Classification of compact transformation groups on complex quadrics with codimension one orbits

Abstract

Let $G$ be a compact connected Lie group and $M$ a rational cohomology complex quadric of real dimension divisible by $4$ (where $\dim M\neq 4$). The aim of this paper is to classify pairs $(G,M)$ such that $G$ acts smoothly on $M$ with codimension one principal orbits. There exist eight such pairs up to essential isomorphism. The underlying manifold $M$ is diffeomorphic to the genuine complex quadric except one pair.