Toroidal compactifications of integral models of Shimura varieties of Hodge type

TL;DR

This paper constructs toroidal and minimal compactifications of integral models of Shimura varieties of Hodge type, extending known cases and providing new proofs for conjectures related to abelian varieties and their reduction properties.

Contribution

It develops a unified approach to compactifications of integral models of Shimura varieties of Hodge type, covering all known cases and introducing new methods for understanding their structure.

Findings

Constructed projective toroidal compactifications for integral models.

Established integral models of the minimal Satake-Baily-Borel compactification.

Provided a new proof of Morita's conjecture on good reduction of certain abelian varieties.

Abstract

We construct projective toroidal compactifications for integral models of Shimura varieties of Hodge type. We also construct integral models of the minimal (Satake-Baily-Borel) compactification. Our results essentially reduce the problem to understanding the integral models themselves. As such, they cover all previously known cases of PEL type, as well as all cases of Hodge type involving parahoric level structures. At primes where the level is hyperspecial, we show that our compactifications are canonical in a precise sense. We also provide a new proof of Y. Morita's conjecture on the everywhere good reduction of abelian varieties whose Mumford-Tate group is anisotropic modulo center. Along the way, we demonstrate an interesting rationality property of Hodge cycles on abelian varieties with respect to p-adic analytic uniformizations.