by W. L. Baily, Jr.

In these talks we consider moduli problems connected with, or possibly connected with, some arithmetic quotients of certain bounded symmetric domains. The investigation of moduli problems possibly connected with a certain "nice" arithmetic quotient of the 27-dimensional exceptional domain associated with the real exceptional Lie group E_{7(-25)} is the central object of interest. It is at the top of (or of highest dimension among) the list of the four irreducible bounded symmetric tube domains of real rank 3. The other three such domains are the Siegel upper half-space of real rank 3 and dimension 6, the tube domain over the cone of positive definite complex 3X3 hermitian matrices of dimension 9, and the tube domain over the cone of positive definite hermitian quaternion 3X3 matrices of dimension 15. Corresponding to these four symmetric tube domains there are four so-called Severi varieties of dimensions 2, 4, 8, and 16, realized explicitly as the Veronese surface in P^5, the product P^2 X P^2 imbedded by the Segre mapping in P^8, the Grassmannian G(2,6) contained in P^{14} via the Plucker imbedding, and the Cayley projective plane contained in P^{26} via a certain imbedding described in terms of the irreducible idempotents in an exceptional Jordan algebra of dimension 27. These two sets of four objects appear to be related in some way, but a completely clear explanation of the relationship seems to be unknown.

Anyway, the 6-dimensional Siegel upper half-space is connected in a well-known way with the moduli of principally polarized Abelian varieties of dimension 3 and thus, via Torelli's theorem, with the moduli of non-hyperelliptic curves of genus 3; the 9-dimensional tube domain over the cone of positive complex hermitian 3X3 matrices is connected with the moduli of principally polarized Abelian varieties with complex multiplication of Weil-type by i and also, somewhat surprisingly, via a tube domain of type IV of dimension 9, with the moduli of nodal Enriques surfaces; the 15-dimensional tube domain over the cone of positive quaternion hermitian 3X3 matrices is connected with the moduli of principally polarized Abelian varieties of dimension 12 having complex multiplicationby the Hurwitz ring of quaternion integers and now, conjecturally (which is where part of the mystery begins) with the moduli of certain Fano 4-folds. At the next, and top, stage, the 27-dimensional exceptional Jordan domain, one can so far only conjecture the existence of something parallel to the earlier cases. This is the main and, so far, quite unexplored area.