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Jun-Muk Hwang and Ngaiming Mok have proved the rigidity of irreducibleHermitian symmetric spaces of compact type under Kaehler degeneration. I adapttheir argument to the algebraic setting in positive characteristic, wherecominuscule homogeneous varieties serve as an analogue of Hermitian symmetricspaces. The main result gives an explicit (computable in terms of Schubertcalculus) lower bound on the characteristic of the base field, guaranteeingthat a smooth projective family with cominuscule homogeneous generic fibre isisotrivial. The bound depends only on the type of the generic fibre, and on thedegree of an invertible sheaf whose extension to the special fibre is veryample. An important part of the proof is a characteristic-free analogue ofHwang and Mok's extension theorem for maps of Fano varieties of Picard number1, a result I believe to be interesting in its own right.

Hwang-Mok rigidity of cominuscule homogeneous varieties in positive characteristic