k-positive symmetrized curvature operators on compact Kähler manifolds and rigidity to complex projective space (Wang–Yang Conjecture 2.4)

Conjecture 2.4 of Wang and Yang (arXiv:2509.00468, 30 August 2025) asks whether a compact Kähler manifold of complex dimension $n$ whose symmetrized curvature operator $\mathcal{R}$ on $\operatorname{Sym}^2T^{1,0}M$ — an operator also called the Calabi curvature operator in the recent literature — is $k$-positive, meaning the sum of the $k$ smallest eigenvalues of $\mathcal{R}$ is positive at every point, for some integer $1\le k\le\lfloor n/2\rfloor$, must be biholomorphic to $\mathbb{CP}^n$. The $k=1$ endpoint is exactly the Siu–Yau theorem; the range $2\le k\le\lfloor n/2\rfloor$ (first open case $n=4$, $k=2$) is open. The formulating paper proves partial results — notably that $\lfloor n/2\rfloor$-positivity already forces the same cohomology ring as $\mathbb{CP}^n$ — and records that the hyperquadric $Q^n$ has $\mathcal{R}$-eigenvalues $2-n,2,\dots,2$, so it becomes $k$-positive only for $k\ge\lfloor n/2\rfloor+1$: the conjecture's range is exactly what excludes it.

Background

A compact Kähler manifold is a closed complex manifold $M^n$ of complex dimension $n$ carrying a Hermitian metric $g$ whose associated fundamental two-form $\omega$ is closed; equivalently the complex structure is parallel, which forces the curvature tensor to satisfy extra symmetries and makes several curvature quantities specific to this setting control the global complex geometry. The most important here is the holomorphic bisectional curvature, introduced by Goldberg and Kobayashi: for unit $(1,0)$ tangent vectors $U,V$ it is the quantity $R(U,\bar U,V,\bar V)$, and positivity of this quantity in all directions is the strongest standard curvature positivity condition on a Kähler manifold.

Frankel conjectured in 1961 that a compact Kähler manifold with positive bisectional curvature is biholomorphic (that is, isomorphic as a complex manifold) to the complex projective space $\mathbb{CP}^n$ with its homogeneous Fubini–Study metric. Siu and Yau proved this in 1980: starting from the fact that the second Betti number (the rank of the second real cohomology group) of such a manifold is $1$ and from a classical characterization of $\mathbb{CP}^n$ through the size of its first Chern class, they produced rational curves (holomorphic images of $\mathbb{CP}^1$) from energy-minimizing harmonic maps from the sphere — showing such maps are holomorphic or anti-holomorphic, then using Grothendieck's splitting theorem (every holomorphic vector bundle over $\mathbb{CP}^1$ splits as a direct sum of holomorphic line bundles) to convert curvature positivity into a degree inequality that forces $\mathbb{CP}^n$. Their paper cites Mori's independent proof of the purely algebro-geometric analogue (a projective manifold with ample, i.e. positive, tangent bundle is $\mathbb{CP}^n$), obtained by reduction to positive characteristic. Siu's companion 1980 papers extended the same circle of ideas: one established the strong rigidity of compact Kähler manifolds through the complex-analyticity of harmonic maps (Annals of Mathematics 112), the other gave curvature characterizations of $\mathbb{CP}^n$ and of the smooth complex hyperquadrics $Q^n$, the nonsingular quadric hypersurfaces in $\mathbb{CP}^{n+1}$ (Duke Mathematical Journal 47).

The subsequent literature weakened the hypothesis step by step, and each weakening is closed. Mok (1988) settled the semipositive version, known as the generalized Frankel conjecture: a compact irreducible Kähler manifold with nonnegative bisectional curvature and second Betti number $b_2=1$ is either $\mathbb{CP}^n$ or an irreducible compact Hermitian symmetric space (a homogeneous Kähler manifold, such as a quadric or a Grassmannian, that is symmetric as a Riemannian space). Gu–Zhang (2010) extended Mok's theorem to the orthogonal-bisectional-curvature condition, and Feng–Liu–Wan (2017) confirmed that positive orthogonal bisectional curvature — positivity of $R(U,\bar U,V,\bar V)$ only for mutually orthogonal unit vectors $U,V$, a strictly weaker condition than positive bisectional curvature — still forces $\mathbb{CP}^n$.

The problem sits on a new weakening axis. The symmetrized curvature operator is the Hermitian endomorphism

$$\mathcal{R}:\Gamma(M,\operatorname{Sym}^2T^{1,0}M)\to\Gamma(M,\operatorname{Sym}^2T^{1,0}M)$$

introduced in this form by Calabi and Vesentini (it is also called the Calabi curvature operator in the recent literature) and determined by the curvature tensor through the natural pairing of symmetric $(2,0)$-tensors: in a unitary frame, $\langle\mathcal{R}(a),b\rangle=\sum_{i,j,k,\ell}R_{ijk\ell}\,a_{ik}\,\overline{b_{j\ell}}$ for symmetric $a,b$, the convention of equation (1.6) of the source preprint. The bundle $\operatorname{Sym}^2T^{1,0}M$ has complex rank $n(n+1)/2$, and $\mathcal{R}$ is called $k$-positive if at every point of $M$ the sum of the $k$ smallest eigenvalues of $\mathcal{R}$ is positive; $k=1$ means positive definiteness. A direct computation shows $\mathcal{R}=2\cdot\mathrm{Id}$ for the Fubini–Study metric on $\mathbb{CP}^n$, and positive definiteness of $\mathcal{R}$ implies positive bisectional curvature, so the $k=1$ endpoint of the question below is exactly the Siu–Yau theorem. The motivation recorded by Wang and Yang is that the holomorphic tangent bundle of a compact complex manifold of dimension at least $2$ can never be Nakano positive — Nakano positivity being vector-bundle positivity where the curvature acts as a positive Hermitian form on $T^{1,0}M$-valued bundle directions, whose vanishing theorem would force $H^{1,1}(M)=0$ — so partial-sum positivity of $\mathcal{R}$ on $\operatorname{Sym}^2T^{1,0}M$ is the natural substitute, and one asks how weak it can be made while the $\mathbb{CP}^n$ conclusion survives. The boundary of the admissible range is visible on the hyperquadrics: Wang–Yang recall the classical computation of Calabi–Vesentini that $Q^n\subset\mathbb{CP}^{n+1}$ with the induced metric has $\mathcal{R}$-eigenvalues $\lambda_1=2-n$ and $\lambda_2=\dots=\lambda_N=2$, where $N=n(n+1)/2$, so $Q^n$ is $k$-positive exactly for $k\ge\lfloor n/2\rfloor+1$; the range $k\le n/2$ in the conjecture is precisely what excludes the quadric, and the conjecture predicts that no other homogeneous alternative enters. Partial results in the direction of the conjecture are known: the formulating paper proves Hodge-number vanishing theorems under $m$-positivity, and its Corollary 1.6 (also obtained independently by Broder–Nienhaus–Petersen–Stanfield–Wink) shows that $\lfloor n/2\rfloor$-positivity already forces the same cohomology ring as $\mathbb{CP}^n$; in 2026 the same circle of authors proved a Kähler–Einstein analogue under a shifted nonnegativity condition, and an American Institute of Mathematics workshop (May 2026) posed related problems about this operator. The biholomorphism conclusion for any $k\ge2$ without an Einstein hypothesis remains open.

Problem Statement

Let $(M^n,\omega)$ be a compact Kähler manifold of complex dimension $n$ and let $\mathcal{R}:\Gamma(M,\operatorname{Sym}^2T^{1,0}M)\to\Gamma(M,\operatorname{Sym}^2T^{1,0}M)$ be its symmetrized curvature operator, the Hermitian endomorphism determined by the curvature tensor on symmetric $(2,0)$-tensors through $\langle\mathcal{R}(a),b\rangle=\sum_{i,j,k,\ell}R_{ijk\ell}a_{ik}\overline{b_{j\ell}}$ (equation (1.6) of the source preprint); the bundle $\operatorname{Sym}^2T^{1,0}M$ has complex rank $n(n+1)/2$, and at each point the eigenvalues of $\mathcal{R}$ are ordered increasingly. Say $\mathcal{R}$ is $k$-positive if at every point of $M$ the sum of the $k$ smallest eigenvalues of $\mathcal{R}$ is positive. Prove, or disprove, the conjecture of Wang and Yang (Conjecture 2.4 of arXiv:2509.00468): if $\mathcal{R}$ is $k$-positive for some integer $1\le k\le\lfloor n/2\rfloor$, then $M$ is biholomorphic to $\mathbb{CP}^n$. The case $k=1$ is the Siu–Yau theorem (positive definite $\mathcal{R}$ implies positive bisectional curvature), so the open content is the range $2\le k\le\lfloor n/2\rfloor$, whose first instance is $n=4$, $k=2$; note that the hyperquadric $Q^n$ is $k$-positive only for $k\ge\lfloor n/2\rfloor+1$ and is therefore not an admissible counterexample.

References

1. Yum-Tong Siu and Shing-Tung Yau, Compact Kähler manifolds of positive bisectional curvature, Inventiones Mathematicae 59(2):189–204 (1980). DOI: 10.1007/BF01390043, https://doi.org/10.1007/BF01390043
2. Theodore Frankel, Manifolds with positive curvature, Pacific Journal of Mathematics 11(1):165–174 (1961). DOI: 10.2140/pjm.1961.11.165, https://doi.org/10.2140/pjm.1961.11.165
3. S. I. Goldberg and S. Kobayashi, Holomorphic bisectional curvature, Journal of Differential Geometry 1(3-4):225–234 (1967). DOI: 10.4310/jdg/1214428090, https://doi.org/10.4310/jdg/1214428090
4. Shigefumi Mori, Projective Manifolds with Ample Tangent Bundles, Annals of Mathematics 110(3):593–606 (1979). DOI: 10.2307/1971241, https://doi.org/10.2307/1971241
5. Yum-Tong Siu, The Complex-Analyticity of Harmonic Maps and the Strong Rigidity of Compact Kähler Manifolds, Annals of Mathematics 112(1):73–111 (1980). DOI: 10.2307/1971321, https://doi.org/10.2307/1971321
6. Yum-Tong Siu, Curvature characterization of hyperquadrics, Duke Mathematical Journal 47(3):641–654 (1980). DOI: 10.1215/s0012-7094-80-04740-7, https://doi.org/10.1215/s0012-7094-80-04740-7
7. Eugenio Calabi and Edoardo Vesentini, On compact, locally symmetric Kähler manifolds, Annals of Mathematics 71(3):472–507 (1960). DOI: 10.2307/1969939, https://doi.org/10.2307/1969939
8. Ngaiming Mok, The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, Journal of Differential Geometry 27(2):179–214 (1988). DOI: 10.4310/jdg/1214441778, https://doi.org/10.4310/jdg/1214441778
9. HuiLing Gu and ZhuHong Zhang, An extension of Mok's theorem on the generalized Frankel conjecture, Science China Mathematics 53(5):1253–1264 (2010). DOI: 10.1007/s11425-010-0013-y, https://doi.org/10.1007/s11425-010-0013-y
10. Huitao Feng, Kefeng Liu and Xueyuan Wan, Compact Kähler manifolds with positive orthogonal bisectional curvature, Mathematical Research Letters 24(3):767–780 (2017). DOI: 10.4310/mrl.2017.v24.n3.a7, https://doi.org/10.4310/mrl.2017.v24.n3.a7
11. Mingwei Wang and Xiaokui Yang, Weitzenböck-Bochner-Kodaira formulas with quadratic curvature terms, arXiv preprint arXiv:2509.00468 (2025). DOI: 10.48550/arXiv.2509.00468, https://arxiv.org/abs/2509.00468 (v1, 30 August 2025)
12. Jiangtao Li, A generalized Frankel conjecture via the Yang-Mills flow, arXiv preprint arXiv:2511.04003 (2025). DOI: 10.48550/arXiv.2511.04003, https://arxiv.org/abs/2511.04003
13. Teng Huang and Jiaogen Zhang, Gårding cones and positivity of curvature operators, arXiv preprint arXiv:2506.15374 (2025). DOI: 10.48550/arXiv.2506.15374, https://arxiv.org/abs/2506.15374
14. Kyle Broder, Jan Nienhaus, Peter Petersen, James Stanfield and Matthias Wink, Vanishing theorems for Hodge numbers and the Calabi curvature operator, Advances in Mathematics 500, article 111077 (2026). DOI: 10.1016/j.aim.2026.111077, https://doi.org/10.1016/j.aim.2026.111077
15. Hasan M. El-Hasan, Xiaolong Li, Jan Nienhaus, Peter Petersen, James Stanfield and Matthias Wink, A characterization of complex projective space via the Calabi curvature operator, arXiv preprint arXiv:2608.23778 (2026). DOI: 10.48550/arXiv.2608.23778, https://arxiv.org/abs/2608.23778