Explicit birational geometry of threefolds of general type, I

Chen, Jungkai A.; Chen, Meng

doi:10.24033/asens.2124

Explicit birational geometry of threefolds of general type, I
[Géométrie birationnelle explicite des variétés de type général de dimension 3, I]

Chen, Jungkai A. ; Chen, Meng

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 3, pp. 365-394

Abstract (VO)
Résumé

Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12} (V) : = dim H^{0} (V, 12 K_{V}) > 0$ and $P_{m_{0}} (V) > 1$ for some positive integer $m_{0} \leq 24$ . A direct consequence is the birationality of the pluricanonical map $ϕ_{m}$ for all $m \geq 126$ . Besides, the canonical volume $Vol (V)$ has a universal lower bound $ν (3) \geq \frac{1}{63 \cdot 126^{2}}$ .

Soit $V$ une variété non singulière complexe de type général et de dimension 3. Nous montrons $P_{12} (V) : = dim H^{0} (V, 12 K_{V}) > 0$ et $P_{m_{0}} (V) > 1$ pour un certain entier $m_{0} \leq 24$ . Une conséquence directe est la birationalité de l’application pluricanonique $ϕ_{m}$ pour tout $m \geq 126$ . De plus, le volume canonique $Vol (V)$ a un minorant universel $ν (3) \geq \frac{1}{63 \cdot 126^{2}}$ .

MR Zbl | 3 citations dans Numdam

DOI : 10.24033/asens.2124

Classification : 14J30, 14B05
Keywords: 3-folds, plurigenus
Mots-clés : variétés de dimension 3, plurigenre

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     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},
     pages = {365--394},
     year = {2010},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 43},
     number = {3},
     doi = {10.24033/asens.2124},
     mrnumber = {2667020},
     zbl = {1194.14060},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2124/}
}

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Chen, Jungkai A.; Chen, Meng. Explicit birational geometry of threefolds of general type, I. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 3, pp. 365-394. doi: 10.24033/asens.2124

Bibliographie
Cité par

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

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

[3] J. A. Chen & M. Chen, The canonical volume of 3-folds of general type with $χ \leq 0$ , J. Lond. Math. Soc. 78 (2008), 693-706. | Zbl | MR

[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. | Zbl | MR

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

[6] M. Chen, Canonical stability of 3-folds of general type with $p_{g} \geq 3$ , Internat. J. Math. 14 (2003), 515-528. | Zbl | MR

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

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

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

[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. | Zbl | MR

[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. | Zbl | MR

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

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

[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. | Zbl | MR

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

[16] Y. Kawamata, On the plurigenera of minimal algebraic $3$ -folds with $K \equiv 0$ , Math. Ann. 275 (1986), 539-546. | Zbl | MR

[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. | Zbl | MR

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

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

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

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

[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. | Zbl | MR

[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. | Zbl | MR

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

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

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

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