Exotic holonomy on moduli spaces of rational curves

Abstract

Bryant [3] proved the existence of torsion free connections with exotic holonomy, i.e., with holonomy that does not occur on the classical list of Berger [1]. These connections occur on moduli spaces Y of rational contact curves in a contact threefold W. Therefore, they are naturally contained in the moduli space Z of all rational curves in W. We construct a connection on Z whose restriction to Y is torsion free. However, the connection on Z has torsion unless both Y and Z are flat. This answers a question of Bryant as to whether the GL(2, C) × SL(2, C)-structures which arise from such a moduli space Z always admit a torsion free connection in the negative. We also show the existence of a new exotic holonomy which is a certain six-dimensional representation of SL(2, C) × SL(2, C). We show that every regular H 3-connection (cf. [3]) is the restriction of a unique connection with this holonomy.