Constant Chern holomorphic sectional curvature conjecture
For every compact Hermitian manifold , if the Chern holomorphic sectional curvature of is identically equal to a constant , then is Kähler when , and the Chern curvature tensor of vanishes identically when ; equivalently, is Chern flat in the zero-curvature case.
References
Primary source
Additional references
- Hermitian threefolds with constant holomorphic sectional curvature — arXiv — Shuwen Chen, Fangyang Zheng
Progress summary
A September 2026 paper claims the remaining three-dimensional case, but the full conjecture remains publicly unverified and open.
The conjecture asks whether compact Hermitian manifolds with constant Chern holomorphic sectional curvature must have the corresponding rigid geometric form. The latest claim concerns only compact Hermitian threefolds and the zero-curvature case; the nonzero case was previously known in that dimension.
Known results
- Balas–Gauduchon (1985): compact complex surfaces with nonpositive constant curvature are Kähler.
- Apostolov–Davidov–Muškarov (1996): treated compact Hermitian surfaces with pointwise constant Chern holomorphic sectional curvature.
- Ma–Nie (2024): obtained results for compact normal balanced threefolds.
- Chen–Li (2026): for compact balanced threefolds, negative curvature is Kähler and zero curvature is Chern flat.
September 2026 threefold claim
Shuwen Chen and Fangyang Zheng report that the zero-curvature case holds for compact Hermitian threefolds, completing the dimension-three claim together with the known nonzero-curvature case. This remains an unverified claim: another recent source explicitly says the general conjecture remains open and does not establish the result for arbitrary Hermitian threefolds.
Current status (as of October 2026): the three-dimensional zero-curvature case is claimed in a recent preprint but is unverified; the full conjecture, especially in higher dimensions, remains open.
Solutions 0
No solutions have been posted yet.