Cone structures and parabolic geometries

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Publikace nespadá pod Pedagogickou fakultu, ale pod Přírodovědeckou fakultu. Oficiální stránka publikace je na webu muni.cz.
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HWANG Jun-Muk NEUSSER Katharina

Rok publikování 2022
Druh Článek v odborném periodiku
Časopis / Zdroj Mathematische Annalen
Fakulta / Pracoviště MU

Přírodovědecká fakulta

Citace
www https://link.springer.com/article/10.1007%2Fs00208-021-02208-4
Doi http://dx.doi.org/10.1007/s00208-021-02208-4
Klíčová slova Cone structures; Rational homogeneous space; Varieties of minimal rational tangents; Cartan connections; Parabolic geometry; Filtered manifolds
Popis A cone structure on a complex manifold M is a closed submanifold C?PTM of the projectivized tangent bundle which is submersive over M. A conic connection on C specifies a distinguished family of curves on M in the directions specified by C. There are two common sources of cone structures and conic connections, one in differential geometry and another in algebraic geometry. In differential geometry, we have cone structures induced by the geometric structures underlying holomorphic parabolic geometries, a classical example of which is the null cone bundle of a holomorphic conformal structure. In algebraic geometry, we have the cone structures consisting of varieties of minimal rational tangents (VMRT) given by minimal rational curves on uniruled projective manifolds. The local invariants of the cone structures in parabolic geometries are given by the curvature of the parabolic geometries, the nature of which depend on the type of the parabolic geometry, i.e., the type of the fibers of C›M. For the VMRT-structures, more intrinsic invariants of the conic connections which do not depend on the type of the fiber play important roles. We study the relation between these two different aspects for the cone structures induced by parabolic geometries associated with a long simple root of a complex simple Lie algebra. As an application, we obtain a local differential-geometric version of the global algebraic-geometric recognition theorem due to Mok and Hong–Hwang. In our local version, the role of rational curves is completely replaced by appropriate torsion conditions on the conic connection.
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