[Paper Review] A survey on the Campana-Peternell Conjecture

Roberto Muñoz, Gianluca Occhetta+3 more|arXiv (Cornell University)|Jan 1, 2014

Algebraic Geometry and Number Theory45 references30 citations

TL;DR

This survey investigates the Campana-Peternell Conjecture, which posits that Fano manifolds with nef tangent bundles are rational homogeneous spaces. The authors analyze the structure of such manifolds using Mori theory, Bott-Samelson varieties, and VMRT theory, proving the conjecture for flag-type Fano manifolds with Dynkin diagrams of type A_n and for F4 and C3 by verifying isomorphisms of VMRTs and constructing recursive homogeneity arguments via smooth Mori contractions.

ABSTRACT

In 1991 Campana and Peternell proposed, as a natural algebro-geometric extension of Mori's characterization of the projective space, the problem of classifying the complex projective Fano manifolds whose tangent bundle is nef, conjecturing that the only varieties satisfying these properties are rational homogeneous. In this paper we review some background material related to this problem, with special attention to the partial results recently obtained by the authors.

Motivation & Objective

• To investigate the Campana-Peternell Conjecture, which asserts that Fano manifolds with nef tangent bundles are rational homogeneous spaces.
• To understand the geometric structure of CP-manifolds—Fano manifolds with nef tangent bundles—by analyzing their Mori contractions and fiber structures.
• To establish homogeneity of flag-type Fano manifolds by comparing their VMRTs (Varieties of Minimal Rational Tangents) with those of rational homogeneous spaces.
• To develop recursive techniques using Bott-Samelson varieties to prove isomorphisms between a given Fano manifold and its homogeneous model.
• To verify the conjecture in low dimensions and for specific Dynkin diagrams, particularly A_n, F4, and C3, using cohomological and birational geometry tools.

Proposed method

• Apply Mori theory to show that every contraction of a CP-manifold is smooth and that the Mori cone is simplicial, mirroring properties of rational homogeneous spaces.
• Use the recursive construction of Bott-Samelson varieties associated with reduced expressions of the longest Weyl group element to compare the geometry of a Fano manifold X with its homogeneous model.
• Establish isomorphisms between intermediate varieties in the recursive sequence by proving that the first cohomology groups h^1(Z_ℓ[s+1], f_ℓ^*[s+1]K_{l_{m-s}}) are 0 or 1 depending on whether a root index has already appeared in the sequence.
• Leverage the fact that if the VMRTs at general points of X and its homogeneous model are projectively isomorphic, then X is isomorphic to the model, as shown via Theorem 2.21.
• Use computational verification in the F4 case via Sage to check that no reduced sequence of the longest Weyl element satisfies the required cohomological conditions unless carefully chosen.
• Employ the fact that smooth Mori contractions of Fano manifolds with nef tangent bundles lead to Fano manifolds with the same properties, enabling inductive arguments on the Dynkin diagram's rank.

Experimental results

Research questions

• RQ1 Are all Fano manifolds with nef tangent bundles rational homogeneous spaces, as predicted by the Campana-Peternell Conjecture?
• RQ2 Can the homogeneity of flag-type Fano manifolds be established by comparing their VMRTs with those of rational homogeneous spaces?
• RQ3 Does the recursive construction of Bott-Samelson varieties via reduced Weyl group expressions yield isomorphisms between a Fano manifold and its homogeneous model?
• RQ4 For which Dynkin diagrams (e.g., A_n, F4, C3) does the conjecture hold, and what are the key cohomological conditions that ensure this?
• RQ5 Can the conjecture be proven for F4-type Fano manifolds using a recursive approach based on smooth Mori contractions and VMRT isomorphisms?

Key findings

• The Campana-Peternell Conjecture holds for Fano manifolds whose Dynkin diagram is of type A_n, as shown by reducing the problem to the base case X_1,n ≅ P(TP^n) and using induction.
• For Fano manifolds with Dynkin diagram F4, the conjecture is proven via a recursive procedure: the smoothness of Mori contractions is established by showing fibers are birational to Bott-Samelson varieties, and the isomorphism to the homogeneous model follows from projective isomorphism of VMRTs.
• The key cohomological condition for constructing isomorphisms between intermediate varieties in the recursive sequence is that h^1(Z_ℓ[s+1], f_ℓ^*[s+1]K_{l_{m-s}}) equals 1 if the root index l_{m-s} has appeared earlier in the sequence, and 0 otherwise.
• For Dynkin diagrams without multiple edges (e.g., A_n), any maximal-length reduced sequence satisfies the required cohomological conditions; for B_n and C_n, a careful choice of sequence is needed.
• In the F4 case, despite 2,144,892 possible reduced sequences for the longest Weyl element, none satisfy the cohomological conditions for all s unless the sequence is specially selected, and the proof relies on the reconstruction argument via families of lines.
• The conjecture is confirmed for the Lagrangian Grassmannian of type C3 via the isomorphism of VMRTs at general points, which are realized as Plücker embeddings of isotropic 3-planes in C^6.