Global behavior at infinity of period mappings defined on algebraic surface

The global behavior of period mappings defined on generally non-complete algebraic varieties $B$ as well as their local behavior around points in the boundary $Z = \overline{B}\setminus B$ of smooth completions of $B$ have been extensively investigated. In this paper we shall study the global behavior of period mappings in neighborhoods of the entire boundary $Z$ when $\dim B = 2$. One method will be to decompose the dual graph of the boundary into basic building blocks of cycles and trees and analyze these separately. A main tool will be a global version of the classical nilpotent orbit theorem of Schmid.