Numerical character of the effectivity of adjoint line bundles
[Le caractère numérique de l’effectivité des systèmes linéaires adjointes]
Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 107-119.

Dans cette note nous montrons que le système linéaire adjoint associé à une paire log-canonique est non-vide dés que la classe de Chern de ce système contient un diviseur effectif dont les coefficients sont rationnels. Nous en déduisons quelques corollaires immédiats.

In this note we show that, for any log-canonical pair (X,Δ), K X +Δ is -effective if its Chern class contains an effective -divisor. Then, we derive some direct corollaries.

DOI : https://doi.org/10.5802/aif.2701
Classification : 14E30
Mots clés : paires log-canoniques, systèmes adjoints, recouvrement ramifié
@article{AIF_2012__62_1_107_0,
     author = {Campana, Fr\'ed\'eric and Koziarz, Vincent and P\u{a}un, Mihai},
     title = {Numerical character of the effectivity of adjoint line bundles},
     journal = {Annales de l'Institut Fourier},
     pages = {107--119},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {1},
     year = {2012},
     doi = {10.5802/aif.2701},
     mrnumber = {2986267},
     zbl = {1250.14009},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2701/}
}
Campana, Frédéric; Koziarz, Vincent; Păun, Mihai. Numerical character of the effectivity of adjoint line bundles. Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 107-119. doi : 10.5802/aif.2701. http://archive.numdam.org/articles/10.5802/aif.2701/

[1] Arapura, Donu Higgs line bundles, Green-Lazarsfeld sets, and maps of Kähler manifolds to curves, Bull. Amer. Math. Soc., Volume 26 (1992) no. 2, pp. 310-314 | Article | MR 1129312 | Zbl 0759.14016

[2] Arapura, Donu Geometry of cohomology support loci for local systems. I, J. Algebraic Geom., Volume 6 (1997) no. 3, pp. 563-597 | MR 1487227 | Zbl 0923.14010

[3] Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; McKernan, James Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., Volume 23 (2010) no. 2, pp. 405-468 | Article | MR 2601039

[4] Budur, Nero Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers, Adv. Math., Volume 221 (2009) no. 1, pp. 217-250 | Article | MR 2509325

[5] Campana, F.; Peternell, T.; Toma, M. Geometric stability of the cotangent bundle and the universal cover of a projective manifold (arXiv:math/0405093, to appear in Bull. Soc. Math. France) | Numdam | MR 2815027

[6] Chen, J.; Hacon, C. On the irregularity of the image of the Iitaka fibration, Comm. in Algebra, Volume 32 (2004) no. 1, pp. 203-215 | Article | MR 2036231

[7] Esnault, Hélène; Viehweg, Eckart Logarithmic de Rham complexes and vanishing theorems, Invent. Math., Volume 86 (1986) no. 1, pp. 161-194 | Article | MR 853449 | Zbl 0603.32006

[8] Fujino, O. On Kawamata’s theorem (arXiv:0910.1156)

[9] Fukuda, S. An elementary semi-ampleness result for log-canonical divisors (arXiv:1003,1388)

[10] Gongyo, Y. Abundance theorem for numerically trivial log canonical divisors of semi-log canonical pairs (arXiv:1005.2796)

[11] Kawamata, Y. On the abundance theorem in the case ν=0 (arXiv:1002.2682)

[12] Kawamata, Y. Pluricanonical systems on minimal algebraic varieties, Invent. Math., Volume 79 (1985) no. 3, pp. 567-588 | Article | MR 782236 | Zbl 0593.14010

[13] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge, 1998 (With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original) | Article | MR 1658959 | Zbl 0926.14003

[14] Nakayama, Noboru Zariski-decomposition and abundance, MSJ Memoirs, 14, Mathematical Society of Japan, Tokyo, 2004 | MR 2104208

[15] Păun, M Relative critical exponents, non-vanishing and metrics with minimal singularities (arXiv:0807.3109)

[16] Shokurov, V. V. A nonvanishing theorem, Izv. Akad. Nauk SSSR Ser. Mat., Volume 49 (1985) no. 3, pp. 635-651 | MR 794958 | Zbl 0605.14006

[17] Simpson, C. Subspaces of moduli spaces of rank one local systems, Ann. Sci. E.N.S. (4), Volume 26 (1993) no. 3, pp. 361-401 | Numdam | MR 1222278 | Zbl 0798.14005