A rigidity theorem for quaternionic Kähler manifolds (Q1917375)

We investigate the cohomology groups \(H^1 (Z, {\mathcal O} (L^{\otimes m}))\), where \(Z\) is the twistor space of a compact quaternionic-Kähler manifold \(M\), of dimension \(4k\), for \(k \geq 1\), \(L\) is the holomorphic contact line bundle on \(Z\), and \(m \geq 0\). 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 \(4k\), for \(k \geq 1\), 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.