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arXiv · 1907.06705

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

Juanyong Wang
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Abstract

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-P\u aun (2017) respectively. We further generalize the later case to the geometric orbifold setting, i.e. prove that $C_{n,m}^{\text{orb}}$ (Campana, 2004) holds when $Y$ is a complex torus.

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Juanyong Wang. 2019-07-15. On the Iitaka Conjecture $C_{n,m}$ for Kähler Fibre Spaces. https://arxiv.org/abs/1907.06705

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