Rational Curves and Parabolic Geometries

The twistor transform of a parabolic geometry has two steps: lift up to a geometry of higher dimension, and then descend to a geometry of lower dimension. The first step is a functor, but the second requires some compatibility conditions. Local necessary conditions were uncovered by Andreas Čap [12]. We uncover necessary and sufficient global conditions for complex analytic geometries: rationality of curves defined by certain differential equations. We apply the theorems to second and third order ordinary differential equations to determine whether their solutions are rational curves. We harness Mori’s bend–and-break to show that any closed Kähler manifold containing a rational curve and bearing a parabolic geometry must inherit its parabolic geometry from a parabolic geometry on a lower dimensional closed Kähler manifold.