Geometric structures on uniruled projective manifolds defined by their varieties of minimal rational tangents. (English) Zbl 1182.14043

Mok, Ngaiming

Hijazi, Oussama (ed.), Géométrie différentielle, physique mathématique, mathématiques et société (II). Volume en l’honneur de Jean Pierre Bourguignon. Paris: Société Mathématique de France (ISBN 978-2-85629-259-4/pbk). Astérisque 322, 151-205 (2008).

Let \(X\) be a complex projective manifold that is uniruled, i.e., there exists a family of rational curves \(f: \mathbb P^1 \rightarrow X\) that dominates \(X\). For a given polarization a minimal rational curve is a rational curve that belongs to such a dominating family and whose degree with respect to the polarization is minimal. If we fix a general point \(x \in X\), the minimal rational curves passing through \(x\) are parametrized by a closed subset \(\mathcal K_x\) of the Hilbert scheme. There is a natural rational map \(\mathcal K_x \dashrightarrow \mathbb P(T_{X,x})\) which associates to a curve its tangent direction at the marked point \(x\). The variety of minimal rational tangents \(\mathcal C_x\) is defined as the strict transform of \(\mathcal K_x\) under this rational map. In a joint research program, J.-M. Hwang and the author investigated the geometry of the variety of minimal rational tangents (VMRTs), in particular in the case where \(X\) is a Fano manifold with Picard number one. This article gives a survey of this research program whose guiding philosophy is to translate differential-geometric notions such as (null) geodesics, curvature and parallel transport to the context of VMRTs and the geometric structures induced by them.
We refer to the very inspiring paper for more details and the numerous applications of these techniques to important problems in algebraic geometry, let us just mention some of the main topics:
- basic properties of VMRTs,
- linearly degenerate VMRTs and their associated distribution,
- Fano contact manifolds,
- rigidity of rational homogeneous manifolds under deformation,
- holomorphic G-structures,
- analytic continuation of VMRT-preserving maps,
- generalized Frankel conjecture,
- projective manifolds with nef tangent bundle.
For the entire collection see [Zbl 1160.53004].

Reviewer: Andreas Höring (Paris)

MSC:

14J45	Fano varieties
14M20	Rational and unirational varieties
32H02	Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32M15	Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
53C10	\(G\)-structures

Keywords:

geometric structure; minimal rational curve; variety of minimal rational tangents; tangent map; analytic continuation; Cauchy characteristic; curvature; prolongation; parallel transport; nef tangent bundle; distribution; differential system; deformation rigidity