Philip Engel (UIC)
-
PDL C-38
Title: E_7 and Calabi-Yau 3-folds
Abstract: Shimura varieties are arithmetic quotients of Hermitian symmetric domains. Famous examples include (1) the upper half-plane modulo SL_2(Z) parameterizing elliptic curves, (2) its generalization, the Siegel upper half-space modulo Sp_{2g}(Z), parameterizing principally polarized abelian varieties of dimension g, and (3) a certain open subset of a quadric, modulo the isometry group of a Z-lattice of signature (2,n).
A Shimura variety is a moduli space of Hodge structures, and as such, carries a tautological "variation of Hodge structure." Famously, variations of Hodge structures also arise from the relative cohomology of a smooth projective morphism. "Deligne's rêve" is that, in fact, the tautological variation of Hodge structure on any Shimura variety comes from geometry.
All three of the above examples (1), (2), (3) embed into A_g and thus in these cases, Deligne's rêve is true. To find examples of Shimura varieties which do not embed into A_g one must look at more exotic/exceptional cases. Attached to the simple Lie group E_7 are some such exceptional Shimura varieties. Not a single VHS on an E_7 Shimura variety is known to come from geometry.
In this talk, I will propose one possible approach to finding this mythical geometric family, which induces a variation of Hodge structure of E_7 Shimura type. This talk will be very speculative.