For Kähler manifolds, two classical pointwise curvature conditions — nonnegative holomorphic bisectional curvature (the hypothesis of the Frankel conjecture and Mok's uniformization theorem) and nonnegative isotropic curvature (the hypothesis of the Micallef–Moore theory and Seshadri's classification) — each imply nonnegative orthogonal bisectional curvature, which is strictly weaker, while the converses from the orthogonal condition fail. In complex dimension two, nonnegative bisectional curvature does imply nonnegative isotropic curvature (Ivey, as cited by Gu–Zhang). Gu and Zhang, in their 2010 extension of Mok's generalized Frankel conjecture, state that in complex dimension at least three it is unknown whether nonnegative holomorphic bisectional curvature implies nonnegative isotropic curvature. This problem records that question in source-faithful form: either prove the implication for all Kähler manifolds (equivalently, all algebraic Kähler curvature tensors) of complex dimension n≥3n\ge 3, or refute it by an explicit Kähler curvature tensor or Kähler metric with nonnegative bisectional curvature but a negative isotropic-curvature direction.

Background

A Kähler manifold is a complex manifold (M,J)(M,J) with a Riemannian metric gg compatible with the complex structure (g(Ju,Jv)=g(u,v)g(Ju,Jv)=g(u,v)) for which JJ is parallel; its curvature tensor then carries extra algebraic structure, and several competing notions of nonnegative curvature are studied.

The most classical condition is nonnegative holomorphic bisectional curvature. For tangent vectors X,YX,Y of type (1,0)(1,0) one requires

R(X,Xˉ,Y,Yˉ)  ≥  0for all X,Y∈T1,0M,R(X,\bar X,Y,\bar Y)\;\ge\;0 \quad\text{for all } X,Y\in T^{1,0}M,

equivalently R(u,Ju,v,Jv)≥0R(u,Ju,v,Jv)\ge 0 for all real tangent vectors u,vu,v; for X=YX=Y this is the holomorphic sectional curvature of the complex line spanned by XX. This is the Kähler analogue of nonnegative sectional curvature: the Frankel conjecture — a compact Kähler manifold with positive bisectional curvature is biholomorphic to complex projective space CPn\mathbb{CP}^n — was proved by Mori and by Siu–Yau, and Mok's uniformization theorem (the generalized Frankel conjecture) settled the nonnegative case: a compact irreducible Kähler manifold with nonnegative bisectional curvature is either biholomorphic to CPn\mathbb{CP}^n or isometric to a compact Hermitian symmetric space (a compact symmetric space with a compatible complex structure, such as a Grassmannian, carrying its canonical Kähler metric).

A weaker Kähler condition is nonnegative orthogonal bisectional curvature: one requires R(u,Ju,v,Jv)≥0R(u,Ju,v,Jv)\ge 0 only for pairs with u⊥vu\perp v and u⊥Jvu\perp Jv.

A different, purely Riemannian condition is nonnegative isotropic curvature (NIC). A Riemannian manifold has NIC if for every orthonormal real four-frame {e1,e2,e3,e4}\{e_1,e_2,e_3,e_4\},

R(e1,e3,e1,e3)+R(e1,e4,e1,e4)+R(e2,e3,e2,e3)+R(e2,e4,e2,e4)−2 R(e1,e2,e3,e4)  ≥  0,R(e_1,e_3,e_1,e_3)+R(e_1,e_4,e_1,e_4)+R(e_2,e_3,e_2,e_3)+R(e_2,e_4,e_2,e_4)-2\,R(e_1,e_2,e_3,e_4)\;\ge\;0,

equivalently the curvature of every totally isotropic (complex null) two-plane in the complexified tangent bundle is nonnegative. Strict positivity (PIC) was introduced by Micallef and Moore, who proved that a compact simply connected PIC manifold is homeomorphic to a sphere. On a Kähler manifold the two theories meet delicately: the identity R(u,v,u,v)+R(u,Jv,u,Jv)=R(u,Ju,v,Jv)R(u,v,u,v)+R(u,Jv,u,Jv)=R(u,Ju,v,Jv) (for u⊥vu\perp v, u⊥Jvu\perp Jv) forces the displayed expression to vanish identically on frames of the form (u,Ju,v,Jv)(u,Ju,v,Jv), so no Kähler manifold can have positive isotropic curvature, and NIC is the strongest isotropic condition available in the Kähler setting.

Micallef and Wang initiated the study of NIC on Kähler manifolds: they proved the four-dimensional classification and conjectured that a compact locally irreducible Kähler manifold with NIC is biholomorphic to CPn\mathbb{CP}^n or isometric to a compact Hermitian symmetric space. Seshadri proved this conjecture, building on Brendle and Schoen's discovery that the Ricci flow (the evolution ∂g/∂t=−2 Ric\partial g/\partial t=-2\,\mathrm{Ric}) preserves NIC; along the way one obtains that NIC implies nonnegative orthogonal bisectional curvature on Kähler manifolds. Gu and Zhang, extending Mok's theorem in another direction, classified compact Kähler manifolds with nonnegative orthogonal bisectional curvature. Their example (Σ,g)×(CPn,g0)(\Sigma,g)\times(\mathbb{CP}^n,g_0) — Σ\Sigma a Riemann surface of Gauss curvature ≥−4\ge -4 attaining −4-4, and g0g_0 the Fubini–Study metric (the standard homogeneous Kähler metric on CPn\mathbb{CP}^n) with sectional curvatures in [1,4][1,4] — has nonnegative orthogonal bisectional curvature but not NIC, so the orthogonal condition is strictly the weakest of the three. By work of Ivey, as cited by Gu and Zhang, in complex dimension two nonnegative bisectional curvature does imply NIC — which is why Seshadri's classification subsumes Mok's theorem on Kähler surfaces. In complex dimension at least three, whether the implication still holds was stated by Gu and Zhang as unknown, and it is the subject of this problem.

Problem Statement

Let (M,g,J)(M,g,J) be a Kähler manifold of complex dimension n≥3n\ge 3 with nonnegative holomorphic bisectional curvature,

R(X,Xˉ,Y,Yˉ)  ≥  0for all X,Y∈T1,0MR(X,\bar X,Y,\bar Y)\;\ge\;0 \qquad\text{for all } X,Y\in T^{1,0}M

(equivalently R(u,Ju,v,Jv)≥0R(u,Ju,v,Jv)\ge 0 for all real tangent vectors u,vu,v). Does it follow that (M,g)(M,g) has nonnegative isotropic curvature, i.e. that

R(e1,e3,e1,e3)+R(e1,e4,e1,e4)+R(e2,e3,e2,e3)+R(e2,e4,e2,e4)−2 R(e1,e2,e3,e4)  ≥  0R(e_1,e_3,e_1,e_3)+R(e_1,e_4,e_1,e_4)+R(e_2,e_3,e_2,e_3)+R(e_2,e_4,e_2,e_4)-2\,R(e_1,e_2,e_3,e_4)\;\ge\;0

holds for every orthonormal real four-frame {e1,e2,e3,e4}\{e_1,e_2,e_3,e_4\} at every point?

Both the hypothesis and the conclusion are pointwise algebraic conditions on the curvature tensor, so the question is equivalent to: is every algebraic curvature tensor satisfying the Kähler symmetries (JJ-invariance) with nonnegative bisectional curvature also of nonnegative isotropic curvature, in complex dimension at least three? A complete answer either (i) proves the implication for all Kähler manifolds — equivalently, all algebraic Kähler curvature tensors — of complex dimension n≥3n\ge 3; or (ii) refutes it by an explicit counterexample: an algebraic Kähler curvature tensor, or a Kähler metric together with a specified point and orthonormal four-frame, whose bisectional curvature is nonnegative while the displayed isotropic expression is negative at that frame. In complex dimension two the implication holds (Ivey, as cited by Gu and Zhang); the open case, explicitly left by Gu and Zhang, is n≥3n\ge 3.

References

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