Ni–Zheng curvature conjecture for Kähler C-spaces

Let $M$ be a Kähler $C$ -space, and let $g$ be its canonical Kähler–Einstein metric. At every point $p\in M$ , for every $X\in T^{1,0}_pM$ and every complex-linear map $T:T^{1,0}_pM\to T^{1,0}_pM$ , the cross quadratic bisectional curvature satisfies $\operatorname{CQB}_g(X,T)\ge 0$ , and the dual cross quadratic bisectional curvature satisfies $\operatorname{dCQB}_g(T)>0$ whenever $T\ne 0$ . In a unitary frame $(E_i)$ , these curvature forms are conventionally written as $\operatorname{CQB}_g(X,T)=\sum_{i,j}R_g(X,\overline X,E_i,\overline{E_j})\langle T(E_j),T(E_i)\rangle$ and $\operatorname{dCQB}_g(T)=\sum_{i,j}R_g(E_i,\overline{E_j},T(E_i),\overline{T(E_j)})$ .

References

The cross quadratic bisectional curvature of Kähler C-spaces of classical type — arXiv — Run-Qiang Jian, Zhuhong Zhang

Source: arXiv

complex-differential-geometry khler-geometry

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Open · progress reported

A new unrefereed paper proves the predicted curvature property for classical examples, but the conjecture for all such spaces remains open.

Ni and Zheng conjectured that every Kähler $C$ -space with its canonical Kähler–Einstein metric has nonnegative $CQB$ and positive $d\mathrm{CQB}$ . Their published work gave supporting examples, not a proof for all Kähler $C$ -spaces.

Known results

Ni and Zheng (2023): irreducible examples with arbitrarily large $b_{2}$ satisfy nonnegative $CQB$ and positive $d\mathrm{CQB}$ ; this includes type $A$ flag manifolds.
Ni and Zheng (2023): positivity of $\mathrm{CQB}_{1}$ or $d\mathrm{CQB}_{1}$ implies positive Ricci curvature and hence the Fano property; a corresponding nonnegative result holds under a no-flat-factor hypothesis.

Classical-type advance (October 28, 2026)

Run-Qiang Jian and Zhuhong Zhang report proving the curvature assertion for Kähler $C$ -spaces of classical type, with the Fano property as a consequence. This is a substantial partial advance, but the unrefereed preprint does not settle the full conjecture, including exceptional types.

Current status (as of October 2026): The conjecture has claimed progress for classical Kähler $C$ -spaces, but its full statement for all Kähler $C$ -spaces remains open and the new claim is unverified.

Sources

No solutions have been posted yet.