Formal principle with convergence for bracket-generating families of rational curves

Establish that for any complex manifold X and any connected open subset K of the space S(X) of smooth rational curves with semipositive normal bundles, such that the natural distribution D ⊂ TK derived from the positive part of the normal bundle is bracket-generating (i.e., successive Lie brackets of local sections of D span TK), a general member C ∈ K satisfies the formal principle with convergence: every formal isomorphism between the formal neighborhoods (C/X)_ and (C′/X′)_ extends to the germ of a biholomorphism between Euclidean neighborhoods.

Background

The paper studies the formal principle with convergence for smooth rational curves in complex manifolds. Prior results show that when the normal bundle of a rational curve is positive, the formal principle with convergence holds (Commichau–Grauert; Hirschowitz), while for semipositive normal bundles only the formal principle (without guaranteed convergence) is known for general deformations (Hwang).

To avoid counterexamples arising from product structures, the authors introduce bracket-generating families of rational curves, defined via a natural distribution D on the parameter space K whose Lie brackets generate all directions. The conjecture extends the positive case (recast as D = O_K) and posits convergence for general members of any bracket-generating family. The paper proves this in the special case of Goursat-type families, leaving the general case open.