The Shafarevich conjecture and some extension theorems for proper hyperbolic polycurves

In this paper, we prove the Shafarevich conjecture for proper hyperbolic polycurves, which is a higher dimensional analogue of that for proper hyperbolic curves. First, we study theories of proper hyperbolic polycurves over regular schemes. For example, we generalize the moduli theory of Kodaira fibrations due to Jost and Yau. We also show the Neron property of proper smooth models of proper hyperbolic polycurves over Dedekind schemes under an assumption on residual characteristics. We then apply these extension theories to the proof of the Shafarevich conjecture for proper hyperbolic polycurves.