The Hermitian Laplace Operator on Nearly Kähler Manifolds

The moduli space of infinitesimal deformations of a nearly Kähler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1, 1) forms (cf. Moroianu et al. in Pacific J Math 235:57–72, 2008). Using the Hermitian Laplace operator and some representation theory, we compute the space on all 6-dimensional homogeneous nearly Kähler manifolds. It turns out that the nearly Kähler structure is rigid except for the flag manifold F (1, 2) = SU 3 / T 2 , which carries an 8-dimensional moduli space of infinitesimal nearly Kähler deformations, modeled on the Lie algebra of the isometry group.