Vanishing theorems for cohomologies of automorphic vector bundles (Q1915939)

Here the author gives a strong criterion for the vanishing of cohomologies of automorphic vector bundles, generalizing the classical results of Calabi-Vesentini and Matsushima. He obtains some vanishing of \(\overline \partial\)-cohomology (or \((q, K^\mathbb{C})\)-cohomology) of unitary \(g\)-modules with coefficients in a \(K\)-module (with \(K\) maximal compact subgroup of a real, semisimple, connected linear Lie group \(G\) without compact factors). In particular he obtains \(H^1 (M, \Theta_M) = 0\) for some manifold \(M\) which is fibered over a locally hermitian symmetric domain and hence the infinitesimal rigidity of such \(M\).