  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.
