A modular symbol is the fundamental class of a totally geodesic submanifold embedded in a locally Riemannian symmetric space , which is defined by a subsymmetric space . In this paper, we consider the modular symbol defined by a semisimple symmetric pair (G,G'), and prove a vanishing theorem with respect to the -component in the Matsushima-Murakami formula based on the discretely decomposable theorem of the restriction . In particular, we determine explicitly the middle Hodge components of certain totally real modular symbols on the locally Hermitian symmetric spaces of type IV.
Cite this article
Toshiyuki Kobayashi, Tadao Oda, A vanishing theorem for modular symbols on locally symmetric spaces. Comment. Math. Helv. 73 (1998), no. 1, pp. 45–70DOI 10.1007/S000140050045