[Lieux de support cohomologiques et systèmes pluricanoniques sur des variétés irrégulières]
Étant donné une variété irrégulière $X$ de type général, nous montrons que si une fibre générale du morphisme d’Albanese de $X$ satisfait certaines conditions de la théorie de Hodge, le lieu $V^0(K_X)$ engendre $\mathrm{Pic} ^0(X)$. Nous montrons ensuite que la condition que $V^0(K_X)$ engendre $\mathrm{Pic} ^0(X)$ peut souvent être appliquée pour prouver la birationalité de certaines applications pluricanoniques de $X$.
Given an irregular variety $X$ of general type, we show that if a general fiber $F$ of the Albanese morphism of $X$ satisfies certain Hodge theoretic conditions, the $0$-th cohomological support locus of $K_X$ generates $\mathrm{Pic}^0(X)$. We then show that the condition that the $0$-th cohomological support locus of $K_X$ generates $\mathrm{Pic} ^0(X)$ can often be applied to prove the birationality of certain pluricanonical maps of $X$.
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Keywords: Cohomological support loci, Pluricanonical maps, Generic vanishing.
Mots-clés : Lieux de support cohomologiques, applications pluricanoniques, annulation générique.
Jiang, Zhi 1
CC-BY-ND 4.0
@article{AIF_2025__75_1_359_0,
author = {Jiang, Zhi},
title = {Cohomological support loci and {Pluricanonical} systems on irregular varieties},
journal = {Annales de l'Institut Fourier},
pages = {359--377},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {75},
number = {1},
year = {2025},
doi = {10.5802/aif.3649},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3649/}
}
TY - JOUR AU - Jiang, Zhi TI - Cohomological support loci and Pluricanonical systems on irregular varieties JO - Annales de l'Institut Fourier PY - 2025 SP - 359 EP - 377 VL - 75 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3649/ DO - 10.5802/aif.3649 LA - en ID - AIF_2025__75_1_359_0 ER -
%0 Journal Article %A Jiang, Zhi %T Cohomological support loci and Pluricanonical systems on irregular varieties %J Annales de l'Institut Fourier %D 2025 %P 359-377 %V 75 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3649/ %R 10.5802/aif.3649 %G en %F AIF_2025__75_1_359_0
Jiang, Zhi. Cohomological support loci and Pluricanonical systems on irregular varieties. Annales de l'Institut Fourier, Tome 75 (2025) no. 1, pp. 359-377. doi : 10.5802/aif.3649. https://aif.centre-mersenne.org/articles/10.5802/aif.3649/
[1] Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften, 267, Springer, 1985, xvi+386 pages | DOI | MR | Zbl
[2] On the bicanonical map of irregular varieties, J. Algebr. Geom., Volume 21 (2012) no. 3, pp. 445-471 | DOI | MR | Zbl
[3] L’application canonique pour les surfaces de type général, Invent. Math., Volume 55 (1979) no. 2, pp. 121-140 | DOI | MR | Zbl
[4] On quint-canonical birationality of irregular threefolds, Proc. Lond. Math. Soc., Volume 122 (2021) no. 2, pp. 234-258 | DOI | MR | Zbl
[5] Pluricanonical maps of varieties of maximal Albanese dimension, Math. Ann., Volume 320 (2001) no. 2, pp. 367-380 | DOI | MR | Zbl
[6] Linear series of irregular varieties, Algebraic geometry in East Asia (Kyoto, 2001), World Scientific, 2002, pp. 143-153 | DOI | MR | Zbl
[7] Pluricanonical systems on irregular 3-folds of general type, Math. Z., Volume 255 (2007) no. 2, pp. 343-355 | DOI | MR | Zbl
[8] Positivity in varieties of maximal Albanese dimension, J. Reine Angew. Math., Volume 736 (2018), pp. 225-253 | DOI | MR | Zbl
[9] On coverings of simple abelian varieties, Bull. Soc. Math. Fr., Volume 134 (2006) no. 2, pp. 253-260 | DOI | Numdam | MR | Zbl
[10] Higher obstructions to deforming cohomology groups of line bundles, J. Am. Math. Soc., Volume 4 (1991) no. 1, pp. 87-103 | DOI | MR | Zbl
[11] A derived category approach to generic vanishing, J. Reine Angew. Math., Volume 575 (2004), pp. 173-187 | DOI | MR | Zbl
[12] On Shokurov’s rational connectedness conjecture, Duke Math. J., Volume 138 (2007) no. 1, pp. 119-136 | DOI | MR | Zbl
[13] On the Iitaka fibration of varieties of maximal Albanese dimension, Int. Math. Res. Not. (2013) no. 13, pp. 2984-3005 | DOI | MR | Zbl
[14] Cohomological support loci of varieties of Albanese fiber dimension one, Trans. Am. Math. Soc., Volume 367 (2015) no. 1, pp. 103-119 | DOI | MR | Zbl
[15] The local Torelli theorem for varieties with divisible canonical class, Math. USSR, Izv., Volume 12 (1978), pp. 53-67 | DOI | Zbl
[16] Higher direct images of dualizing sheaves. I, Ann. Math., Volume 123 (1986) no. 1, pp. 11-42 | DOI | MR | Zbl
[17] Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, 1998, viii+254 pages | DOI | MR | Zbl
[18] Pushforwards of pluricanonical bundles under morphisms to abelian varieties, J. Eur. Math. Soc., Volume 22 (2020) no. 8, pp. 2511-2536 | DOI | MR | Zbl
[19] Regularity on abelian varieties. I, J. Am. Math. Soc., Volume 16 (2003) no. 2, pp. 285-302 | DOI | MR | Zbl
[20] Hodge modules on complex tori and generic vanishing for compact Kähler manifolds, Geom. Topol., Volume 21 (2017) no. 4, pp. 2419-2460 | DOI | MR | Zbl
[21] The local Torelli theorem. I. Complete intersections, Math. Ann., Volume 217 (1975) no. 1, pp. 1-16 | DOI | MR | Zbl
[22] Decomposition theorem for proper Kähler morphisms, Tôhoku Math. J., Volume 42 (1990) no. 2, pp. 127-147 | DOI | MR | Zbl
[23] Subspaces of moduli spaces of rank one local systems, Ann. Sci. Éc. Norm. Supér., Volume 26 (1993) no. 3, pp. 361-401 | DOI | Numdam | MR | Zbl
[24] On the tetracanonical map of varieties of general type and maximal Albanese dimension, Collect. Math., Volume 63 (2012) no. 3, pp. 345-349 | DOI | MR | Zbl
[25] Local Torelli theorem for non-singular complete intersections, Jpn. J. Math., New Ser., Volume 2 (1976) no. 2, pp. 411-418 | DOI | MR | Zbl
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