Positive orthogonal holomorphic bisectional curvature on compact complex manifolds: is the Hermitian Frankel property rigid beyond the conformally Kähler setting?

Kalafat and Koca (Geom. Dedicata 174, 2015) proved that a compact complex surface admitting a conformally Kähler metric of positive orthogonal holomorphic bisectional curvature is biholomorphic to the complex projective plane, and that an Einstein Hermitian metric with this property is, up to rescaling, the Fubini–Study metric. The Kähler-case analogue is known in every complex dimension $n\ge 2$ (Chen, Gu–Zhang, Wilking, Feng–Liu–Wan), and since positivity of the orthogonal bisectional curvature is invariant under conformal rescaling of a Hermitian metric, the conformally Kähler case is likewise settled in all dimensions. These results leave open the following generalization: if the conformally Kähler hypothesis is dropped entirely, must a compact complex $n$-manifold ($n\ge 2$) admitting an arbitrary Hermitian metric whose orthogonal holomorphic bisectional curvature is positive still be biholomorphic to $\mathbb{CP}_n$? Even the case of complex surfaces without the conformally Kähler or Einstein assumption is open.

Background

A Riemannian metric $g$ on a complex manifold $(M,J)$ is Hermitian if the complex structure $J$ acts orthogonally at every point. Its holomorphic sectional curvature $\mathrm{H}(X)=\mathrm{Rm}(X,JX,X,JX)$ is the sectional curvature of the complex line (the $J$-invariant real plane) spanned by a unit tangent vector $X$, and its holomorphic bisectional curvature $\mathrm{H}(X,Y)=\mathrm{Rm}(X,JX,Y,JY)$ compares two complex lines. The metric is Kähler if the Kähler form $\omega(\cdot,\cdot)=g(J\cdot,\cdot)$ is closed; for Kähler metrics the bisectional curvature is a sum of two sectional curvatures, but this fails for a general Hermitian metric. The classical Frankel conjecture - a compact Kähler manifold of positive holomorphic bisectional curvature is biholomorphic to complex projective space $\mathbb{CP}_n$ - was proved by Siu–Yau by harmonic-map methods and independently by Mori by algebraic-geometric methods; the four-dimensional case goes back to Andreotti. The Fubini–Study metric is the standard homogeneous (Kähler–Einstein) metric on $\mathbb{CP}_n$.

The orthogonal holomorphic bisectional curvature (orthogonal bisectional curvature, OBC) is the restriction of $\mathrm{H}(X,Y)$ to pairs of orthogonal complex lines, i.e. $\mathrm{Rm}(X,JX,Y,JY)$ for unit vectors with $X\perp Y$ and $X\perp JY$. The condition was introduced by Cao and Hamilton in Kähler–Ricci flow studies (the Kähler–Ricci flow is the parabolic evolution of a Kähler metric by its Ricci tensor). For Kähler metrics, positive OBC is strictly weaker than positive bisectional curvature, and the corresponding Frankel-type theorem was established in all dimensions: Chen proved it under the extra assumption of positive first Chern class $c_1>0$ (positivity of the anticanonical class), Gu–Zhang showed $c_1>0$ is automatic, Wilking gave a Ricci-flow reproof via his Lie-algebraic machinery, and Feng–Liu–Wan gave a direct Siu–Yau-style proof. A metric is conformally Kähler if some positive rescaling $ug$ is Kähler. Kalafat–Koca proved that a compact complex surface with a conformally Kähler metric of positive OBC is $\mathbb{CP}_2$; their key Lemma 2.1 states that positivity of OBC is invariant under conformal rescaling of a Hermitian metric, so the OBC condition passes to the Kähler representative in the conformal class. They also proved, using the classical fact that a compact Hermitian–Einstein surface is conformally Kähler, that an Einstein Hermitian metric (Einstein meaning the Ricci tensor is proportional to the metric) of positive OBC on a compact complex surface is the Fubini–Study metric up to rescaling. Their argument is otherwise four-dimensional: it uses the splitting of the trace-free Weyl curvature tensor into self-dual and anti-self-dual parts (a splitting available only in real dimension four), a Weitzenböck formula (an identity expressing a Laplacian through covariant derivatives plus curvature terms), and the Enriques–Kodaira classification of compact complex surfaces.

In the genuinely non-Kähler Hermitian world the landscape is different. Curvature conditions are often imposed on the Chern connection (the canonical connection compatible with both $J$ and the Hermitian metric), which differs from the Levi-Civita connection off the Kähler locus. Yang's RC-positivity (for every nonzero $(1,0)$ vector $v$ there exists $w$ with positive bisectional pairing $\mathrm{H}(v,w)$) and the Hermitian curvature flow of Streets–Tian (an analogue of Ricci flow for non-Kähler Hermitian metrics) yield structural results — e.g. every holomorphic map from an RC-positive compact complex manifold into a manifold whose cotangent bundle is nef (numerically effective, i.e. nonnegative on every curve) is constant, and preservation of positivity of the Chern OBC along the flow — but no classification. Tong exhibited generalized Hopf manifolds (compact non-Kähler complex manifolds diffeomorphic to $S^1\times S^{2n-1}$) and Vaisman manifolds (a class of compact locally conformally Kähler manifolds — Hermitian manifolds whose Kähler form is locally proportional to a Kähler form) satisfying a positivity condition that coincides in the Kähler case with nonnegativity of the quadratic orthogonal bisectional curvature (a trace-type curvature condition on orthogonal complex-line pairs), showing that non-Kähler examples populate conditions arbitrarily close to this one.

Problem Statement

Let $(M,J)$ be a compact complex manifold of complex dimension $n\ge 2$, and let $g$ be a Hermitian metric on $(M,J)$ - not assumed Kähler, conformally Kähler, balanced (Hermitian with co-closed Kähler form), or Einstein (Ricci tensor proportional to the metric) - whose orthogonal holomorphic bisectional curvature is positive: at every point and for all unit tangent vectors $X,Y$ with $X\perp Y$ and $X\perp JY$,

$$\mathrm{Rm}_g(X, JX, Y, JY) > 0,$$

where $\mathrm{Rm}_g$ is the $(0,4)$ curvature tensor of the Levi-Civita connection of $g$ (the convention of the source work; for Kähler metrics this coincides with the Cao–Hamilton orthogonal bisectional curvature). Must $M$ be biholomorphic to $\mathbb{CP}_n$? The question is open already for complex surfaces ($n=2$) when the metric is neither conformally Kähler nor Einstein; the conformally Kähler case in all dimensions and the Kähler case in all dimensions are known (both affirmative). A negative answer would be a compact complex manifold not biholomorphic to $\mathbb{CP}_n$ together with a Hermitian metric satisfying the displayed positivity condition. The Chern-connection variant of the same positivity condition is a closely related but distinct question.

References

1. Mustafa Kalafat and Caner Koca, Conformally Kähler surfaces and orthogonal holomorphic bisectional curvature, Geometriae Dedicata 174 (2015), no. 1, 401–408, DOI 10.1007/s10711-014-0025-9 (arXiv:1410.5001), https://doi.org/10.1007/s10711-014-0025-9
2. Yum-Tong Siu and Shing-Tung Yau, Compact Kähler manifolds of positive bisectional curvature, Inventiones Mathematicae 59 (1980), 189–204, DOI 10.1007/BF01390043, https://doi.org/10.1007/BF01390043
3. Shigefumi Mori, Projective manifolds with ample tangent bundles, Annals of Mathematics 110 (1979), 593–606, DOI 10.2307/1971241, https://doi.org/10.2307/1971241
4. Aldo Andreotti, On the complex structures of a class of simply-connected manifolds, in: Algebraic Geometry and Topology (A Symposium in Honor of S. Lefschetz), Princeton University Press (1957), 53–77, DOI 10.1515/9781400879915-005, https://doi.org/10.1515/9781400879915-005
5. X. X. Chen, On Kähler manifolds with positive orthogonal bisectional curvature, Advances in Mathematics 215 (2007), 427–445, DOI 10.1016/j.aim.2006.11.006, https://doi.org/10.1016/j.aim.2006.11.006
6. Hui-Ling Gu and Zhu-Hong Zhang, An extension of Mok's theorem on the generalized Frankel conjecture, Science China Mathematics 53 (2010), no. 5, 1253–1264, DOI 10.1007/s11425-010-0013-y, https://doi.org/10.1007/s11425-010-0013-y
7. Burkhard Wilking, A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities, Journal für die reine und angewandte Mathematik 679 (2013), 223–247, DOI 10.1515/crelle.2012.018, https://doi.org/10.1515/crelle.2012.018
8. Huitao Feng, Kefeng Liu and Xueyuan Wan, Compact Kähler manifolds with positive orthogonal bisectional curvature, Mathematical Research Letters 24 (2017), no. 3, 767–780, DOI 10.4310/MRL.2017.v24.n3.a7, https://doi.org/10.4310/MRL.2017.v24.n3.a7
9. Yury Ustinovskiy, Lie-algebraic curvature conditions preserved by the Hermitian curvature flow, Mathematische Annalen 379 (2021), no. 3-4, 1713–1745, DOI 10.1007/s00208-020-01965-y, https://doi.org/10.1007/s00208-020-01965-y
10. Freid Tong, A new positivity condition for the curvature of Hermitian manifolds, Mathematische Zeitschrift 298 (2021), no. 3-4, 1175–1185, DOI 10.1007/s00209-020-02647-w, https://doi.org/10.1007/s00209-020-02647-w
11. Xiaokui Yang, RC-positivity, vanishing theorems and rigidity of holomorphic maps, Journal of the Institute of Mathematics of Jussieu 20 (2021), 1023–1038, arXiv:1807.02601, https://arxiv.org/abs/1807.02601
12. Huitao Feng, Kefeng Liu, Xueyuan Wan and Xiaokui Yang, Rational curves on Hermitian manifolds, arXiv:1409.2404 (preprint, withdrawn; version 2 is a withdrawal notice), https://arxiv.org/abs/1409.2404