On the Hodge embedding for integral models of Shimura varieties

We give a simple, down-to-earth, proof that Kottwitz’s (resp. Rapoport–Zink) PEL type integral models of Shimura varieties admit closed embeddings into Siegel integral models. We also show that Rapoport’s and Kottwitz’s integral models agree with Kisin’s integral models for relevant Shimura data. Our result also applies to certain exotic PEL type integral models not covered by Kottwitz (or Kisin, Kisin–Pappas etc). Moreover, combined with a result of Lan’s, we obtain closed embeddings for toroidal compactifications of integral models.