H^1(X, -L)=0 holds for nef Cartier L on projective klt threefolds X in char p>5 under the conditions that either K_X is not big with L big, or -K_X is nef with L of numerical dimension two.
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Mumford vanishing for threefolds in positive characteristic
H^1(X, -L)=0 holds for nef Cartier L on projective klt threefolds X in char p>5 under the conditions that either K_X is not big with L big, or -K_X is nef with L of numerical dimension two.