Special holonomy of compact simply connected Ricci-flat manifolds

Determine whether every simply connected compact Ricci-flat manifold has special holonomy.

Background

The paper places this question in the context of compact Einstein manifolds with zero Einstein constant. Under the assumption of special holonomy, the second-order Einstein equation reduces to a first-order equation, and many examples with special holonomy are known, including Ricci-flat Kähler metrics furnished by Yau’s solution of the Calabi conjecture.

The unresolved issue is whether special holonomy is forced for every simply connected compact Ricci-flat manifold, rather than merely occurring in the presently known constructions.

References

In the case that λ=0, it remains an open question whether every simply connected compact Ricci-flat manifold has special holonomy.

— Inhomogeneous Einstein metrics on complex projective spaces  (2608.16880 - Cao-Labora et al., 17 Aug 2026) in Section 1, paragraph beginning “In the case that λ=0”