Calabi-yau Varieties with Fibre Structures I

Motivated by the Strominger-Yau-Zaslow conjecture, we study fibre spaces whose total space has trivial canonical bundle. Especially, we are interest in CalabiYau varieties with fibre structures. 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 some results on parabolic-semi-positivity. We then apply these results to study nonisotrivial Calabi-Yau varieties fibred by Abelian varieties (or fibred by hyperkähler 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.