The Campana–Peternell conjecture in dimension six

1 Introduction

A smooth connected complex projective variety X is Fano if its anticanonical line bundle \omega_X^{-1} is ample. A vector bundle E is nef if the tautological line bundle \mathcal O_{\mathbb P(E)}(1) has nonnegative degree on every curve. We use the quotient convention \mathbb P(E)=\operatorname{Proj}\operatorname{Sym}E. A variety is rational homogeneous if it is isomorphic to G/P, where G is a connected semisimple complex algebraic group and P is a parabolic subgroup.

The tangent bundle of a rational homogeneous variety is generated by global sections and is therefore nef. Mori’s characterization of projective space by ampleness of its tangent bundle [25] makes the corresponding question for nef tangent bundles natural. The Campana–Peternell conjecture asks whether every smooth complex Fano variety with nef tangent bundle is rational homogeneous [5]. Thus it predicts that a numerical positivity condition determines a variety with a transitive algebraic group action. We prove the conjecture in dimension six.

The Picard number \rho(X) is the rank of the group of divisor classes modulo numerical equivalence, and the pseudoindex is the least anticanonical degree of a rational curve on X. These two invariants organize much of the progress on the conjecture. Campana and Peternell established it in dimensions at most three and for fourfolds of Picard number greater than one [5, 6]. The remaining fourfold case was settled through Mok’s and Hwang’s study of the tangent directions of minimal rational curves [22, 16]. Watanabe proved the fivefold case of Picard number greater than one [29, Theorem 1.2], and Kanemitsu completed the fivefold classification [19, Theorem 0.2].

Kanemitsu also proved that a smooth complex Fano n-fold with nef tangent bundle is rational homogeneous whenever \rho(X)>n-5 [18, Theorem 0.2]. For a sixfold, this covers every \rho(X)\geq2, leaving only Picard number one. In that range, standard characterization results reduce the problem to pseudoindices four and five. Watanabe treated the pseudoindex-four sixfold case [30, Theorem 2.4]. We include an independent intersection argument for it; the principal new step is pseudoindex five, where a characteristic-number constraint leads to the Grassmannian.

The difficulty is to pass from positivity of T_X to a specific projective model for the tangent directions of minimal rational curves. Nefness gives strong smoothness and deformation properties, but these properties do not by themselves identify the threefold of directions that occurs in pseudoindex five. We extract additional information by using the universal family in two ways: as a family of rational curves over their parameter space, and as a family of threefolds over X. The compatibility of these two maps yields a characteristic-number identity strong enough to determine the projective variety of tangent directions.

For a covering family of rational curves of least anticanonical degree, the closure of the tangent directions to its members smooth at a general point is called the variety of minimal rational tangents (VMRT). Characteristic classes and the geometry of minimal curves already interact in Mok’s work on one-dimensional varieties of minimal rational tangents [22]. Under an additional fourth-Betti-number hypothesis, his argument uses the two rational-curve fibrations of the universal family and characteristic-class inequalities for a rank-two bundle on the parameter space to constrain the tangent image. Hwang removed that hypothesis [16, Section 4]; the resulting classification is stated in [23, Theorem 15]. A related structural approach studies elementary Mori contractions, the contractions of extremal rays: if all of these contractions of a Fano manifold are smooth \mathbb P^1-fibrations, the manifold is a complete flag variety [26, Theorem 1.2]. These earlier results show how multiple curve fibrations can constrain both characteristic classes and homogeneous structure. The calculation here concerns a three-dimensional tangent image. It uses the two fibrations to produce explicit linear intersection identities, then combines them with the product identities furnished by the projection formula.

1.1 The argument

Assume \rho(X)=1. Standard characterization theorems leave pseudoindices four and five; the other possibilities are excluded or give projective space \mathbb P^6 and the smooth quadric Q^6. The proof is organized around the universal family of rational curves of least degree. Its evaluation map and its projection to the parameter space are both smooth. Along every fibre of the latter map, the relative tangent bundle of evaluation is a sum of copies of \mathcal O_{\mathbb P^1}(-1).

For pseudoindex four, Kanemitsu’s factorization theorem reduces the problem to a variety admitting two fibrations. We rule out the possible fibres by Hard Lefschetz and intersection theory. The last case is an odd-dimensional quadric bundle: its orthogonal description forces a vanishing characteristic number, whereas the second fibration would make an integral divisor have nonintegral degree.

For pseudoindex five and a general point x\in X, the evaluation fibre F is a smooth threefold parametrizing minimal rational curves marked at x. Sending a marked curve to its tangent direction at x gives a finite morphism F\to\mathbb P(T_x^*X)\simeq\mathbb P^5. Its image is the variety of minimal rational tangents, and the pullback L of \mathcal O(1) is ample. At this point the degree of the map and the projective type of its image are both undetermined. The central numerical step is

We derive this identity from the cohomology of the universal family, Deligne’s fixed-part theory [10], and Grothendieck–Riemann–Roch [3]. All resulting characteristic-number constraints are written explicitly. A finite rational calculation then extracts the quadratic relation.

This method separates two useful ingredients. The two fibrations give linear relations among intersection numbers; integration along evaluation also makes selected numbers factor as products. Multiplying these relations forces a nonlinear constraint on the polarized fibre. The proof explains both ingredients and gives an exact certificate for the finite elimination.

Finally, this image spans \mathbb P^5. The sectional-genus formula excludes u=4 and forces L^3=3 when u=8/3. The variety of minimal rational tangents is consequently the Segre threefold \mathbb P^1\times\mathbb P^2\subset\mathbb P^5. The VMRT method of Hwang and Mok [17, Propositions 1.2.1–1.2.2] supplies the nondegeneracy criterion, and Mok’s recognition theorem [24] identifies X with \operatorname{Gr}(2,5), the Grassmannian of two-dimensional subspaces of \mathbb C^5.

1.2 Consequences of the classification

The classification also has a consequence for compact Kähler manifolds that need not be projective. A differential-geometric antecedent is Mok’s uniformization theorem for compact Kähler manifolds with nonnegative holomorphic bisectional curvature [21]. For nef tangent bundles, the structure theorem of Demailly–Peternell–Schneider relates the compact Kähler problem to the Fano conjecture through the fibres of the Albanese map [12, Main Theorem and Theorem 3.14]. To state the consequence precisely, a holomorphic vector bundle is nef in the analytic sense when its tautological line bundle admits smooth Hermitian metrics whose curvature is bounded below by -\varepsilon\omega for every \varepsilon>0, for a fixed Hermitian form \omega on its projectivization. This agrees with the preceding numerical definition in the projective setting [12, Section 1.A and Definitions 1.2 and 1.9]. For a connected compact Kähler manifold X with nef tangent bundle, put

This maximum exists and is at most \dim_{\mathbb C}X by the Demailly–Peternell–Schneider Albanese theorem [12, Proposition 3.9 and Theorem 3.14].

The first conclusion applies the Fano classification fibrewise to the Demailly–Peternell–Schneider fibration. The second also uses local deformation rigidity of rational homogeneous manifolds, the Fischer–Grauert theorem, and the universal-cover splitting theorem of Claudon–Höring–Kollár. The torus A may be nonprojective, and the finite-cover bundle may remain twisted: the product is asserted only for the ordinary universal cover. We prove the corollary after the sixfold theorem in Section 7.

The Fano classification also gives a positive Kähler–Einstein metric on every Fano manifold of positive dimension at most six with nef tangent bundle; Corollary 7.1 records the normalization of this metric.

Organization. Section 2 establishes the common geometry, including the rational-curve fibration calculation in Section 2.1. Section 3 settles pseudoindex four. Sections 4 and 5 derive the quadratic relation. Section 6 recognizes the Grassmannian, and Section 7 combines the cases with Kanemitsu’s Picard-number theorem and derives Corollary 1.2. Appendix A includes the complete executable certificate for the finite calculation.

2 Minimal rational curves

We now assume that X has Picard number one. The first task is to construct a smooth universal family of rational curves and determine its relative tangent bundle along every member. These facts will provide the common geometric input for the two remaining pseudoindices.

All cohomology groups in the intersection calculations have rational coefficients. A class in H^{2i} has weight i. For a smooth morphism a we write T_a=\ker(da) and -K_a=c_1(T_a). Kodaira vanishing gives H^i(X,\mathcal O_X)=0 for i>0. The exponential sequence and the Lefschetz (1,1) theorem therefore give

We also use that smooth complex Fano manifolds are rationally connected and simply connected; see [7, Theorems 2.11 and 5.3.2].

The pseudoindex of X is the positive integer

Choose an irreducible component V of the normalized space of rational curves whose members have degree d. Let

be its universal family and evaluation map. Thus a fibre of \pi is the normalization of the corresponding curve, and its map to X is birational onto that curve.

We can now isolate the cases that require further work.

2.1 Cohomology of a rational-curve fibration

The two remaining cases use integration along a smooth \mathbb P^1-fibration. The following lemma expresses this operation through an involution and a product rule, without requiring the fibration to be the projectivization of a vector bundle.

In particular, if \bar h=\ell-h and \ell=2h-c, repeated use of (2.6) gives the polynomial identities

These formulas are obtained by induction, so do not require division by a cohomology class.

3 Pseudoindex four

Watanabe treats this case in [30, Theorem 2.4]. We give an intersection-theoretic argument using the factorization of the universal family established by Kanemitsu. His theorem gives either rational homogeneity or a factorization of the evaluation map; we exclude the latter possibility. The relative-degree calculation below reproves Kanemitsu’s Lemma 4.3, and the exclusion of projective-plane fibres adapts the Betti-number argument in his proof of Theorem 0.2 [19]. In dimension six, the remaining quadric-bundle case is excluded by the orthogonal-bundle calculation given here.

Let X be a smooth connected complex projective Fano sixfold with nef tangent bundle, Picard number one and pseudoindex four. Let \pi:U\to V and e:U\to X be the universal family and evaluation map of a minimal rational component. By Proposition 2.1, we have

4 Characteristic-class constraints in pseudoindex five

We now assume that X is a Fano sixfold with nef tangent bundle, \rho(X)=1, and pseudoindex five. Let

be its smooth universal family. Thus \dim U=9, \dim V=8, the fibres of \pi are projective lines, and e has relative dimension three. The preceding geometric argument gives T_e|_C\simeq\mathcal O_{\mathbb P^1}(-1)^{\oplus3} for every \pi-fibre C. Our goal is to constrain the polarization of an evaluation fibre by combining the two fibrations.

Put

The class h has degree one on a \pi-fibre. Lemma 2.3 therefore shows that \ell comes from V and that \bar h=\ell-h.

4.1 The polarized evaluation fibre

4.2 Classes descending to the parameter space

Define r,s,t to be the elementary symmetric polynomials in the three formal roots obtained by adding h to the Chern roots of E. Equivalently,

This definition uses only rational cohomology; it does not assert that h is the first Chern class of a line bundle.

For the fibre F and polarization L of Lemma 4.1, the definitions and h|_F=0 give

We shall constrain the ratio

Its denominator is positive. The intersection identities below will force a quadratic equation for u. In Section 6, this equation and the sectional-genus formula will determine the degree of L.

4.3 Two direct-image identities

We first record why the Hodge-theoretic direct images appearing in Riemann–Roch contribute no positive-degree Chern character.

The base X is simply connected, so the lemma applies to e. Relative Grothendieck–Riemann–Roch [3, §7, p. 113] for \mathcal O_U and \Omega^1_{U/X}=E^\vee gives

Indeed these are the components of weight i-3>0 of

respectively.

These two choices capture the equations from all relative holomorphic forms. Indeed, for a rank-three bundle and p=0,1, \bigwedge^{3-p}E^\vee=\det(E^\vee)\otimes\bigwedge^pE and \operatorname{td}(E^\vee)=e^{-c_1(E)}\operatorname{td}(E). Replacing the Chern roots by their negatives therefore gives

Thus \Omega^2_{U/X} and \Omega^3_{U/X} repeat, up to sign in each weight, the equations from \Omega^1_{U/X} and \mathcal O_U.

4.4 Explicit polynomial representatives

We now turn the tangent-sequence and Riemann–Roch identities into linear relations among weight-nine intersection numbers on U. The projection formula will then identify selected normalized numbers as products involving u. Set

Then x_1=5h and x_i=0 for i\ge7, by dimension. Work in the polynomial ring

with respective weights (1,2,3,4,5,6,1,1,2,3). Evaluation of these generators as cohomology classes defines a graded homomorphism R\to H^{2*}(U,\mathbb Q).

Our unknowns are the integrals of weight-nine monomials in R. We obtain linear relations among them from three sources: classes descending to the eight-dimensional space V, classes pulled back from the six-dimensional space X and their \pi-integrals, and the two Riemann–Roch vanishings. The tangent sequences make the \pi-integrals explicit. Recall that \delta=\pi^*\pi_* lowers weight by one. The recurrences below express these operations as polynomials, before we collect the resulting relations into a matrix.

Let p_j be the Newton power sum of the three formal roots with elementary symmetric functions r,s,t, and put p_0=3. For complete specificity, if s_1=r,s_2=s,s_3=t, Newton’s identities read

Define

These represent i!\operatorname{ch}_i(E), with q_0=3. The two tangent exact sequences on U give, for every i\ge1,

Applying \delta=\pi^*\pi_* and using (2.7) gives \delta x_i=D_i, where

In particular D_1=5, consistently with x_1=5h, and D_i evaluates to zero for i\ge7. This calculation uses the polynomial product rule for \delta, so remains valid even when the evaluated classes are zero divisors.

For a monomial m in the six base generators (h,x_2,x_3,x_4,x_5,x_6) define \Delta m recursively by

In the last formula, remove the first generator g dividing m=gn in the displayed order. By (2.6), this recursively specified polynomial evaluates to \delta m. No independence of different formal recursion orders is needed.

Finally, write \theta_i for a polynomial representing \operatorname{td}_i(E). Through weight nine, put

with the remaining \beta_j zero. To obtain this formula, apply the logarithmic derivative to the product of the Todd factors:

Since \operatorname{ch}(E^\vee)=\sum_{j\ge0}(-1)^jq_j/j!, the second polynomial in (4.4) is represented by

4.5 The linear constraints and their geometric meaning

Let \mathcal B_k be the weight-k monomials in (h,x_2,x_3,x_4,x_5,x_6), let \mathcal I_k be those in (\ell,r,s,t), and let \mathcal T_k be all weight-k monomials in R. A negative subscript means the empty list. Monomials are ordered recursively: for an ordered generator list ending in a variable g of weight a, list g^j times the monomials of weight k-aj in the preceding variables, in increasing order of j.

Form a rational matrix A whose columns are indexed by \mathcal T_9. Its rows are coefficient vectors of the following polynomials. We group them according to their geometric source.

First, for a base monomial m, both m-h\delta m and \delta m descend to V by Lemma 2.3, as does \ell. The classes r,s,t descend by Lemma 4.2. Products of these classes of weight nine vanish because \dim V=8. This gives the rows

Second, a base monomial m of weight greater than six vanishes, as does \delta m. Also x_i=0 for i\ge7, so D_i=\delta x_i evaluates to zero. Their multiples give

Finally, the Riemann–Roch identities (4.4) and the projection formula give zero integrals for

Zero rows may be omitted. Let

The preceding vanishings prove Av=0. The first five row families come from zero cohomology classes; the final family only requires the stated integrals to vanish.

The constraints are linear in the intersection numbers on U. To recover a useful condition on u, we next use the product structure forced by integration along e.

We use invariant monomials of weights three and four because their pushforwards along e lie in H^0(X,\mathbb Q) and H^2(X,\mathbb Q)=\mathbb Qh_X, respectively. Each pushforward is determined by one scalar. These two cohomology groups therefore supply scalar product identities. Write the ordered lists M=\mathcal B_6 and G=(\mathcal I_3,\mathcal I_4) explicitly as

Indices begin at zero, and w_j denotes the weight of G_j. Put D=\int_F c_1(L)^3>0 and define

The first expression uses the corresponding base classes on X. The second defines a scalar because w_j-3 is zero or one and H^2(X,\mathbb Q)=\mathbb Qh_X. We have \xi_0=\eta_0=1 and \eta_1=u.

Use the index set

for the selected invariant monomials. This subcollection suffices for the elimination in Section 5 and reduces its size; no vanishing is asserted for the omitted monomials r^3, r^4, and r^2s. Let

Order C_1 by increasing i and then increasing j. For each such pair the monomial M_i/h^{w_j-3} is defined, and the projection formula gives

The common factor on the right is \int_U h^6\ell^3>0. We now retain all linear relations that Av=0 induces on these selected coordinates. The following construction expresses them as a matrix N. If K is any matrix whose columns form a rational basis of \ker A, let P consist of its rows indexed, with repetitions if necessary, by the monomials on the left of (4.18). Let N be the nonzero rows in the reduced row echelon form of a matrix whose rows form a basis of \ker(P^{\mathsf T}). Thus N is determined by the ordered coordinates, independently of the chosen basis K. Then

Indeed v\in\ker A, so its selected coordinates lie in the column space of P, and division by the single positive scalar in (4.18) gives the displayed equation. This completes the geometric derivation of the finite constraints.

5 The numerical constraint

The geometric argument has produced the bilinear equations (4.19). We now prove that they allow only two values of u. This section is a finite calculation over \mathbb Q: its input is the explicitly defined matrix A and its selected coordinates C_1, and its output is a quadratic equation for \eta_1.

We give the finite certificate, including the distinction between the modular rank bound and the rational identities needed to finish the proof. The exact recurrences and elimination procedure are implemented in Appendix A.

5.1 Multiplying the constraints

Write y_{i,j}=\xi_i\eta_j for the selected bilinear coordinates. Multiplying each equation of Ny=0 by \eta_a\eta_b, with a,b\in J, gives linear equations in the cubic products

Here \operatorname{sort} arranges the three indices in nondecreasing order, identifying equal products. We will impose those linear equations without requiring an arbitrary solution v to have this product form. In the resulting larger solution space, we prove that

For the product vector, \xi_0=\eta_0=1 makes the left side 32-20\eta_1+3\eta_1^2. Thus a linear statement on cubic coordinates will prove the desired quadratic relation. The following spaces make this relaxation precise; the intermediate quadratic space also makes its computation smaller.

For n=2,3, form C_n from C_{n-1} by appending an index j\in J to its tuple and sorting. Repeated coordinates are identified. Equivalently, (i,z)\in C_n means that z is a size-n multiset on J, and either h\mid M_i or z contains an index of weight three. Let \mathcal K_1=\ker N. Recursively define \mathcal K_n\subseteq\mathbb Q^{C_n} by requiring, for every j\in J, that the slice

Thus \mathcal K_3 is the kernel of a single rational matrix: for each unordered pair a,b\in J, allowing a=b, impose N on the slice

There are \binom{16}{2}=120 such pairs. In particular,

belongs to \mathcal K_3: every double slice is a scalar multiple of the given vector in \ker N.

Seven monomials M_i contain h, and six of the fifteen indices in J have weight three. Therefore

which gives |C_1|=129, |C_2|=1140 and |C_3|=6820.

5.2 An upper bound from finite-field elimination

The coefficient recurrences in Section 4 give 1785 nonzero rows and 581 columns for A. Exact elimination gives a matrix K of size 581\times65 with AK=0. The selected-row matrix P has size 129\times65. Choose N in reduced row echelon form. The following checks give, in particular, complete bases rather than merely lists of relations:

The identities AK=0 and NP=0 hold over \mathbb Q. Every denominator in these matrices is prime to 10007. Consequently the first two ranks prove that K is a basis of \ker A, and the last two prove that N is the full left annihilator of P. Write \mathcal K_n(\mathbb F_{10007}) for the spaces defined by the same slice equations using the reduction of N modulo 10007.

Here is the elimination procedure used for the larger spaces. Suppose that the columns of Z are a basis for the solutions on a union S of slices, and the columns of B are a basis for the solutions on the next slice T. On the overlap, the two parameter vectors a,b must satisfy

If the columns of \binom{V_1}{V_2} form the kernel of this matrix, then the columns of

parametrize all solutions on S\cup T. This parametrization is injective: zero on the union is zero on each piece, and the old parametrizations were injective. Appending a fixed index to a multiset is itself injective, so there are no hidden identifications within an individual slice. Ordinary Gaussian elimination and successive application of this construction give

For the rational conclusion, use the single double-slice matrix defining \mathcal K_3. It has 69\cdot120=8280 rows and 6820 columns, and its denominators are units modulo 10007. Its modular rank is 6814, so its rational rank is at least 6814. Thus

This argument does not assume that intermediate rational kernel bases specialize to bases.

5.3 Six rational columns

It remains to exhibit six independent rational solutions and show that the functional giving the desired quadratic vanishes on their span. These columns are witnesses for the linear system: no assertion that they arise from varieties is needed. Three have product form, and three are specified by their small supports. The relaxed system allows both kinds without imposing the original normalization or requiring a product representation. Define the following rational vectors:

For (B,H)=(B(0),H^a),(B(1),H^a),(B',H^b), let

Direct rational substitution gives N(B_iH_j)_{C_1}=0, so these three columns lie in \mathcal K_3. Three further columns s_1,s_2,s_3 have the following entries and are zero elsewhere:

Applying N to each of their 120 double slices gives zero over \mathbb Q. These are exact rational equalities, separate from the finite-field calculation.

The six columns are independent. Indeed, restrict them to the rows

The upper-right 3\times3 block is zero. The upper-left block has determinant -8/15, and the lower-right block is \left(\begin{smallmatrix}3&0&0\\0&2&0\\0&4&1\end{smallmatrix}\right), with determinant 6. The full minor is -16/5. Together with (5.5), this proves that the six displayed columns form a rational basis of \mathcal K_3.

Returning to the evaluation fibre, (4.19) verifies the hypothesis of the lemma and \eta_1=u. Hence

It remains to identify which value is compatible with the tangent map and to recognize the resulting variety of minimal rational tangents.

6 Recognizing the Grassmannian

We finish the pseudoindex-five case by interpreting the numerical constraint on the fibres of the evaluation map. Throughout this section, X is a smooth complex Fano sixfold with nef tangent bundle, Picard number one and pseudoindex five. We use its minimal rational component V, universal family \pi\colon U\to V, and evaluation e\colon U\to X. Thus e is smooth with connected three-dimensional fibres. For a general point x\in X, let F=e^{-1}(x) and let L be the tangent-map polarization of Lemma 4.1. Our goal is to recover X from the projective geometry of the tangent directions to its minimal rational curves. For this polarization, Lemma 5.1 gives

We first show that the tangent image spans \mathbb P^5. Its degree bound and the sectional-genus formula will then determine L^3 and the degree of the tangent map. Finally, normality of the image will let us use the smoothness of F to select the Segre threefold among the varieties of minimal degree.

7 The sixfold theorem and its consequences

7.1 A Kähler–Einstein consequence

7.2 Albanese fibres and the ordinary universal cover

We now combine the classification with the analytic structure theorems for compact Kähler manifolds with nef tangent bundle.

A Exact arithmetic certificate

The following complete Python program verifies the finite calculations in Section 5. It requires SymPy and NumPy. Run python3 verification/verify.py with assertions enabled to perform the checks described below. The monomial order, coefficient recurrences and projection are exactly those defined in Section 4.

The calculation has three logically separate parts. First, exact rational identities AK=0 and NP=0, together with the four modular ranks in Section 5, certify the kernel and annihilator bases. Reduction of a rational entry explicitly inverts its denominator modulo 10007, so a noninvertible denominator would stop the calculation. Second, the slice-merging procedure computes the finite-field spaces and checks their dimensions. The final basis is also tested directly against every double-slice constraint. Third, rational substitution checks the six displayed columns, their nonzero minor -16/5, and the final linear functional.

SymPy performs the polynomial and rational operations exactly. The finite-field arrays use signed 64-bit integers. After each reduction, their entries lie between 0 and 10006. The inner dimension of every matrix product is at most 6820, since a basis has no more columns than the number of coordinates of its space. Every accumulated dot product is therefore at most

The products and subtractions in a pivot step have smaller absolute values. Thus no arithmetic operation can wrap around before modular reduction.