Abstract:

We shall begin with a brief review on Shimura varieties and their toroidal compactifications, and investigate the question of whether a closed immersion between Shimura varieties defined by some morphism of Shimura data extends to a closed immersion (rather than just a morphism) between their toroidal compactifications.  We shall pin down some sufficient conditions on the associated cone decompositions, and explain whether they can be satisfied up to refinements.  If time permits, we shall also discuss the analogous question for integral models.