A reduction of canonical stability index of 4 and 5 dimensional projective varieties with large volume
Annales de l'Institut Fourier, Volume 67 (2017) no. 5, p. 2043-2082

We study the canonical stability index of nonsingular projective varieties of general type with either large canonical volume or large geometric genus. As applications of a general extension theorem established in the first part, we prove some optimal results in dimensions 4 and 5, which are parallel to some well-known results on surfaces and 3-folds.

Nous étudions l’indice de stabilité canonique d’une variété projective lisse de type général avec un grand volume canonique ou un grand genre géométrique. Comme applications d’un théorème général d’extension établi dans la première partie, nous prouvons des résultats optimaux en dimensions 4 et 5 similaires à certains résultats bien connus sur les surfaces et les variétés de dimension 3.

Received : 2016-05-12
Revised : 2016-12-09
Accepted : 2017-01-24
Published online : 2017-11-17
DOI : https://doi.org/10.5802/aif.3129
Classification:  14E05,  14J35,  14J40
Keywords: canonical volumes, pluricanonical systems, extension theorems
@article{AIF_2017__67_5_2043_0,
     author = {Chen, Meng and Jiang, Zhi},
     title = {A reduction of canonical stability index of 4 and 5 dimensional projective varieties with large volume},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {5},
     year = {2017},
     pages = {2043-2082},
     doi = {10.5802/aif.3129},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2017__67_5_2043_0}
}
Chen, Meng; Jiang, Zhi. A reduction of canonical stability index of 4 and 5 dimensional projective varieties with large volume. Annales de l'Institut Fourier, Volume 67 (2017) no. 5, pp. 2043-2082. doi : 10.5802/aif.3129. http://www.numdam.org/item/AIF_2017__67_5_2043_0/

[1] Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; Mckernan, James Existence of minimal models for varieties of log general type, J. Am. Math. Soc., Tome 23 (2010) no. 2, pp. 405-468 | Article | MR 2601039 | Zbl 1210.14019

[2] Bombieri, Enrico Canonical models of surfaces of general type, Publ. Math., Inst. Hautes Étud. Sci. (1972) no. 42, pp. 171-219 | MR 0318163 | Zbl 0259.14005

[3] Brown, Gavin; Kasprzyk, Alexander Four-dimensional projective orbifold hypersurfaces, Exp. Math., Tome 25 (2016) no. 2, pp. 176-193 | Article | MR 3463567 | Zbl 1343.14039

[4] Chen, Jungkai A.; Chen, Meng Explicit birational geometry of 3-folds of general type, II, J. Differ. Geom., Tome 86 (2010) no. 2, pp. 237-271 http://projecteuclid.org/euclid.jdg/1299766788 | Article | MR 2772551 | Zbl 1218.14026

[5] Chen, Jungkai A.; Chen, Meng Explicit birational geometry of 3-folds and 4-folds of general type, III, Compos. Math., Tome 151 (2015) no. 6, pp. 1041-1082 | Article | MR 3357178 | Zbl 1329.14030

[6] Chen, Meng Canonical stability of 3-folds of general type with p g 3, Int. J. Math., Tome 14 (2003) no. 5, pp. 515-528 | Article | MR 1993794 | Zbl 1070.14009

[7] Chen, Meng On pluricanonical systems of algebraic varieties of general type, Algebraic geometry in East Asia—Seoul 2008, Math. Soc. Japan, Tokyo (Advanced Studies in Pure Mathematics) Tome 60 (2010), pp. 215-236 | MR 2761928 | Zbl 1214.14009

[8] Chen, Meng On an efficient induction step with Nklt(X,D) – notes to Todorov, Commun. Anal. Geom., Tome 20 (2012) no. 4, pp. 765-779 | Article | MR 2981839 | Zbl 1261.14022

[9] Debarre, Olivier Systèmes pluricanoniques sur les variétés de type général (d’après Hacon-McKernan, Takayama, Tsuji), Séminaire Bourbaki. Vol. 2006/2007, Société Mathématique de France (Astérisque) Tome 317 (2008), pp. 119-140 (Exp. No. 970, vii) | MR 2487732 | Zbl 1151.14031

[10] Di Biagio, Lorenzo Pluricanonical systems for 3-folds and 4-folds of general type, Math. Proc. Camb. Philos. Soc., Tome 152 (2012) no. 1, pp. 9-34 | Article | MR 2860415 | Zbl 1232.14024

[11] Hacon, Christopher D.; Mckernan, James Boundedness of pluricanonical maps of varieties of general type, Invent. Math., Tome 166 (2006) no. 1, pp. 1-25 | Article | MR 2242631 | Zbl 1121.14011

[12] Hacon, Christopher D.; Mckernan, James; Xu, Chenyang On the birational automorphisms of varieties of general type, Ann. Math., Tome 177 (2013) no. 3, pp. 1077-1111 | Article | MR 3034294 | Zbl 1281.14036

[13] Iano-Fletcher, Anthony R. Working with weighted complete intersections, Explicit birational geometry of 3-folds, Cambridge University Press. (London Mathematical Society Lecture Note Series) Tome 281 (2000), pp. 101-173 | MR 1798982 | Zbl 0960.14027

[14] Kawamata, Yujiro A generalization of Kodaira-Ramanujam’s vanishing theorem, Math. Ann., Tome 261 (1982) no. 1, pp. 43-46 | Article | MR 675204 | Zbl 0476.14007

[15] Kawamata, Yujiro On Fujita’s freeness conjecture for 3-folds and 4-folds, Math. Ann., Tome 308 (1997) no. 3, pp. 491-505 | Article | MR 1457742 | Zbl 0909.14001

[16] Kawamata, Yujiro On the extension problem of pluricanonical forms, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), 193–207, American Mathematical Society (Contemporary Mathematics) Tome 241 (1999), pp. 193-207 | Zbl 0972.14005

[17] Kobayashi, Masanori On Noether’s inequality for threefolds, J. Math. Soc. Japan, Tome 44 (1992) no. 1, pp. 145-156 | Article | MR 1139663 | Zbl 0766.14033

[18] Lazarsfeld, Robert Positivity in algebraic geometry. I Classical setting: line bundles and linear series, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3., Tome 48 (2004), xviii+387 pages | Article | MR 2095471 | Zbl 1093.14501

[19] Lazarsfeld, Robert Positivity in algebraic geometry. II Positivity for vector bundles, and multiplier ideals., Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3., Tome 49 (2004), xviii+385 pages (Positivity for vector bundles, and multiplier ideals) | Article | MR 2095472 | Zbl 1093.14500

[20] Mckernan, James Boundedness of log terminal Fano pairs of bounded index (2002) (https://arxiv.org/abs/math/0205214 )

[21] Reider, Igor Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. Math., Tome 127 (1988) no. 2, pp. 309-316 | Article | MR 932299 | Zbl 0663.14010

[22] Siu, Yum-Tong Invariance of plurigenera, Invent. Math., Tome 134 (1998) no. 3, pp. 661-673 | Article | MR 1660941 | Zbl 0955.32017

[23] Siu, Yum-Tong Finite generation of canonical ring by analytic method, Sci. China, Ser. A, Tome 51 (2008) no. 4, pp. 481-502 | Article | MR 2395400 | Zbl 1153.32021

[24] Takayama, Shigeharu Pluricanonical systems on algebraic varieties of general type, Invent. Math., Tome 165 (2006) no. 3, pp. 551-587 | Article | MR 2242627 | Zbl 1108.14031

[25] Todorov, Gueorgui Tomov Pluricanonical maps for threefolds of general type, Ann. Inst. Fourier, Tome 57 (2007) no. 4, pp. 1315-1330 http://aif.cedram.org/item?id=AIF_2007__57_4_1315_0 | Article | MR 2339333 | Zbl 1122.14031

[26] Tsuji, Hajime Pluricanonical systems of projective varieties of general type. I, Osaka J. Math., Tome 43 (2006) no. 4, pp. 967-995 http://projecteuclid.org/euclid.ojm/1165850044 | MR 2303558 | Zbl 1142.14012

[27] Viehweg, Eckart Vanishing theorems, J. Reine Angew. Math., Tome 335 (1982), pp. 1-8 | Article | MR 667459 | Zbl 0485.32019

[28] Xu, Jinsong The third and fourth pluricanonical maps of threefolds of general type, Math. Proc. Camb. Philos. Soc., Tome 157 (2014) no. 2, pp. 209-220 | Article | MR 3254589 | Zbl 1328.14023