A survey on the Campana-Peternell conjecture (Q2809910)

As the title suggests, the paper under review is a survey on the Campana-Peternell conjecture and related questions. In [Math. Ann. 289, No. 1, 169--187 (1991; Zbl 0729.14032)], \textit{F. Campana} and \textit{T. Peternell} predicted that every Fano manifold with nef tangent bundle (\textit{CP-manifold} for short) is a rational homogeneous manifold. This conjecture is a natural extension of Mori's characterization of a projective space as a smooth projective variety with ample tangent bundle.NEWLINENEWLINEAfter the introduction, the paper starts by recalling basic facts on rational homogeneous manifolds and deformation theory of rational curves on CP-manifolds.NEWLINENEWLINEThe next section focuses on Mori theory of CP-manifolds. The following statements are proven in this section: (1) The Mori cone of a CP-manifold is simplicial. (2) Every extremal contraction of a CP-manifold is of fiber type, and the fibers and the target are also CP-manifolds.NEWLINENEWLINEIn the fourth section, positive answers to the Campana-Peternell conjecture in low dimensions are given. The classification of CP-manifolds with two \(\mathbb{P}^1\)-fibrations is also discussed.NEWLINENEWLINEIn Section 5, the proof of the Camapana-Peternell conjecture for Fano manifolds with big and \(1\)-ample tangent bundle is presented. This section follows [\textit{L. E. Solá Conde} and \textit{J. A. Wiśniewski}, Proc. Lond. Math. Soc., III. Ser. 89, No. 2, 273--290 (2004; Zbl 1061.14019)].NEWLINENEWLINEThe final section presents a strategy to attack the Campana-Peternell conjecture: (Step 1) Prove the conjecture for CP-manifolds flag type manifolds. (Step 2) Prove that every CP-manifold is dominated by flag type manifolds. In [\textit{G. Occhetta} et al., ``Fano manifolds whose elementary contractions are smooth \(\mathbb{P}^1\)-fibrations: a geometric characterization of flag varieties'', Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), to appear], the first step was completed.