Convex separably rationally connected complete intersections (Q2814391)

It was conjectured by \textit{F. Campana} and \textit{T. Peternell} [Math. Ann. 289, No. 1, 169--187 (1991; Zbl 0729.14032)] that any complex projective Fano manifold \(X\) with nef tangent bundle is rational homogeneous. As at the time of this review, the conjecture is known to be true if \(\dim X \leq 5\), thanks to the contribution of different authors. In [Proc. Am. Math. Soc. 141, No. 5, 1539--1543 (2013; Zbl 1264.14020)], \textit{R. Pandharipande} verified the conjecture for complex Fano complete intersections of arbitrary dimension, by proving the more general result that if a manifold \(X\) is a convex (i.e. any morphism \(\mu : \mathbb{P}^1\rightarrow X\) satisfies \(H^1(\mathbb{P}^1, \mu^* T_X)=0\)) and rationally connected complete intersection, then \(X\) is rational homogeneous.NEWLINENEWLINEIn the paper under review, the author proves that if \(X\) is a convex, smooth complete intersection in \(\mathbb{P}^N\), over an algebraically closed field of arbitrary characteristic, that admits an immersion \(\mathbb{P}^1\rightarrow X\), then \(X\) has degree at most two (hence \(X\) is rational homogeneous). The result generalizes Pandharipande's result to arbitrary characteristic, since if \(X\) is Fano or separably rationally connected complete intersection then \(X\) admits an immersion \(\mathbb{P}^1\rightarrow X\).