We establish a procedure for constructing compatible mixed Hodge structures for the cohomology of various topological compactifications of locally symmetric varieties, notably ones that are not algebraic varieties. This is carried out in full for the case of the reductive Borel-Serre compactification, and conditionally for the excentric compactifications.
"On the Reductive Borel-Serre Compactification, III: Mixed Hodge Structures." Asian J. Math. 8 (4) 881 - 912, December, 2004.