A New Approach to Compactifying Locally Symmetric Varieties

SUPPOSE D IS a bounded symmetric domain and I' c Aut (D) is a discrete group of arithmetic type. Then Borel and Baily [2] have shown that DIP can be canonically embedded as a Zariski-open subset in a projective variety Dir. However, Igusa [6] and others have found that the singularities of D/F are extraordinarily complicated and this presents a significant obstacle to using algebraic geometry on D/P in order to derive information on automorphic forms on D, etc. Igusa [7] for D = E2 and 9N, (En = Siegel's n x n upper half-space) and F commensurable with Sp(2n, Z), and Hirzebruch [4] for D = SJJ21 x Ei and F commensurable with SL(2, R) (B = integers in a real quadratic field) have given explicit resolutions of Dir. Independently, Satake [9] and I working in collaboration with Y. Tai, M. Rapaport and A. Ash have attacked the general case, using closely related methods. The purpose of this paper is to give a very short outline of my approach. It builds in an essential way on the construction of -torus embeddings". a theory which has been published in the Springer Lecture Notes [8] (by G. Kempf, F. Knudsen, B. Saint-I)onat and myself; some of which has been worked out independently by M. Demazure [3] and M. Hochster [5]). We intend to publish full details of the present research as soon as possible in a sequel "Toroidal Embeddings II" to the Notes [8]. At the present time, however, we cannot claim to have written down complete proofs of our "Main Theorem" and although I definitely believe it is true and not difficult, it is more accurate to describe the ideas below only as a suggested approach to the problem of constructing a non-singular compactification of D/F.