Vanishing theorems for quaternionic Kähler manifolds (Q2715755)

On a manifold with special holonomy, one can define geometric vector bundles corresponding to the various representations of the holonomy group. For each such geometric vector bundle the authors define an elliptic, self-adjoint operator acting on its sections. Via representation theory, the Hodge-Laplace, Dirac or twisted Dirac operators and, correspondingly, the Weitzenböck and Lichnerowicz formulae can be interpreted in this way. Vanishing theorems for the Betti numbers of quaternionic Kähler manifolds may now be derived. In particular, for positive quaternionic Kähler manifolds the authors simplify the proof of the vanishing of odd Betti numbers [cf. \textit{S. Salamon}, Invent. Math. 67, 143-171 (1982; Zbl 0486.53048)] and prove a strong Lefschetz theorem. NEWLINENEWLINENEWLINEThe same method works for negative quaternionic Kähler manifolds and they can prove a weak Lefschetz theorem. This is an exceptionally important paper, not only because of the beauty and power of the method, but also because it offers a really unified view on apparently different phenomena.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00032].