Generalized Tanaka prolongation and convergence of formal equivalence between embeddings

TL;DR

This paper extends the convergence of formal equivalences between embeddings of complex manifolds to cases with semi-positive normal bundles, using a generalized Tanaka prolongation approach.

Contribution

It introduces a generalized Tanaka prolongation method for geometric structures, enabling convergence results under weaker positivity assumptions.

Findings

Convergence of formal equivalences holds under semi-positive normal bundles with geometric conditions.

Generalized Tanaka prolongation provides an absolute parallelism for certain deformation families.

Application to minimal rational curves on hypersurfaces demonstrates the theory's utility.

Abstract

The works of Commichau--Grauert and Hirschowitz showed that a formal equivalence between embeddings of a compact complex manifold is convergent, if the embeddings have sufficiently positive normal bundles in a suitable sense. We show that the convergence still holds under the weaker assumption of semi-positive normal bundles if some geometric conditions are satisfied. Our result can be applied to many examples of general minimal rational curves, including general lines on a smooth hypersurface of degree less than $n$ in the $(n+1)$-dimensional projective space. As a key ingredient of our arguments, we formulate and prove a generalized version of Tanaka's prolongation procedure for geometric structures subordinate to vector distributions, a result of independent interest. When applied to the universal family of the deformations of the compact submanifolds satisfying our geometric conditions, the generalized Tanaka prolongation gives a natural absolute parallelism on a suitable fiber space. A formal equivalence of embeddings must preserve these absolute parallelisms, which implies its convergence.