Homogeneity of Hermitian–Einstein metrics on complex projective spaces

Determine whether every Hermitian–Einstein metric on complex projective space P^n for n>2 is homogeneous.

Background

The paper proves that the constructed inhomogeneous Einstein metrics on complex projective spaces are not Hermitian. It also recalls that in complex dimension two, a Hermitian–Einstein metric on P2 must be isometric to the Fubini–Study metric and is therefore homogeneous.

The authors identify the higher-dimensional analogue as unresolved: although the question is settled in complex dimension two, it remains unknown whether Hermitian–Einstein metrics on Pn with n>2 must all be homogeneous.

References

To our knowledge, it remains an open question whether every Hermitian--Einstein metric on Pn with n > 2 is homogeneous.

— Inhomogeneous Einstein metrics on complex projective spaces  (2608.16880 - Cao-Labora et al., 17 Aug 2026) in Section 1, paragraph beginning “Our examples have real dimension strictly greater than 4”