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doi: 10.3934/era.2021040
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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 Early access 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

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