The Shafarevich conjecture and some extension theorems for proper hyperbolic polycurves

TL;DR

This paper proves the Shafarevich conjecture for proper hyperbolic polycurves, extending results from curves to higher dimensions, and develops related moduli and extension theories.

Contribution

It establishes the Shafarevich conjecture for higher-dimensional proper hyperbolic polycurves and generalizes moduli theory for these objects.

Findings

Proved the Shafarevich conjecture for proper hyperbolic polycurves.

Generalized the moduli theory of Kodaira fibrations.

Established the Neron property for models over Dedekind schemes.

Abstract

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.