gpt 6 astra ultra attempt on Math-472

Campana–Peternell Conjecture for Fano Manifolds

Definitions and mathematical statement

A smooth complex projective variety $X$ is Fano when its anticanonical line bundle $K_X^{-1}$ is ample. A vector bundle $E$ is nef if the tautological quotient line bundle $\mathcal O_{{\mathbb{P}}(E)}(1)$ has nonnegative degree on every complete algebraic curve in ${\mathbb{P}}(E)$. The conjecture states \[ X\text{ Fano},\quad T_X\text{ nef} \quad\Longrightarrow\quad X\cong G/P, \] where $G$ is a connected complex semisimple algebraic group and $P$ a parabolic subgroup, so $G/P$ is projective and homogeneous. The isomorphism is algebraic. Nefness is a numerical condition weaker than ampleness; assuming an ample tangent bundle would substantially narrow the problem.

Formulation and concept references: [S1].

Short English statement

Must a smooth Fano variety with numerically nonnegative tangent bundle be a rational homogeneous space?

Research attempt

Scope and a source-status caution (27 September 2026).

The hypothesis is nefness of the tautological quotient line bundle on $\mathbb P(T_X)$, not ampleness of $T_X$ and not global generation. Those stronger properties would remove major parts of the difficulty. The supplied survey [S1] states the conjecture and its equivalent global-generation formulation, and records established low-dimensional and structural results. The elementary proofs below are reconstructions of standard special cases and reductions, not proposed new classifications.

A more recent primary source requires particular care. Wang's arXiv:2408.03799v2, dated 3 December 2025, retains an abstract claiming that nefness of $T_X$ on a Fano manifold implies bigness and semiampleness, but explicitly adds a notice that a gap in its proof invalidates the main theorem's proof [E1]. The inspected version record still lists v2. A promised corrected revision is not itself a proof. This attempt does not use that claimed abundance result.

The approach is to understand restrictions to curves, prove the surface case using the classical del Pezzo classification, and compare moving rational curves with global vector fields. The final comparison produces an exact missing implication rather than a false passage from numerical positivity to symmetry.

Curve tests and elementary stability properties.

We use the following form of the definition: a vector bundle $E$ on a projective variety is nef if, for every morphism $f:C\to X$ from a smooth projective curve and every quotient line bundle $f^*E\twoheadrightarrow Q$, one has $\deg Q\geq0$. A quotient line defines a map to $\mathbb P(E)$ and pulls back its tautological bundle. Conversely, normalizing a curve in $\mathbb P(E)$ gives such a quotient, which proves equivalence with the definition in the statement.

Pullbacks and quotient bundles of nef bundles are nef, directly from this criterion. A globally generated bundle is nef: every quotient line on a curve is globally generated, and a nonzero section of a line bundle has an effective zero divisor, so its degree cannot be negative. These observations establish the easy direction for homogeneous spaces: the infinitesimal action of a transitive algebraic group supplies global vector fields spanning every tangent space.

The corresponding statement for subbundles is false. On $\mathbb P^1$ the exact sequence \[ 0\longrightarrow\mathcal O(-1)\longrightarrow\mathcal O^{\oplus2} \longrightarrow\mathcal O(1)\longrightarrow0 \] has a nef middle term and a non-nef subbundle. Thus in the relative tangent sequence of a smooth morphism one may immediately use its quotient, but may not deduce nefness of its kernel merely by inspection. Any theorem asserting that fibres in the present setting again have nef tangent bundle needs an additional argument.

Every rational curve has nonnegative splitting degrees.

Lemma 468.1. If $T_X$ is nef and $f:\mathbb P^1\to X$ is nonconstant, write \[ f^*T_X\cong\bigoplus_{i=1}^n\mathcal O(a_i) \] using the splitting theorem for bundles on $\mathbb P^1$. Then all $a_i\geq0$, at least one $a_i\geq2$, and $-K_X\cdot f_*[\mathbb P^1]\geq2$.

Proof. Each summand is a quotient line bundle of $f^*T_X$, so nefness gives $a_i\geq0$. Since the ground field has characteristic zero and $f$ is nonconstant, its differential $\mathcal O(2)=T_{\mathbb P^1}\to f^*T_X$ is not the zero map. If every $a_i<2$, all its component homomorphisms would vanish because $H^0(\mathbb P^1,\mathcal O(a_i-2))=0$. Thus one $a_i\geq2$. Finally the anticanonical degree is the degree of $\det f^*T_X$, namely $\sum_i a_i\geq2$.$\square$

This degree counts the parametrized curve with its mapping degree. For a normalization map to an irreducible rational curve it gives the ordinary anticanonical intersection. One cannot assume a splitting of the special form $\mathcal O(2)\oplus\mathcal O(1)^a\oplus\mathcal O^b$ for every rational map: multiple covers already produce larger degrees. For example a degree-$d$ map to $\mathbb P^1$ pulls its tangent bundle back to $\mathcal O(2d)$.

The same splitting gives \[ H^1(\mathbb P^1,f^*T_X)=0, \qquad H^1(\mathbb P^1,f^*T_X(-p))=0. \] Under the standard deformation theory of morphisms, the first vanishing makes the Hom-scheme smooth at $f$. The second says that evaluation at $p$ is a submersion there: the differential is the evaluation map from $H^0(f^*T_X)$, and its cokernel embeds into $H^1(f^*T_X(-p))$ by the evaluation exact sequence. Thus these curves move in families covering a neighbourhood of their points. The deformation-theoretic interpretation is an established input, also used in [S1, Lemma 2.8]; the displayed cohomology vanishing is completely accounted for by the nonnegative splitting degrees.

Negative normal directions rule out many candidates.

If $C\subset X$ is a smooth embedded curve, there is an exact sequence \[ 0\longrightarrow T_C\longrightarrow T_X|_C \longrightarrow N_{C/X}\longrightarrow0. \] Nefness of $T_X$ forces $N_{C/X}$ to be nef as a quotient. For a surface, $N_{C/X}$ is a line bundle of degree $C^2$, so \[ T_X\text{ nef}\quad\Longrightarrow\quad C^2\geq0 \quad\text{for every smooth embedded curve }C. \tag{468.1} \] In particular a $(-1)$-curve on a surface is an immediate obstruction. Its tangent bundle can have positive total degree and still fail nefness: along an exceptional curve of a point blow-up the normal quotient is $\mathcal O(-1)$, and in fact \[ T_X|_C\cong\mathcal O(2)\oplus\mathcal O(-1). \] The splitting follows because the extension class lies in $H^1(\mathbb P^1,\mathcal O(3))=0$. Its determinant has degree one, which makes clear that positivity of $-K_X$ on the curve does not control every quotient of $T_X|_C$.

The same test rules out the tangent bundle of the blow-up of a smooth $n$-fold at a point, for $n\geq2$. Its exceptional divisor is $E\cong\mathbb P^{n-1}$ with normal line $\mathcal O_E(-1)$. Restricting the quotient $T_X|_E\twoheadrightarrow N_{E/X}$ to a line in $E$ gives a negative quotient of the restricted tangent bundle. This statement concerns that specific blow-up geometry, not an assertion that every birational morphism is a point blow-up.

The complete dimension-one and surface cases.

In dimension one, a Fano curve has $\deg(-K)=2-2g>0$, so $g=0$ and it is $\mathbb P^1$. Its tangent line is $\mathcal O(2)$ and the conclusion follows.

For dimension two we use the classical classification of smooth complex del Pezzo surfaces as an explicit external input: each is $\mathbb P^2$, $\mathbb P^1\times\mathbb P^1$, or the blow-up of $\mathbb P^2$ at between one and eight points in the configurations giving an ample anticanonical divisor. In the blow-up cases an exceptional $(-1)$-curve exists. Equation $468.1$ excludes every one of them. Hence a Fano surface with nef tangent bundle is one of the first two surfaces.

Both remaining surfaces satisfy all hypotheses. The Euler sequence \[ 0\longrightarrow\mathcal O_{\mathbb P^2} \longrightarrow\mathcal O_{\mathbb P^2}(1)^{\oplus3} \longrightarrow T_{\mathbb P^2}\longrightarrow0 \] shows that $T_{\mathbb P^2}$ is globally generated and hence nef. Its anticanonical line is $\mathcal O(3)$. On the product, \[ T_{\mathbb P^1\times\mathbb P^1} =\mathcal O(2,0)\oplus\mathcal O(0,2),\qquad -K=\mathcal O(2,2). \] Both summands are globally generated and the anticanonical line is ample. These are homogeneous spaces for $\operatorname{PGL}_3$ and $\operatorname{PGL}_2\times\operatorname{PGL}_2$, respectively. This proof is a standard classification-based special case, not a replacement for the much deeper results known in larger dimensions.

The Hirzebruch surfaces provide a useful independent numerical check. For $\mathbb F_m=\mathbb P_{\mathbb P^1}(\mathcal O\oplus\mathcal O(m))$ with $m\geq0$, the minimal section $C_0$ has $C_0^2=-m$ and a ruling fibre $F$ satisfies $F^2=0$, $C_0\cdot F=1$. The canonical formula gives \[ -K_{\mathbb F_m}=2C_0+(m+2)F,\qquad -K\cdot C_0=2-m,\qquad -K\cdot F=2. \] Thus the Fano members are $m=0,1$, while nef tangent forces $m=0$ by the normal quotient along $C_0$. The Fano surface $\mathbb F_1$ is therefore a concrete example showing why the tangent hypothesis adds real content beyond ampleness of the determinant.

Why enough global vector fields would finish the proof.

Proposition 468.2. If $X$ is a smooth complex Fano variety and $T_X$ is globally generated, then $X$ is a rational homogeneous space $G/P$ with $G$ connected semisimple.

Proof. Let $G=\operatorname{Aut}^0(X)$. We use the standard identification $\operatorname{Lie}(G)=H^0(X,T_X)$. The differential of the orbit map $G\to X$, $g\mapsto g(x)$, at the identity is precisely evaluation of global vector fields at $x$. Global generation therefore makes every orbit open. Distinct orbits would partition the connected variety into disjoint open subsets, so there is just one orbit.

For completeness, the Fano hypothesis removes any non-linear algebraic group issue. Choose $m$ so that $-mK_X$ is very ample. Its canonical functorial action embeds $\operatorname{Aut}(X)$ in the projective linear group of its global sections: an automorphism acting trivially on the embedded variety is the identity. Hence $G$ is a connected linear algebraic group. Its connected solvable radical $R$ has a fixed point on the complete variety $X$ by the Borel fixed-point theorem. Because $R$ is normal and $G$ is transitive, it then fixes every point: if $x$ is fixed, $r(gx)=g(g^{-1}rg)x=gx$. The action is faithful, so $R$ is trivial. Thus $G$ is semisimple. The stabilizer $P$ of a point has projective quotient $G/P\cong X$, and is therefore parabolic.$\square$

The algebraic-group facts invoked here are standard: representability of the automorphism group, the Lie-algebra identification, Borel's fixed-point theorem and the projective-quotient characterization of parabolic subgroups. This gives a complete reduction using those facts; the unsolved part has not been hidden in a claim that all nef bundles are generated by sections. The same reduction appears as the global-generation version of the conjecture in [S1].

Local curve sections are not global vector fields.

For a rational map $f:\mathbb P^1\to X$, nefness gives a surjective evaluation map \[ H^0(\mathbb P^1,f^*T_X)\longrightarrow T_{X,f(p)}. \] To use Proposition 468.2 one instead needs \[ H^0(X,T_X)\longrightarrow T_{X,f(p)} \] to be onto. These spaces of sections are different. The restriction map from global vector fields on $X$ to sections along $f$ has not been shown surjective. Deforming one map $\mathbb P^1\to X$ changes that curve; it does not produce an automorphism of the ambient variety.

There is no general numerical theorem repairing this distinction. Even on a Fano surface a nef line bundle need not be globally generated. Take a del Pezzo surface with $K_X^2=1$ and put $A=-K_X$. It is ample and hence nef. Kodaira vanishing applied to $A=K_X+2A$ gives $H^i(X,A)=0$ for $i>0$, while the same theorem applied to $\mathcal O_X=K_X+A$ gives $\chi(\mathcal O_X)=1$. Surface Riemann--Roch now gives \[ h^0(X,A)=\chi(A)=1+\tfrac12 A\cdot(A-K_X)=2. \] If this pencil were basepoint-free, it would define a morphism to $\mathbb P^1$ with $A$ the pullback of $\mathcal O(1)$. Fibres over two distinct points are disjoint, forcing $A^2=0$, contrary to $A^2=1$. Thus $A$ is nef but not globally generated. This is a line-bundle diagnostic on a Fano base, not a counterexample involving that surface's tangent bundle; the latter is already ruled out by its exceptional curves.

Likewise, semiampleness of $\mathcal O_{\mathbb P(T_X)}(1)$ would give sections of a symmetric power after taking a multiple. It would not, without an extra argument, yield the degree-one vector fields needed to generate $T_X$. A proof of the weaker abundance assertion, even if available, would still need a further homogeneity step.

Products and the limit of the reduction.

Known examples are closed under products: if $X,Y$ have globally generated tangent bundles, then $T_{X\times Y}=p_X^*T_X\oplus p_Y^*T_Y$ is globally generated; if both are Fano, the anticanonical line of the product is the exterior tensor product of their anticanonical lines and is ample. If $X=G/P$ and $Y=H/Q$, their product is $(G\times H)/(P\times Q)$. This checks consistency of the predicted class with the most elementary constructions.

It gives no decomposition theorem for an arbitrary Fano variety with nef tangent bundle. Numerical splitting of intersection data is not splitting of the variety, and an invariant family of rational curves does not by itself construct the required algebraic group action.

Outcome and exact first gap.

The attempt proves the dimension-one and surface cases using the stated classical inputs, excludes negative normal quotients and point blow-ups, derives the free-rational-curve vanishing, and proves that global generation of $T_X$ gives the desired semisimple homogeneous description. The examples test the differences between positive determinant, nefness, generation along a curve and generation on the whole variety.

The first gap is the passage from all those curvewise nonnegative quotients to enough global sections of $T_X$ at every point, or to an independent construction of a transitive algebraic group action. The supplied arguments establish neither passage in arbitrary dimension. The main conjecture remains unresolved by this attempt.

Sources

• S1 A survey on the Campana-Peternell Conjecture. https://arxiv.org/abs/1407.6483. Formulation source.
• E1 Juanyong Wang, An abundance-type result for the tangent bundles of smooth Fano varieties, arXiv:2408.03799v2, 3 December 2025. https://arxiv.org/abs/2408.03799; https://arxiv.org/html/2408.03799v2. Version record and the author's opening gap notice inspected 27 September 2026. The affected main theorem is not used as an established input.
• E2 Roberto Muñoz, Gianluca Occhetta, Luis E. Solá Conde, Kiwamu Watanabe and Jarosław A. Wiśniewski, full text of [S1], revised 11 November 2015. https://arxiv.org/html/1407.6483. Conjecture 1.3, Proposition 2.1 and the rational-curve discussion in Section 2.3 inspected 27 September 2026. Standard classification, deformation and algebraic-group inputs are identified in the body.