By applying the positivity theorem of direct images and a pluricanonical version of the structure theorem on the cohomology jumping loci à la Green–Lazarsfeld–Simpson, we show that the klt Kähler version of the Iitaka conjecture (Ueno, 1975) for (surjective morphism between compact Kähler manifolds with connected general fibre) holds true when the determinant of the direct image of some power of the relative canonical bundle is big on or when is a complex torus. These generalize the corresponding results of Viehweg (1983) and of Cao-Păun (2017) respectively. We further generalize the later case to the geometric orbifold setting, i.e. prove that (Campana, 2004) holds when is a complex torus.
En appliquant la positivité des images directes et une version pluricanonique du théorème de structure des lieux de saut cohomologique à la Green–Lazarsfeld–Simpson, nous démontrons que la version klt kählérienne de la conjecture d’Iitaka (Ueno, 1975) pour (morphisme surjectif entre variétés kählériennes compactes à fibre générale connexe) est vraie si le déterminant de l’image directe d’une certaine puissance du fibré canonique relative est gros sur ou si est un tore complexe. Ceci généralisent les résultats correspondants de Viehweg (1983) et de Cao-Păun (2017) respectivement. De plus nous généralisons le deuxième résultat ci-dessus au cadre des orbifoldes géométriques, c-à-d., nous démontrons que (Campana, 2004) est vraie quand est un tore complexe.
Accepted:
Published online:
@article{AFST_2021_6_30_4_813_0, author = {Wang, Juanyong}, title = {On the {Iitaka} {Conjecture~}$C_{n,m}$ for {K\"ahler} {Fibre} {Spaces}}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {813--897}, publisher = {Universit\'e Paul Sabatier, Toulouse}, volume = {Ser. 6, 30}, number = {4}, year = {2021}, doi = {10.5802/afst.1690}, language = {en}, url = {http://www.numdam.org/articles/10.5802/afst.1690/} }
TY - JOUR AU - Wang, Juanyong TI - On the Iitaka Conjecture $C_{n,m}$ for Kähler Fibre Spaces JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2021 SP - 813 EP - 897 VL - 30 IS - 4 PB - Université Paul Sabatier, Toulouse UR - http://www.numdam.org/articles/10.5802/afst.1690/ DO - 10.5802/afst.1690 LA - en ID - AFST_2021_6_30_4_813_0 ER -
%0 Journal Article %A Wang, Juanyong %T On the Iitaka Conjecture $C_{n,m}$ for Kähler Fibre Spaces %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2021 %P 813-897 %V 30 %N 4 %I Université Paul Sabatier, Toulouse %U http://www.numdam.org/articles/10.5802/afst.1690/ %R 10.5802/afst.1690 %G en %F AFST_2021_6_30_4_813_0
Wang, Juanyong. On the Iitaka Conjecture $C_{n,m}$ for Kähler Fibre Spaces. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 4, pp. 813-897. doi : 10.5802/afst.1690. http://www.numdam.org/articles/10.5802/afst.1690/
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