Explicit birational geometry of threefolds of general type, I
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 3, p. 365-394

Let V be a complex nonsingular projective 3-fold of general type. We prove P 12 (V):=dimH 0 (V,12K V )>0 and P m 0 (V)>1 for some positive integer m 0 24. A direct consequence is the birationality of the pluricanonical map ϕ m for all m126. Besides, the canonical volume Vol(V) has a universal lower bound ν(3)1 63·126 2 .

Soit V une variété non singulière complexe de type général et de dimension 3. Nous montrons P 12 (V):=dimH 0 (V,12K V )>0 et P m 0 (V)>1 pour un certain entier m 0 24. Une conséquence directe est la birationalité de l’application pluricanonique ϕ m pour tout m126. De plus, le volume canonique Vol(V) a un minorant universel ν(3)1 63·126 2 .

DOI : https://doi.org/10.24033/asens.2124
Classification:  14J30,  14B05
Keywords: 3-folds, plurigenus
@article{ASENS_2010_4_43_3_365_0,
     author = {Chen, Jungkai A. and Chen, Meng},
     title = {Explicit birational geometry of threefolds of general type, I},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 43},
     number = {3},
     year = {2010},
     pages = {365-394},
     doi = {10.24033/asens.2124},
     zbl = {1194.14060},
     mrnumber = {2667020},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2010_4_43_3_365_0}
}
Chen, Jungkai A.; Chen, Meng. Explicit birational geometry of threefolds of general type, I. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 3, pp. 365-394. doi : 10.24033/asens.2124. http://www.numdam.org/item/ASENS_2010_4_43_3_365_0/

[1] A. Beauville, Complex algebraic surfaces, London Mathematical Society Lecture Note Series 68, Cambridge Univ. Press, 1983. | MR 732439 | Zbl 0512.14020

[2] E. Bombieri, Canonical models of surfaces of general type, Publ. Math. I.H.É.S. 42 (1973), 171-219. | Numdam | MR 318163 | Zbl 0259.14005

[3] J. A. Chen & M. Chen, The canonical volume of 3-folds of general type with χ0, J. Lond. Math. Soc. 78 (2008), 693-706. | MR 2456899 | Zbl 1156.14009

[4] J. A. Chen, M. Chen & D.-Q. Zhang, The 5-canonical system on 3-folds of general type, J. reine angew. Math. 603 (2007), 165-181. | MR 2312557 | Zbl 1121.14029

[5] J. A. Chen & C. D. Hacon, Pluricanonical systems on irregular 3-folds of general type, Math. Z. 255 (2007), 343-355. | MR 2262735 | Zbl 1195.14018

[6] M. Chen, Canonical stability of 3-folds of general type with p g 3, Internat. J. Math. 14 (2003), 515-528. | MR 1993794 | Zbl 1070.14009

[7] M. Chen, On the -divisor method and its application, J. Pure Appl. Algebra 191 (2004), 143-156. | MR 2048311 | Zbl 1049.14034

[8] M. Chen, A sharp lower bound for the canonical volume of 3-folds of general type, Math. Ann. 337 (2007), 887-908. | MR 2285742 | Zbl 1124.14038

[9] M. Chen & K. Zuo, Complex projective 3-fold with non-negative canonical Euler-Poincaré characteristic, Comm. Anal. Geom. 16 (2008), 159-182. | MR 2411471 | Zbl 1149.14034

[10] L. Ein & R. Lazarsfeld, Global generation of pluricanonical and adjoint linear series on smooth projective threefolds, J. Amer. Math. Soc. 6 (1993), 875-903. | MR 1207013 | Zbl 0803.14004

[11] A. R. Fletcher, Contributions to Riemann-Roch on projective 3-folds with only canonical singularities and applications, in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46, Amer. Math. Soc., 1987, 221-231. | MR 927958 | Zbl 0662.14026

[12] A. R. Fletcher, Inverting Reid's exact plurigenera formula, Math. Ann. 284 (1989), 617-629. | MR 1006376 | Zbl 0661.14013

[13] C. D. Hacon & J. Mckernan, Boundedness of pluricanonical maps of varieties of general type, Invent. Math. 166 (2006), 1-25. | MR 2242631 | Zbl 1121.14011

[14] A. R. Iano-Fletcher, Working with weighted complete intersections, in Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser. 281, Cambridge Univ. Press, 2000, 101-173. | MR 1798982 | Zbl 0960.14027

[15] Y. Kawamata, A generalization of Kodaira-Ramanujam's vanishing theorem, Math. Ann. 261 (1982), 43-46. | MR 675204 | Zbl 0476.14007

[16] Y. Kawamata, On the plurigenera of minimal algebraic 3-folds with K0, Math. Ann. 275 (1986), 539-546. | MR 859328 | Zbl 0582.14015

[17] Y. Kawamata, K. Matsuda & K. Matsuki, Introduction to the minimal model problem, in Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math. 10, North-Holland, 1987, 283-360. | MR 946243 | Zbl 0672.14006

[18] J. Kollár, Higher direct images of dualizing sheaves. I, Ann. of Math. 123 (1986), 11-42. | MR 825838 | Zbl 0598.14015

[19] J. Kollár & S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge Univ. Press, 1998. | MR 1658959 | Zbl 0926.14003

[20] T. Luo, Global 2-forms on regular 3-folds of general type, Duke Math. J. 71 (1993), 859-869. | MR 1240606 | Zbl 0838.14032

[21] M. Reid, Canonical 3-folds, in Journées de Géométrie Algébrique d'Angers, juillet 1979, Sijthoff & Noordhoff, 1980, 273-310. | MR 605348 | Zbl 0451.14014

[22] M. Reid, Minimal models of canonical 3-folds, in Algebraic varieties and analytic varieties (Tokyo, 1981), Adv. Stud. Pure Math. 1, North-Holland, 1983, 131-180. | MR 715649 | Zbl 0558.14028

[23] M. Reid, Young person's guide to canonical singularities, in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46, Amer. Math. Soc., 1987, 345-414. | MR 927963 | Zbl 0634.14003

[24] S. Takayama, Pluricanonical systems on algebraic varieties of general type, Invent. Math. 165 (2006), 551-587. | MR 2242627 | Zbl 1108.14031

[25] H. Tsuji, Pluricanonical systems of projective varieties of general type. I, Osaka J. Math. 43 (2006), 967-995. | MR 2303558 | Zbl 1142.14012

[26] E. Viehweg, Vanishing theorems, J. reine angew. Math. 335 (1982), 1-8. | MR 667459 | Zbl 0485.32019