Abstract : A complex submanifold in a complex manifold satisfies the formal principle if its formal neighborhood determines its biholomorphic germ. A smooth rational curve in a complex manifold satisfies the formal principle if its normal bundle is positive. It is unknown whether a rational curve with semi-positive normal bundle satisfies the formal principle. We discuss the simplest unknown case of a smooth rational curve in a threefold whose normal bundle is the direct sum of a trivial line bundle and a line bundle of degree 1.

Keywords : formal principle; rational curves; affine connection