The Hartogs extension problem for holomorphic parabolic and reductive geometries

Abstract

Every holomorphic effective parabolic or reductive geometry on a domain over a Stein manifold is the pullback of a unique such geometry on the envelope of holomorphy of the domain. We use this result to classify the Hopf manifolds which admit holomorphic reductive geometries, and to classify the Hopf manifolds which admit holomorphic parabolic geometries. Every Hopf manifold which admits a holomorphic parabolic geometry with a given model admits a flat one. We classify flat holomorphic parabolic geometries on Hopf manifolds. For every generalized flag manifold there is a Hopf manifold with a flat holomorphic parabolic geometry modelled on that generalized flag manifold.

Holomorphic Cartan geometries and rational curves

“…For example, if f : Z → Y is the blowup of a subvariety Y 0 ⊂ Y , then over smooth points of Y 0 , Z is smooth and f is not a local isomorphism, so Z admits no holomorphic Cartan geometry. Proposition 2.1 of [45] is a similar result. For proof, see [39, p. 50, Cor.…”

Symmetries of holomorphic geometric structures on tori

“…Proof. Holomorphically split g = W ⊕ h for some H-module W ; this H-module is effective [17] p. 9 lemma 6.1. At each point of B, the Cartan connection splits into a 1-form valued in W and a connection 1-form, say ω = σ + γ.…”

“…At each point of the total space B of the Cartan geometry, the 1-form σ is semibasic, so defines a 1-form σ on the corresponding point of the base manifold, a coframe. Because H acts effectively on W , the map σ identifies the total space of the Cartan geometry with a subbundle of the frame bundle of the base manifold [17] corollary 6.2. Hence the Cartan geometry is precisely an H-reduction of the frame bundle with a holomorphic connection.…”