Cone structures and parabolic geometries

J 2022

Cone structures and parabolic geometries

HWANG, Jun-Muk a Katharina NEUSSER

Základní údaje

Originální název

Cone structures and parabolic geometries

Autoři

HWANG, Jun-Muk a Katharina NEUSSER (40 Rakousko, garant, domácí)

Vydání

Mathematische Annalen, Germany, Springer Berlin Heidelberg, 2022, 0025-5831

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10101 Pure mathematics

Stát vydavatele

Německo

Utajení

není předmětem státního či obchodního tajemství

Odkazy

URL

Impakt faktor

Impact factor: 1.400

Kód RIV

RIV/00216224:14310/22:00125176

Organizační jednotka

Přírodovědecká fakulta

DOI

http://dx.doi.org/10.1007/s00208-021-02208-4

UT WoS

000659801200001

EID Scopus

2-s2.0-85107502551

Klíčová slova anglicky

Cone structures; Rational homogeneous space; Varieties of minimal rational tangents; Cartan connections; Parabolic geometry; Filtered manifolds

Štítky

rivok

Příznaky

Mezinárodní význam, Recenzováno

Změněno: 27. 6. 2022 14:48, Mgr. Marie Novosadová Šípková, DiS.

Anotace

V originále

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.

Návaznosti

GX19-28628X, projekt VaV

Název: Homotopické a homologické metody a nástroje úzce související s matematickou fyzikou

Investor: Grantová agentura ČR, Homotopické a homologické metody a nástroje úzce související s matematickou fyzikou

Citovat

HWANG, Jun-Muk a Katharina NEUSSER. Cone structures and parabolic geometries. Mathematische Annalen. Germany: Springer Berlin Heidelberg, 2022, roč. 383, 1-2, s. 715-759. ISSN 0025-5831. Dostupné z: https://dx.doi.org/10.1007/s00208-021-02208-4.

@article{1821657,
   author = {Hwang, JunandMuk and Neusser, Katharina},
   article_location = {Germany},
   article_number = {1-2},
   doi = {http://dx.doi.org/10.1007/s00208-021-02208-4},
   keywords = {Cone structures; Rational homogeneous space; Varieties of minimal rational tangents; Cartan connections; Parabolic geometry; Filtered manifolds},
   language = {eng},
   issn = {0025-5831},
   journal = {Mathematische Annalen},
   title = {Cone structures and parabolic geometries},
   url = {https://link.springer.com/article/10.1007%2Fs00208-021-02208-4},
   volume = {383},
   year = {2022}
}

TY  - JOUR
ID  - 1821657
AU  - Hwang, Jun-Muk - Neusser, Katharina
PY  - 2022
TI  - Cone structures and parabolic geometries
JF  - Mathematische Annalen
VL  - 383
IS  - 1-2
SP  - 715-759
EP  - 715-759
PB  - Springer Berlin Heidelberg
SN  - 00255831
KW  - Cone structures
KW  - Rational homogeneous space
KW  - Varieties of minimal rational tangents
KW  - Cartan connections
KW  - Parabolic geometry
KW  - Filtered manifolds
UR  - https://link.springer.com/article/10.1007%2Fs00208-021-02208-4
N2  - 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.
ER  -

HWANG, Jun-Muk a Katharina NEUSSER. Cone structures and parabolic geometries. \textit{Mathematische Annalen}. Germany: Springer Berlin Heidelberg, 2022, roč.~383, 1-2, s.~715-759. ISSN~0025-5831. Dostupné z: https://dx.doi.org/10.1007/s00208-021-02208-4.

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