A reduction of canonical stability index of 4 and 5 dimensional projective varieties with large volume

Chen, Meng; Jiang, Zhi

Annales de l'Institut Fourier, Tome 67 (2017) no. 5, pp. 2043-2082.

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.

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.

Classification : 14E05, 14J35, 14J40
Keywords: canonical volumes, pluricanonical systems, extension theorems
Mot clés : volumes canoniques, systèmes pluricanoniques, théorèmes d’extension

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

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

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

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

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

[6] Chen, Meng Canonical stability of 3-folds of general type with $p_g\ge 3$, Int. J. Math., Volume 14 (2003) no. 5, pp. 515-528 | DOI | MR | Zbl

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

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

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

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

[11] Hacon, Christopher D.; McKernan, James Boundedness of pluricanonical maps of varieties of general type, Invent. Math., Volume 166 (2006) no. 1, pp. 1-25 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

[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., Volume 127 (1988) no. 2, pp. 309-316 | DOI | MR | Zbl

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

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

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

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

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

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

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