Nef tangent bundles force homogeneity in dimension six

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Family 067 · Algebraic and complex geometry
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-12

1. The question

Some algebraic varieties look the same at every point because a large symmetry group moves any point to any other. Projective spaces, quadrics and Grassmannians are familiar examples. Can a numerical positivity condition force a variety to have this much symmetry?

The Campana–Peternell conjecture predicts that a smooth complex Fano variety with nef tangent bundle is rational homogeneous. The hypothesis involves degrees of bundles on curves. The conclusion identifies the entire variety as a quotient of an algebraic group.

The manuscript in this family claims the conjecture in complex dimension six. Its main new case is recognized as the Grassmannian of two-dimensional subspaces of a five-dimensional vector space. The proposed bridge from positivity to symmetry is the geometry of tangent directions to minimal rational curves.

2. Background and essential definitions

Fano varieties and nef tangent bundles

A smooth projective variety $X$ is Fano when its anticanonical line bundle $-K_X$ is ample. Here $K_X$ is the bundle of top-degree holomorphic differential forms. Ampleness means that a sufficiently high tensor power supplies an embedding into projective space.

The tangent bundle $T_X$ collects the tangent spaces at all points. A vector bundle $E$ is nef, or numerically effective, when the tautological line bundle $\mathcal O_{\mathbb P(E)}(1)$ has nonnegative degree on every curve in its projectivization. In this definition, $\mathbb P(E)$ parametrizes one-dimensional quotients. Equivalently, after pulling $E$ back along a map from a smooth projective curve, every quotient line bundle has nonnegative degree.

Nefness permits zero degrees. It is substantially weaker than ampleness and does not say that one has a Hermitian metric with a particular nonnegative curvature tensor. For tangent bundles, it nevertheless places strong restrictions on how rational curves deform.

A rational homogeneous variety is a space $G/P$, where $G$ is a connected semisimple complex algebraic group and $P$ is a parabolic subgroup, so the quotient is projective. The group acts transitively. Its infinitesimal action generates the tangent bundle by global sections, and that implies nefness. The conjecture asks for a converse under the Fano hypothesis.

Two numerical guides

The Picard number $\rho(X)$ is the number of independent divisor classes modulo numerical equivalence. When it equals one, all divisor classes point in the same numerical direction, making independent contractions especially restrictive.

The pseudoindex is

$$\iota_X=\min\{-K_X\cdot C:C\subset X\text{ is an irreducible rational curve}\}.$$

The intersection number is the anticanonical degree of the curve. A rational curve has normalization $\mathbb P^1$. The pseudoindex records the smallest degree, rather than the divisibility of $-K_X$ in the line-bundle group; the latter is the different invariant called the index.

Choose a covering family of rational curves of this smallest degree. At a general point $x$, collect their tangent directions and take the closure inside the projective space of tangent lines. The result is the variety of minimal rational tangents, often abbreviated VMRT. Think of it as the set of directions in which the simplest algebraic curves can leave the point.

3. Development history and previous partial results

Mori’s theorem that a projective manifold with ample tangent bundle is projective space gave a powerful model for positivity-based classification. Campana and Peternell proposed the nef analogue in their 1991 paper, with homogeneous spaces replacing projective space as the expected class.

Campana–Peternell established the conjecture in dimensions at most three and for fourfolds with Picard number greater than one. Mok and Hwang’s methods using minimal rational tangents completed the remaining four-dimensional case. Watanabe treated fivefolds with higher Picard number, and Kanemitsu completed the fivefold classification.

A second theorem of Kanemitsu covers an $n$-dimensional Fano manifold with nef tangent bundle whenever

$$\rho(X)>n-5.$$

For a sixfold this already settles every Picard number at least two. Thus the new theorem does not begin with all sixfolds as unexplored territory: its remaining work lies at Picard number one.

Earlier characterization results then reduce the possibilities to projective space, a quadric, and pseudoindex four or five. Watanabe’s coindex-three paper includes the pseudoindex-four sixfold case. The present manuscript gives another argument for that case, while identifying pseudoindex five as its principal new contribution.

There is also a broader analytic background. Demailly, Peternell and Schneider’s structure theorem for compact Kähler manifolds with nef tangent bundle relates them, after finite unramified cover, to Fano fibers over a complex torus. That reduction explains why one additional Fano dimension can yield consequences for compact Kähler manifolds that are not themselves projective.

4. What the new manuscripts claim

The sixfold classification

The main theorem states that every smooth connected complex projective Fano variety $X$ satisfying

$$\dim_{\mathbb C}X=6,\qquad T_X\text{ nef}$$

is rational homogeneous. There is no additional assumption on Picard number, Betti numbers, or the existence of a particular group action.

The decisive specialized statement says that when $\rho(X)=1$ and $\iota_X=5$,

$$X\cong\operatorname{Gr}(2,5).$$

The Grassmannian on the right parametrizes two-dimensional linear subspaces of $\mathbb C^5$. Its dimension is $2(5-2)=6$, so it has exactly the required size. Combining this case with the previous classification and the manuscript’s treatment of pseudoindex four gives the full theorem.

A universal-cover consequence

For a connected compact Kähler manifold, define its augmented irregularity by

$$\widetilde q(X)=\max_{X'\to X}h^1(X',\mathcal O_{X'}),$$

where the maximum runs over connected finite étale covers, meaning finite unramified holomorphic covers. The quantity measures the largest torus direction visible after such a cover.

Assume the holomorphic tangent bundle is analytically nef and

$$0\leq\dim_{\mathbb C}X-\widetilde q(X)\leq6.$$

The manuscript claims that a suitable finite cover has a smooth, holomorphically locally trivial fibration over a complex torus, with rational homogeneous fiber $F$. Moreover, the ordinary universal cover satisfies

$$\widetilde X\cong F\times\mathbb C^{\widetilde q(X)}.$$

The fiber is allowed to be a point. This covers every such compact Kähler manifold of dimension at most six, and also higher-dimensional ones whose remaining fiber dimension is at most six. The finite-cover bundle can stay twisted; the displayed product belongs to the universal cover.

The classification also yields positive Kähler–Einstein metrics on Fano manifolds with nef tangent bundle in dimensions one through six. With the manuscript’s normalization, the Kähler form satisfies $\operatorname{Ric}(\omega)=\omega$ and represents $2\pi c_1(-K_X)$.

5. Why the advance matters

Importance: 4/5. This is a major new-dimensional case of a longstanding classification conjecture. It would show that weak numerical positivity still forces complete homogeneity at the first dimension beyond the established fivefold theorem.

The proposed method also has broader interest: it extracts a rigid geometric model from compatibility between two different fibrations of a space of curves. The Kähler consequence translates the classification into a product description of universal covers.

The score is below the collection’s highest tier because the main classification remains dimension-specific. The all-dimensional Campana–Peternell conjecture is not claimed here. As elsewhere in the series, the score concerns significance if the stated result is correct.

6. The main idea

Look at the same family in two ways

Let $V$ parametrize minimal rational curves, and let $U$ consist of a curve together with a marked point on it. There are two maps:

$$V\ \xleftarrow{\ \pi\ }\ U\ \xrightarrow{\ e\ }\ X.$$

The map $\pi$ forgets the marked point, while $e$ remembers only its location on $X$. A fiber of $\pi$ is a projective line. Nefness of $T_X$ gives the deformation properties needed for these families and evaluation maps to be smooth.

In the pseudoindex-five case, a fiber $F=e^{-1}(x)$ is a smooth threefold. It parametrizes the minimal curves through $x$. Their tangent directions give a finite map from $F$ into $\mathbb P^5$. Pulling back the hyperplane bundle gives an ample line bundle $L$ on $F$.

At this stage, neither the degree of that map nor the identity of its image is known. The core task is to recover them from the two compatible fibrations of $U$.

A numerical identity narrows the geometry

Chern classes are cohomological invariants of vector bundles; their top-degree products can be integrated to give characteristic numbers. The manuscript combines identities for these numbers from both maps, using Riemann–Roch and Hodge-theoretic fixed-part information.

Integration along a fiber supplies linear relations. The projection formula supplies additional relations in which numbers factor as products. The combination is claimed to force

$$32-20u+3u^2=0, \qquad u=\frac{c_1(T_F)\cdot L^2}{L^3}.$$

Thus $u=4$ or $u=8/3$. The manuscript includes an exact arithmetic certificate for this finite elimination. The role of that calculation is to extract the displayed identity from geometric constraints; the classification also needs the geometric steps before and after it.

Take the curve obtained by intersecting two general divisors of $L$. Adjunction computes its genus as

$$g=1+\left(1-\frac u2\right)L^3.$$

The tangent image spans $\mathbb P^5$, so its degree is at least three, and consequently $L^3\geq3$. The choice $u=4$ would make the genus negative. The other choice forces $L^3=3$, a degree-one tangent map and a degree-three image.

Recognize the tangent directions, then the manifold

The classification of varieties of minimal degree, together with smoothness and normality, identifies this image as

$$\mathbb P^1\times\mathbb P^2\hookrightarrow\mathbb P^5$$

in its Segre embedding. The Segre map sends two vectors to their rank-one tensor; its coordinates are all pairwise products of their coordinates.

This is precisely the tangent-direction geometry of $\operatorname{Gr}(2,5)$. At a two-plane $A\subset\mathbb C^5$, the tangent space is $\operatorname{Hom}(A,\mathbb C^5/A)$. Tangents to lines correspond to rank-one maps, whose projectivization is the displayed product. Mok’s recognition theorem then converts equality of these projective tangent models into an isomorphism of the original manifolds.

The separate pseudoindex-four argument uses a factorization theorem to isolate a possible nonhomogeneous case. Its alternative fibrations impose incompatible cohomological and integral-degree conditions. It rules out that alternative, rather than ruling out homogeneous pseudoindex-four sixfolds themselves.

7. Scope, limitations, and open questions

The principal theorem is for smooth projective Fano varieties over $\mathbb C$, in dimension six. Dropping smoothness, changing characteristic, or replacing the tangent bundle by another bundle requires separate results.

Nef tangent bundle alone does not make a compact Kähler manifold rational homogeneous: complex tori already show why a torus factor must be allowed. The analytic corollary uses the earlier Albanese structure theorem and the condition on the remaining fiber dimension. Its torus base can be nonprojective.

The universal cover is holomorphically convex: enough holomorphic functions remain after contracting its compact homogeneous fibers. It is generally not Stein when $F$ has positive dimension, because a Stein space cannot contain such a compact positive-dimensional complex subvariety. Nor does the asserted universal-cover product imply that some finite cover splits as a global product.

The natural next classification question is higher dimension. There the minimal-tangent varieties have more possibilities, and this particular characteristic-number identity does not automatically extend. The manuscript settles its six-dimensional configuration rather than supplying an all-dimensional recognition formula.

8. Sources and further reading

The preceding frontier is described in Kanemitsu’s fivefold theorem, his higher-Picard-number theorem, and Watanabe’s coindex-three work. The source manuscript cites the original Hwang–Mok tangent-recognition results and the Demailly–Peternell–Schneider structure theorem, and gives the precise characteristic-class calculation used in the remaining case.

Source manuscripts

• M1. OpenAI, The Campana–Peternell conjecture in dimension six (September 25, 2026).

Sources consulted on 12 October 2026 at repository revision fd4aeeb. Download the manuscript citations.