Kodaira-Spencer isomorphisms and degeneracy maps on Iwahori-level Hilbert modular varieties: the saving trace

TL;DR

This paper establishes a Kodaira-Spencer isomorphism for Hilbert modular varieties with Iwahori level structure, analyzes degeneracy maps, and applies these results to simplify the construction of Hecke operators at primes over p.

Contribution

It provides a new Kodaira-Spencer isomorphism and vanishing theorems for fibers of degeneracy maps, improving understanding of the structure and Hecke operators on Hilbert modular varieties.

Findings

Proved a Kodaira-Spencer isomorphism for Iwahori-level Hilbert modular varieties.

Established vanishing of higher direct images of structure and dualizing sheaves.

Simplified the construction of Hecke operators at primes over p.

Abstract

We consider integral models of Hilbert modular varieties with Iwahori level structure at primes over p, first proving a Kodaira-Spencer isomorphism that gives a concise description of their dualizing sheaves. We then analyze fibres of the degeneracy maps to Hilbert modular varieties of level prime to p and deduce the vanishing of higher direct images of structure and dualizing sheaves, generalizing prior work with Kassaei and Sasaki (for p unramified in the totally real field F). We apply the vanishing results to prove flatness of the finite morphisms in the resulting Stein factorizations, and combine them with the Kodaira-Spencer isomorphism to simplify and generalize the construction of Hecke operators at primes over p on Hilbert modular forms (integrally and mod p).