On the Iitaka Conjecture $C_{n,m}$ for Kähler Fibre Spaces

Wang, Juanyong

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 4, pp. 813-897.

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 $f:X\to Y$ (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 $C_{n,m}^{\mathrm{orbifold}}$ (Campana, 2004) est vraie quand $Y$ est un tore complexe.

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 $f:X\to Y$ (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 $C_{n,m}^{\mathrm{orbifold}}$ (Campana, 2004) holds when $Y$ is a complex torus.

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