An application of Cartan's equivalence method to Hirschowitz's conjecture on the formal principle

A conjecture of Hirschowitz's predicts that a globally generated vector bundle W on a compact complex manifold A satisfies the formal principle; i.e., the formal neighborhood of its zero section determines the germ of neighborhoods in the underlying complex manifold of the vector bundle W. By applying Cartan's equivalence method to a suitable differential system on the universal family of the Douady space of the complex manifold, we prove that this conjecture is true if A is a Fano manifold, or if the global sections of W separate points of A. Our method shows more generally that for any unobstructed compact submanifold A in a complex manifold, if the normal bundle is globally generated and its sections separate points of A, then a sufficiently general deformation of A satisfies the formal principle. In particular, a sufficiently general smooth free rational curve on a complex manifold satisfies the formal principle.