Holomorphic Cartan geometries and rational curves

Abstract

A b s t r ac t . We prove that any compact Kähler manifold bearing a holomorphic Cartan geometry contains a rational curve just when the Cartan geometry is inherited from a holomorphic Cartan geometry on a lower dimensional compact Kähler manifold. This shows that many complex manifolds admit no or few holomorphic Cartan geometries.

The Hartogs extension problem for holomorphic parabolic and reductive geometries

“…Every 3-dimensional projective Klein geometry M = Γ\U for which U ⊂ P 3 contains a projective line is known [18]. On any connected complex manifold containing a rational curve, any holomorphic projective connection is flat [4] p. 9 corollary 4, has a Zariski dense family of deformations of the rational curve, and admits no holomorphic deformations, as the Schwarzian derivative has to vanish along the rational curves [24], so each of these Klein geometries is the unique holomorphic projective connection on each of those complex 3-folds.…”

Projective structures and $ρ$-connections

“…Finally, as any Fano manifold is compact simply-connected and admits rational curves as in Corollary 3.2 (see [10] and the references therein) from Corrolary 3.1 we obtain the following fact [3] : the projective space is the only Fano manifold which admits a projective structure (compare [7, (5.3)] , [6] , [10] ).…”

Holomorphic G-structures and foliated Cartan geometries on compact complex manifolds

“…The methods used in [BD2] to prove Theorem 3.4 and Theorem 3.5 does not use the results in [BM,HM,KO2,KO3,Ye]: they are specific to the case of GL(2, C)-geometry and unify the twisted holomorphic symplectic case (even dimensional case) and the holomorphic conformal case (odd dimensional case).…”