Deformation theory of nearly K\"ahler manifolds

TL;DR

This paper investigates the deformation theory of nearly K"ahler manifolds, revealing that such deformations are generally obstructed, with specific focus on the homogeneous structure on the flag manifold.

Contribution

It demonstrates that the infinitesimal deformations of the homogeneous nearly K"ahler structure on the flag manifold are obstructed at second order.

Findings

Deformations of nearly K"ahler manifolds are generally obstructed.

Infinitesimal deformations on the flag manifold are obstructed to second order.

The study links the deformation theory to the holonomy of the Riemannian cone.

Abstract

Nearly K\"ahler manifolds are the Riemannian 6-manifolds admitting real Killing spinors. Equivalently, the Riemannian cone over a nearly K\"ahler manifold has holonomy contained in G2. In this paper we study the deformation theory of nearly K\"ahler manifolds, showing that it is obstructed in general. More precisely, we show that the infinitesimal deformations of the homogeneous nearly K\"ahler structure on the flag manifold are all obstructed to second order.