关于Kähler纤维空间的Iitaka猜想$C_{n,m}$
On the Iitaka Conjecture $C_{n,m}$ for Kähler Fibre Spaces
摘要：
通过应用直接象的正性定理和Green-Lazarsfeld-Simpson上同调跳跃轨迹的结构定理的复正则化形式，我们证明了Iitaka猜想的klt Kähler形式$C_{n,m}$（上野，1975）用于$f:X\to Y$（具有连通一般纤维的紧致Kähler流形之间的满射态射）当相对典范丛的某个幂的直像的行列式在$Y$上较大或当$Y$是复环面时成立。这些结果分别推广了Viehweg（1983）和Cao-P \u aun（2017）的相应结果。我们进一步推广后一种情况的几何orbifold设置，即证明$C_{n,m}^{\text{orb}}$（Campana，2004年）持有时，$Y$是一个复杂的环面。
英文摘要：
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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