Non-abelian LVM manifolds}

Abstract

We extend the tangency construction of López de Medrano, Verjovsky and Meersseman from diagonal abelian actions to non-abelian Samelson groups. For these groups the tangency locus is a moment-map zero set: the commutator term in the leafwise Hessian vanishes there, and under an admissibility hypothesis, Kempf--Ness convexity gives a global transversal whose projectivization is a compact complex manifold. The construction includes compact rotation groups and groups with non-closed central rotations. In the latter setting we construct explicit families of complex dimension $2m-3$, $m\geq5$, which are $4$-connected and admit no symplectic form. Their meromorphic function fields are rational of transcendence degree $m-2$, although every holomorphic map to a projective space is constant. We compute their connected automorphism groups and show that they admit no maximal torus action. They are Hopf-surface bundles over compact complex quotients of stable Grassmannian loci; these bases give related examples already in complex dimension five. The families demonstrate that the non-abelian tangency construction reaches beyond compact complex manifolds with maximal torus actions.