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

TL;DR

This paper proves Hirschowitz's conjecture on the formal principle for certain vector bundles on Fano manifolds and introduces a method using Cartan's equivalence to analyze deformations of submanifolds.

Contribution

It applies Cartan's equivalence method to confirm the conjecture for Fano manifolds and extends the result to general unobstructed submanifolds with globally generated normal bundles.

Findings

Hirschowitz's conjecture holds for vector bundles on Fano manifolds.

General unobstructed submanifolds with globally generated normal bundles satisfy the formal principle.

Sufficiently general deformations of free rational curves satisfy the formal principle.

Abstract

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.