  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.
