Abstract
We investigate the cohomology groups H 1( Z, O( L ⊗ m )), where Z is the twistor space of a compact quaternionic-Kähler manifold M, of dimension 4 k, for k ⩾ 1, L is the holomorphic contact line bundle on Z, and m ⩾ 0. The Penrose transform is used to prove a vanishing theorem for this cohomology group when M has negative scalar curvature. This theorem implies that if ( M, g) is a compact quaternionic-Kähler manifold of dimension 4 k, for k ⩾ 1, then ( M, g) has no nontrivial deformations through quaternionic-Kähler manifolds. The vanishing theorem is also used to show that the first Betti number of M is zero, when k > 1 and M has negative scalar curvature.
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