Horospherical 2-Fano varieties (Q6592697)

Fano manifolds are complex projective manifolds whose tangent bundles have ample first Chern class, and satisfy good properties, such as being rationally connected. A special subclass is that of 2-Fano manifolds, namely the Fano manifolds with positive second Chern character. They are expected to satisfy stronger versions of the good properties of Fano manifolds. In Section 4 of this article, the authors review all known examples.\N\NThe main goal of this paper is to take a step forward in the classification of 2-Fano manifolds, by classifying those that are horospherical varieties of Picard rank 1. Horospherical varieties are projective manifolds endowed with a \(G\)-action, where \(G\) is a reductive group, and with an open orbit of the form \(G/H\), where \(H\) contains a maximal unipotent subgroup. They form a special class of spherical varieties. The horospherical varieties with Picard number 1 have been classified by Pasquier. The authors compute the second Chern character for each of such manifolds and check whether the 2-Fano condition holds.\N\NThe main result is the following (Theorem 1.1): the only non-homogeneous 2-Fano horospherical varieties with Picard number 1 are\N\[\N(B_3, \omega_1,\omega_3) \quad \text{and} \quad (C_n, \omega_m, \omega_{m-1}), \, (n,m) = (3k,2k+1), \, k \geq 1,\N\]\Nwhere the datum \((\mathrm{type}(G), \omega_Y,\omega_X)\) -- of the semisimple type of the reductive group \(G\) together with fundamental weights -- uniquely determines an associated horospherical manifold. More details on these manifolds are given in Section 2.\N\NThe paper concludes with Section 4, where the authors investigate the relation of the 2-Fano condition with \(K\)-stability and slope stability of the tangent bundle.