Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs

Birkar, Caucher; Zhang, De-Qi

Publications Mathématiques de l'IHÉS, Tome 123 (2016), pp. 283-331.

For every smooth complex projective variety $W$ of dimension $d$ and nonnegative Kodaira dimension, we show the existence of a universal constant $m$ depending only on $d$ and two natural invariants of the very general fibres of an Iitaka fibration of $W$ such that the pluricanonical system $|m{K}_W|$ defines an Iitaka fibration. This is a consequence of a more general result on polarized adjoint divisors. In order to prove these results we develop a generalized theory of pairs, singularities, log canonical thresholds, adjunction, etc.

[1.] Birkar, C. On existence of log minimal models, Compos. Math., Volume 145 (2009), pp. 1442-1446 | DOI | MR | Zbl

[2.] Birkar, C. Existence of log canonical flips and a special LMMP, Publ. Math. Inst. Hautes Études Sci., Volume 115 (2012), pp. 325-368 | DOI | Numdam | MR | Zbl

[3.] Birkar, C.; Cascini, P.; Hacon, C.; McKernan, J. Existence of minimal models for varieties of log general type, J. Am. Math. Soc., Volume 23 (2010), pp. 405-468 | DOI | MR | Zbl

[4.] Birkar, C.; Hu, Z. Log canonical pairs with good augmented base loci, Compos. Math., Volume 150 (2014), pp. 579-592 | DOI | MR | Zbl

[5.] Di Cerbo, G. Uniform bounds for the Iitaka fibration, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume 13 (2014), pp. 1133-1143 | MR | Zbl

[6.] Chen, J.; Chen, M. Explicit birational geometry of threefolds of general type, I, Ann. Sci. Éc. Norm. Super., Volume 43 (2010), pp. 365-394 | DOI | Numdam | MR | Zbl

[7.] Fujino, O.; Mori, S. A canonical bundle formula, J. Differ. Geom., Volume 56 (2000), pp. 167-188 | DOI | MR | Zbl

[8.] Hacon, C.; McKernan, J. Boundedness of pluricanonical maps of varieties of general type, Invent. Math., Volume 166 (2006), pp. 1-25 | DOI | MR | Zbl

[9.] Hacon, C. D.; McKernan, J.; Xu, C. On the birational automorphisms of varieties of general type, Ann. Math. (2), Volume 177 (2013), pp. 1077-1111 | DOI | MR | Zbl

[10.] Hacon, C. D.; McKernan, J.; Xu, C. ACC for log canonical thresholds, Ann. Math. (2), Volume 180 (2014), pp. 523-571 | DOI | MR | Zbl

[11.] C. D. Hacon and C. Xu, Boundedness of log Calabi-Yau pairs of Fano type, Math. Res. Lett. (to appear), | arXiv

[12.] Iitaka, S. Deformations of compact complex surfaces, II, J. Math. Soc. Jpn., Volume 22 (1970), pp. 247-261 | DOI | MR | Zbl

[13.] Jiang, X. On the pluricanonical maps of varieties of intermediate Kodaira dimension, Math. Ann., Volume 356 (2013), pp. 979-1004 | DOI | MR | Zbl

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

[15.] Kawamata, Y. On the length of an extremal rational curve, Invent. Math., Volume 105 (1991), pp. 609-611 | DOI | MR | Zbl

[16.] Kawamata, Y. Subadjunction of log canonical divisors. II, Am. J. Math., Volume 120 (1998), pp. 893-899 | DOI | MR | Zbl

[17.] J. Kollár, et al., Flips and abundance for algebraic threefolds, Astérisque, 211 (1992).

[18.] Kollár, J.; Mori, S. Birational geometry of algebraic varieties (1998) | DOI | Zbl

[19.] Pacienza, G. On the uniformity of the Iitaka fibration, Math. Res. Lett., Volume 16 (2009), pp. 663-681 | DOI | MR | Zbl

[20.] Shokurov, V. V. 3-fold log flips, With an appendix by Yujiro Kawamata, Russ. Acad. Sci. Izv. Math., Volume 40 (1993), pp. 95-202 | MR | Zbl

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

[22.] Todorov, G.; Xu, C. On Effective Log Iitaka Fibration for 3-folds and 4-folds, Algebra Number Theory, Volume 3 (2009), pp. 697-710 | DOI | MR | Zbl

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

[24.] Viehweg, E.; Zhang, D.-Q. Effective Iitaka fibrations, J. Algebraic Geom., Volume 18 (2009), pp. 711-730 | DOI | MR | Zbl