scattering problems within general polyhedral geometry. Electronic Research Archive, 2021, 29 (1) : 1753-1782. doi: 10.3934/era.2020090

[17]

Y. Kabeya, Eiji Yanagida, Shoji Yotsutani. Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems. Communications on Pure & Applied Analysis, 2002, 1 (1) : 85-102. doi: 10.3934/cpaa.2002.1.85

[18]

Nanhee Kim. Uniqueness and Hölder type stability of continuation for the linear thermoelasticity system with residual stress. Evolution Equations & Control Theory, 2013, 2 (4) : 679-693. doi: 10.3934/eect.2013.2.679

[19]

Hong Man, Yibin Yu, Yuebang He, Hui Huang. Design of one type of linear network prediction controller for multi-agent system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 727-734. doi: 10.3934/dcdss.2019047

[20]

Zhiying Qin, Jichen Yang, Soumitro Banerjee, Guirong Jiang. Border-collision bifurcations in a generalized piecewise linear-power map. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 547-567. doi: 10.3934/dcdsb.2011.16.547

 Impact Factor: 0.263

Metrics

  • PDF downloads (11)
  • HTML views (35)
  • Cited by (0)

Other articles
by authors

[Back to Top]