Holomorphic G-Structures and Foliated Cartan Geometries on Compact Complex Manifolds

This is a survey dealing with holomorphic G-structures and holomorphic Cartan geometries on compact complex manifolds. Our emphasis is on the foliated case. We investigate holomorphic foliations with a transverse holomorphic Cartan geometry, and also with the more general structure of branched transverse holomorphic Cartan geometry. The first part of the chapter presents the geometric notion of holomorphic G-structure whose origin was motivated by various classical examples. We explain some classification results for compact complex manifolds endowed with holomorphic G-structures highlighting the special case of GL ( 2 , ℂ ) $$\mathrm {GL}(2,{\mathbb {C}})$$ -structures and SL ( 2 , ℂ ) $$\mathrm {SL}(2,{\mathbb {C}})$$ -structures. The second part of the survey deals with holomorphic Cartan geometry in the classical case and in the branched and generalized case. Two definitions of foliated (branched, generalized) Cartan geometry are described and shown to be equivalent. We provide some classification results of compact complex manifolds with foliated (branched or generalized) Cartan geometries. At the end some related open problems are formulated.