On the Partial Compactification of the Arithmetic Quotient of a Period

In his survey paper [4], P. A. Griffiths conjectured (§9.2) the existence of a partial compactification for the arithmetic quotients of period matrix domains. In this paper we want to announce some results concerning the topological aspects of this conjecture for the case of the periods of 2-forms on a polarized Hodge manifold V. The period matrix domain D of all possible period matrices for the primitive harmonic 2-forms on V can be described as follows (cf. [3], [6]). Let h=h, k=h, where h denotes the dimension of the space of primitive harmonic forms of bidegree (p,q). Let H be a complex vector space of dimension m=2h+k, HR^H a real form, and HZ^HR SL lattice. We also fix a rationally defined, nondegenerate, bilinear symmetric form Q on H, We denote by X the set of all W e Gr (h, H) satisfying