On the Convergence of Formal Equivalence between Embeddings

Abstract

Let j: X --* W denote the morphism embedding the compact complex manifold X into the manifold W. The formal principle holds (for j) if, given any other embedding j': X'-W' together with a formal isomorphism between j and j', there is an actual isomorphism between j and j' (for definitions see ? 2.1). For instance, the formal principle holds for exceptional curves of the first kind on surfaces. On the other hand, Arnold [1] recently gave the example of an embedding (with a topologically trivial normal bundle) of an elliptic curve into a surface for which the formal principle fails. A number of articles dealing with the formal principle in algebraic as well as in analytic geometry have been published (Nirenberg-Spencer [19], Grauert [7], Hironaka-Rossi [16], Griffiths [9], [10], Hironaka [14], Hironaka-Matsumura [15], Hartshorne [11], Artin [2], Gieseker [6].) Moreover, a few weeks after the present work was written, CommichauGrauert [23] proved that the formal principle holds under an assumption (one-positivity) which seems much weaker than ours. We hope that the present article retains some interest because of its method which may work in the algebraic and/or singular case. We are interested, in the complex-analytic case, in the situation where the normal bundle is fairly positive or non-negative: this means there are enough submanifolds in W around X. We propose a conjecture and then give our result: we let D denote the germ of the space of submanifolds near X (an open subset in the Douady space), UD denote the universal submanifold of D x W, and LID denote the projection from UD into W.