The Hermitian Laplace Operator on Nearly K\"ahler Manifolds

TL;DR

This paper characterizes the space of infinitesimal deformations of nearly K"ahler structures on 6-dimensional manifolds using the Hermitian Laplace operator, revealing rigidity except for the flag manifold with an 8-dimensional deformation space.

Contribution

It computes the moduli space of infinitesimal nearly K"ahler deformations on all 6-dimensional homogeneous examples, highlighting the unique flexibility of the flag manifold.

Findings

Most nearly K"ahler structures are rigid.

The flag manifold has an 8-dimensional deformation space.

The moduli space is described via eigenspaces of the Laplace operator.

Abstract

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.