Semi-stable and splitting models for unitary Shimura varieties over ramified places. II

TL;DR

This paper constructs and analyzes integral models of certain unitary Shimura varieties over ramified primes, demonstrating their normality, flatness, Cohen-Macaulay property, and semi-stable reduction after a single blow-up.

Contribution

It introduces explicit semi-stable models for unitary Shimura varieties at ramified primes, extending previous splitting model constructions with detailed geometric properties.

Findings

Models are normal, flat, Cohen-Macaulay, with reduced special fiber.

A single blow-up yields semi-stable models.

Explicit calculations confirm the geometric properties.

Abstract

We consider Shimura varieties associated to a unitary group of signature $(n-1, 1)$. For these varieties, we construct $p$-adic integral models over odd primes $p$ which ramify in the imaginary quadratic field with level subgroup at $p$ given by the stabilizer of a vertex lattice in the hermitian space. Our models are given by a variation of the construction of the splitting models of Pappas-Rapoport and they have a simple moduli theoretic description. By an explicit calculation, we show that these splitting models are normal, flat, Cohen-Macaulay and with reduced special fiber. In fact, they have relatively simple singularities: we show that a single blow-up along a smooth codimension one subvariety of the special fiber produces a semi-stable model. This also implies the existence of semi-stable models of the corresponding Shimura varieties.