Signature of quaternionic Kaehler manifolds (Q1078832)

Quaternionic Kähler manifolds have analogues of the Hard-Lefschetz theorem and the Hodge index theorem. The latter on the signature is simpler and clear cut. Let M be a compact connected quaternionic Kähler manifold of positive scalar curvature with dimension 4n. Theorem 1. \[ H^{2p}(M)=\sum_{2k\leq p}(L_{\Omega})^ k P^{2p-4k}(M),\quad p\leq n, \] and \((L_{\Omega})^{n-p}\) is an isomorphism of \(H^{2p}(M)\) onto \(H^{4n-2p}(M)\), where \(L_{\Omega}\) is the multiplication by the canonical 4-form \(\Omega\) and \(P^{2q}(M)\) are the primitive cohomology classes of degree 2q; \(H^{2p+1}(M)=\{0\}.\) Theorem 2. The signature of M equals the Betti number \(b_{2n}\). The method is to use the Penrose construction due to \textit{S. Salamon} and his results [Invent. Math. 67, 143-171 (1982; Zbl 0486.53048)], connecting M to a Kähler manifold. Example. The signature is 8 if M is the compact symmetric space EIX.