Converse of the pseudoindex inequality for m-positive bisectional curvature on compact Kähler manifolds

Jiangtao Li (arXiv:2511.04003, 2025) introduced $m$-positive bisectional curvature — positivity of the sum of the $m$ smallest eigenvalues of the curvature operator $R(U,\bar U)$ on a compact Kähler manifold — and proved that an $n$-dimensional compact Kähler manifold with $m$-positive bisectional curvature ($m<n$) has pseudoindex $i(M)\ge n-m+2$, where the pseudoindex is the minimal anticanonical degree of rational curves; combined with Fujita-type classifications this yields a full characterization in the case $m=2$ (del Pezzo surfaces in dimension 2; $\mathbb{CP}^n$ and the smooth quadric $Q^n$ in dimension $n\ge 3$). Whether the converse holds for $n>m>2$ is open: does every compact Kähler $n$-manifold with pseudoindex at least $n-m+2$ (in practice a Fano manifold) carry a Kähler metric with $m$-positive bisectional curvature? The converse is known only for $m=1$ and $m=2$. The first open case, $m=3$, reduces via Fujita's classification of Fano $n$-folds with pseudoindex $n-1$ to a metric-existence question on the members of that list, most of which are not homogeneous. The problem asks for a proof of the converse for all $n>m>2$, or an explicit counterexample — i.e. completion of the pseudoindex characterization of the $m$-positivity hierarchy, a natural generalization of the Frankel–Siu–Yau uniformization theorem.

Background

On a compact Kähler manifold — a closed complex manifold whose Riemannian metric is compatible with the complex structure — the curvature tensor is probed by the bisectional curvature $R(U,\bar U,V,\bar V)$, the sectional curvature of the complex two-plane spanned by holomorphic tangent vectors $U,V$. Fixing $U$, the operator $R(U,\bar U)$ on the holomorphic tangent space is Hermitian with real eigenvalues $\lambda_1(x,U)\le\cdots\le\lambda_n(x,U)$; its trace is the Ricci curvature $\mathrm{Ric}(U,\bar U)$. The classical positivity conditions are $\lambda_1>0$ for all $U$ (positive bisectional curvature) and $\lambda_1+\cdots+\lambda_n>0$ (positive Ricci curvature).

Frankel conjectured, and Siu–Yau proved in 1980, that a compact Kähler manifold of positive bisectional curvature is biholomorphic to complex projective space $\mathbb{CP}^n$; Mori proved the algebraic analogue (Hartshorne's conjecture: a smooth projective variety whose tangent bundle is ample — ample is the algebro-geometric positivity notion under which sufficiently high powers of a line bundle have enough sections to give a projective embedding — is $\mathbb{CP}^n$) in 1979. Mok (1988) extended this to nonnegative bisectional curvature: up to a finite unramified cover, such a manifold is a product of projective spaces and compact irreducible Hermitian symmetric spaces (Kähler manifolds that are symmetric spaces with compatible complex structure, e.g. quadrics and Grassmannians). Siu (1980) gave a curvature characterization of $\mathbb{CP}^n$ and the smooth complex hyperquadrics $Q^n$ and introduced his own, different notion of $m$-positivity of bisectional curvature, which requires nonnegative bisectional curvature. Variants of the Frankel conclusion under positive orthogonal bisectional curvature (positivity of $R(U,\bar U,V,\bar V)$ for orthogonal holomorphic directions) were settled by Chen, Gu–Zhang, and Feng–Liu–Wan.

In arXiv:2511.04003 Jiangtao Li introduced a new curvature condition: a compact Kähler manifold $(M,g)$ of dimension $n$ has $m$-positive bisectional curvature if

$$\lambda_1(x,U)+\cdots+\lambda_m(x,U)>0\quad\text{for all }x\in M\text{ and all nonzero }U\in T^{1,0}_xM,$$

a spectral condition interpolating between positive bisectional curvature ($m=1$) and positive Ricci curvature ($m=n$), and deliberately different from Siu's earlier notion. The main theorem of the note settles $m=2$: an $n$-dimensional compact Kähler manifold admits a Kähler metric with 2-positive bisectional curvature if and only if $n=2$ and it is a del Pezzo surface (a two-dimensional Fano manifold, i.e. a smooth projective surface with ample anticanonical bundle), or $n\ge 3$ and it is biholomorphic to $\mathbb{CP}^n$ or the smooth quadric $Q^n$ (the smooth hypersurface in $\mathbb{CP}^{n+1}$ cut out by a nondegenerate quadratic form).

The bridge to algebraic geometry is the pseudoindex. For a Fano manifold $M$ (a compact complex manifold whose anticanonical bundle $-K_M$ — the dual of the canonical bundle, the determinant line bundle of holomorphic one-forms — is ample; by Yau's solution of the Calabi conjecture, such manifolds carry Kähler metrics of positive Ricci curvature), the pseudoindex is the minimal anticanonical degree of a rational curve (the image of a holomorphic map from $\mathbb{CP}^1$):

$$i(M)=\inf\{-K_M\cdot C:\ C\ \text{a rational curve on}\ M\}$$

A theorem of Miyaoka (2004), completed by Dedieu–Höring (2017), states that an $n$-dimensional Fano manifold with $i(M)\ge n$ is $\mathbb{CP}^n$ or $Q^n$. Li proves the analytic counterpart (Proposition 5.3): if $M$ is a compact Kähler $n$-manifold with $m$-positive bisectional curvature and $m<n$, then $i(M)\ge n-m+2$ (the case $m=2$ recovers $i(M)\ge n$, which drives the main theorem). The proof pulls the tangent bundle back to $\mathbb{CP}^1$ along a rational curve and deforms the pulled-back connection by the Yang–Mills flow (the $L^2$-gradient flow of the Yang–Mills energy $\int|F_A|^2$, where $F_A$ is the curvature of a unitary connection $A$); a new tensor maximum principle, derived from Hamilton's maximum principle for parabolic systems (which ensures that a closed convex cone of tensors preserved by the reaction terms is preserved along the flow), shows that $m$-positivity of the Hermitian endomorphism $\sqrt{-1}\Lambda_\omega F_A$ (the curvature $F_A$ contracted with the Kähler form $\omega$) is preserved along the flow, and the limiting Yang–Mills connection splits as a direct sum of line bundles (by the Kobayashi–Hitchin correspondence on $\mathbb{CP}^1$, the theorem identifying Yang–Mills connections with polystable holomorphic bundles), forcing the degree estimate.

The converse of Proposition 5.3 is the open question. For $m=1$ it holds: $i(M)\ge n+1$ forces $M\cong\mathbb{CP}^n$, which has positive bisectional curvature for the Fubini–Study metric (the Kähler metric on $\mathbb{CP}^n$ induced by the standard Hermitian inner product on $\mathbb{C}^{n+1}$). For $m=2$ it holds: $i(M)\ge n$ forces $\mathbb{CP}^n$ or $Q^n$, and both carry 2-positive metrics — for $Q^n$ this is an explicit computation on the Hermitian symmetric metric. For $m=3$ the condition becomes $i(M)\ge n-1$, and Fujita's classification (1990) of $n$-dimensional Fano manifolds with pseudoindex $n-1$ gives a finite explicit list; but nothing is known about whether the non-homogeneous members of that list — the del Pezzo manifolds, i.e. Fano $n$-folds of pseudoindex $n-1$ such as the smooth cubic hypersurfaces, for which $-K_X\cdot\ell=n-1$ on a line $\ell$ — admit Kähler metrics with 3-positive bisectional curvature. For general $m$, no algebraic classification of Fano manifolds with pseudoindex at least $n-m+2$ is available, and the question is wide open.

The question also connects to the program around orthogonal Ricci curvature, $\mathrm{Ric}^\perp(X)=\mathrm{Ric}(X,\bar X)-H(X)$, where $H(X)=R(X,\bar X,X,\bar X)$ is the holomorphic sectional curvature, the bisectional curvature in the single direction $X$. Ni–Wang–Zheng (2021) proved that a compact Kähler threefold with $\mathrm{Ric}^\perp>0$ is biholomorphic to $\mathbb{CP}^3$ or $Q^3$ and obtained partial results in dimension 4, with higher dimensions open; positivity of $\mathrm{Ric}^\perp$ is weaker than $m$-positivity for every $m<n$, so progress on the pseudoindex converse is expected to inform that classification program.

Problem Statement

Fix integers $n>m>2$. Let $M$ be a compact Kähler manifold of dimension $n$ whose pseudoindex satisfies

$$i(M)\ge n-m+2,$$

where the pseudoindex $i(M)=\inf\{-K_M\cdot C:\ C\text{ a rational curve on }M\}$ is the minimal anticanonical degree of a rational curve. The pseudoindex is defined for Fano manifolds — compact complex manifolds with ample anticanonical bundle (Definition 4.1 of Li, arXiv:2511.04003) — and the question is aimed at that class; in particular, the hypothesis is not to be read as holding vacuously for compact Kähler manifolds that contain no rational curves. Does $M$ necessarily admit a Kähler metric with $m$-positive bisectional curvature, i.e. a metric whose curvature tensor $R$ is such that the sum $\lambda_1(x,U)+\cdots+\lambda_m(x,U)$ of the $m$ smallest eigenvalues — ordered increasingly, $\lambda_1(x,U)\le\cdots\le\lambda_n(x,U)$ — of the Hermitian operator $R(U,\bar U)$ on $T^{1,0}_xM$ is strictly positive for every point $x\in M$ and every nonzero holomorphic tangent vector $U$?

This is the converse of Proposition 5.3 of Li (arXiv:2511.04003), asked there for every pair $(n,m)$ with $n>m>2$: a general proof would settle the question, and so would a counterexample, i.e. a compact Kähler manifold $M^n$ with $i(M)\ge n-m+2$ for some such pair, together with a proof that no Kähler metric on $M$ has $m$-positive bisectional curvature. The first open case is $m=3$: by Fujita's classification of $n$-dimensional Fano manifolds with pseudoindex $n-1$, the question there becomes whether every member of Fujita's list — including its non-homogeneous del Pezzo manifolds (Fano $n$-folds of pseudoindex $n-1$, such as smooth cubic hypersurfaces) — carries a Kähler metric with 3-positive bisectional curvature. The answer is already known to be affirmative for $m=1$ ($\mathbb{CP}^n$) and for $m=2$ ($\mathbb{CP}^n$ and $Q^n$ in dimension $n\ge3$; all del Pezzo surfaces in dimension 2).

References

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