Varieties of minimal rational tangents on uniruled projective manifolds.

*(English)*Zbl 0978.53118
Schneider, Michael (ed.) et al., Several complex variables. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 37, 351-389 (1999).

Summary: On a polarized uniruled projective manifold we pick an irreducible component \({\mathcal K}\) of the Chow space whose generic members are free rational curves of minimal degree. The normalized Chow space of minimal rational curves marked at a generic point is nonsingular, and its strict transform under the tangent map gives a variety of minimal rational tangents, or VMRT. In this survey we present a systematic study of VMRT by means of techniques from differential geometry (distributions, \(G\)-structures), projective geometry (the Gauss map, tangency theorems), the deformation theory of (rational) curves, and complex analysis (Hartogs phenomenon, analytic continuation).

We give applications to a variety of problems on uniruled projective manifolds, especially on irreducible Hermitian symmetric manifolds \(S\) of the compact type and more generally on rational homogeneous manifolds \(G/P\) of Picard number 1, including the deformation rigidity of \(S\) and the same for homogeneous contact manifolds of Picard number 1, the characterization of \(S\) of rank at least 2 among projective uniruled manifolds in terms of \(G\)-structures, solution of Lazarsfeld’s Problem for finite holomorphic maps from \(G/P\) of Picard number 1 onto projective manifolds, local rigidity of finite holomorphic maps from a fixed projective manifold onto \(G/P\) of Picard number 1 other than \(\mathbb{P}^n\), and a proof of the stability of tangent bundles of certain Fano manifolds.

For the entire collection see [Zbl 0933.00014].

We give applications to a variety of problems on uniruled projective manifolds, especially on irreducible Hermitian symmetric manifolds \(S\) of the compact type and more generally on rational homogeneous manifolds \(G/P\) of Picard number 1, including the deformation rigidity of \(S\) and the same for homogeneous contact manifolds of Picard number 1, the characterization of \(S\) of rank at least 2 among projective uniruled manifolds in terms of \(G\)-structures, solution of Lazarsfeld’s Problem for finite holomorphic maps from \(G/P\) of Picard number 1 onto projective manifolds, local rigidity of finite holomorphic maps from a fixed projective manifold onto \(G/P\) of Picard number 1 other than \(\mathbb{P}^n\), and a proof of the stability of tangent bundles of certain Fano manifolds.

For the entire collection see [Zbl 0933.00014].

##### MSC:

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

14J45 | Fano varieties |

53C30 | Differential geometry of homogeneous manifolds |

53C10 | \(G\)-structures |

53C12 | Foliations (differential geometric aspects) |

32G05 | Deformations of complex structures |