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.

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 C n,m (Ueno, 1975) for f:XY (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 Y or when Y 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 C n,m orbifold (Campana, 2004) holds when Y 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 C n,m (Ueno, 1975) pour f:XY (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 Y ou si Y 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 C n,m orbifold (Campana, 2004) est vraie quand Y est un tore complexe.

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DOI: 10.5802/afst.1690
Wang, Juanyong 1

1 Academy of Mathematics and Systems Science, Chinese Academy of Sciences
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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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