Hwang-Mok rigidity of cominuscule homogeneous varieties in positive characteristic

TL;DR

This paper extends the rigidity results of Hermitian symmetric spaces to positive characteristic algebraic varieties, providing explicit bounds to ensure isotriviality in families of cominuscule homogeneous varieties.

Contribution

It adapts Hwang and Mok's rigidity argument to positive characteristic, introducing a computable characteristic bound and a characteristic-free extension theorem for Fano varieties.

Findings

Explicit lower bound on characteristic for rigidity

Extension theorem for Fano varieties of Picard number 1

Rigidity of cominuscule homogeneous varieties in positive characteristic

Abstract

Jun-Muk Hwang and Ngaiming Mok have proved the rigidity of irreducible Hermitian symmetric spaces of compact type under Kaehler degeneration. I adapt their argument to the algebraic setting in positive characteristic, where cominuscule homogeneous varieties serve as an analogue of Hermitian symmetric spaces. The main result gives an explicit (computable in terms of Schubert calculus) lower bound on the characteristic of the base field, guaranteeing that a smooth projective family with cominuscule homogeneous generic fibre is isotrivial. The bound depends only on the type of the generic fibre, and on the degree of an invertible sheaf whose extension to the special fibre is very ample. An important part of the proof is a characteristic-free analogue of Hwang and Mok's extension theorem for maps of Fano varieties of Picard number 1, a result I believe to be interesting in its own right.