Complex versus real sectional curvature for compact symmetric spaces

Determine whether, for every compact symmetric space, the maximal complex sectional curvature equals the maximal sectional curvature.

Background

The paper proves that the maximal complex sectional curvature of a compact simple Lie group with a bi-invariant metric equals its maximal sectional curvature. For compact symmetric spaces, it derives an upper bound for complex sectional curvature and identifies several cases in which the bound is attained, including the symmetric spaces SU(n)/SO(n) and Hermitian symmetric spaces. It also notes examples—such as spheres SO(n+1)/SO(n) for n ≥ 4, SU(2n)/Sp(n), and E6/F4—where the maximal sectional curvature is strictly below the general bound. The unresolved question asks whether equality between maximal complex and real sectional curvature nevertheless holds for every compact symmetric space.

References

We do not know whether, for every compact symmetric space, the maximal complex sectional curvature equals the maximal sectional curvature, as Corollary \ref{cor:groupcurvature} shows for compact Lie groups.

— Sharp commutator bounds on complex simple Lie algebras: the Böttcher--Wenzel inequality and the comass of the Cartan $3$-form  (2610.11800 - Fox, 8 Oct 2026) in Section “Complex sectional curvature,” immediately following the discussion of compact symmetric spaces