Calabi-Yau Varieties with Fibre Structures

Motivated by the Strominger-Yau-Zaslow conjecture, we study fibre spaces whose total space has trivial canonical bundle. Especially, we are interest in Calabi-Yau varieties with fibre structure. In this paper, we only consider semi-stable families. We use Hodge theory and the generalized Donaldson-Simpson-Uhlenbeck-Yau correspondence to study the parabolic structure of higher direct images over higher dimensional quasi-projective base, and obtain an important result on parabolic-semi-positivity. We then apply this result to study nonisotrivial Calabi-Yau varieties fibred by Abelian varieties (or fibred by hyperk\"ahler varieties), we obtain that the base manifold for such a family is rationally connected and the dimension of a general fibre depends only on the base manifold.