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Hwang-Mok rigidity of cominuscule homogeneous varieties in positive characteristic

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.