The Hermitian Laplace Operator on Nearly K\"ahler Manifolds

The moduli space NK of infinitesimal deformations of a nearly K"ahler 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. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6-dimensional homogeneous nearly K"ahler manifolds. It turns out that the nearly K"ahler 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 K"ahler deformations, modeled on the Lie algebra su_3 of the isometry group.