On the infinitesimal automorphisms of quaternionic structures
Let \(G\) be any of the quaternionic linear groups \(\text{GL} (n,\mathbb{H})\), \(\text{GL} (n,\mathbb{H}) \text{Sp} (1)\) or of their compact subgroups \(\text{Sp} (n)\), \(\text{Sp} (n) \text{Sp}(1)\); let \(a (M)\) denote the infinitesimal automorphisms for any of the first two \(G\)-structures and \(a(M,g)\) for any of the last two on a \(4n\)-dimensional manifold \(M\). The comparison between \(a(M)\) and \(a(M,g)\) is discussed on hyperkähler and quaternion Kähler manifolds \(M\). In particular the vanishing of \(a(M)\) is proved on compact quaternion Kähler manifolds with negative scalar curvature and on irreducible compact hyperkähler manifolds. On compact quaternion Kähler \(M\) with positive scalar curvature a formula of Bott type is proved relating the behaviour around the zeros of the infinitesimal automorphisms with the symplectic Pontryagin characteristic numbers.
scientific article; zbMATH DE number 811436