MathDB

Tang's constant mixed curvature conjecture for compact Hermitian manifolds

From papers

Let (Mn,g)(M^{n},g) be a compact Hermitian manifold with n≥2n\geq2. For a (1,0)(1,0)-vector XX, let Cα,β(X)\mathcal{C}_{\alpha,\beta}(X) denote the mixed curvature, formed from the relevant Chern Ricci curvature and holomorphic sectional curvature. Tang's constant mixed curvature conjecture. If Cα,β=c\mathcal{C}_{\alpha,\beta}=c is constant and c≠0c\neq0, then gg is Kähler. This conjecture was proposed for the non-Kähler case; the source notes that it was proved in complex dimension 22 and verified for several special classes in higher dimensions, but leaves the general case open.

Progress summary

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Tang's constant mixed curvature conjecture for compact Hermitian manifolds

    Let (Mn,g)(M^n,g) be a compact Hermitian manifold, and let Cα,β\mathcal{C}_{\alpha,\beta} denote its mixed curvature. Suppose that

    Cα,β=c\mathcal{C}_{\alpha,\beta}=c

    for a constant cc. Tang's conjecture. If c≠0c\neq 0, then gg is Kähler. This conjecture extends known positivity and negativity results for mixed curvature from the Kähler setting to compact Hermitian manifolds; the paper verifies it for several special classes, but it remains open in general.

    source: Shuwen Chen and Fangyang Zheng, “On Hermitian manifolds with constant mixed curvature”, arXiv:2503.12432 (2025).

Sources & referencesView supporting material

Primary source

Weiguo Chen and Kai Tang, “Constant kth-mixed curvature”, arXiv:2510.05546 (2025).

Additional references

2 papers in this index state this conjecture (2025). The statement above is taken from the most recent of them; the others are arXiv:2503.12432.

chern-geometrycurvaturehermitian-geometrykahler-geometrymixed-curvature

Solutions 0

No solutions have been posted yet.