On some formal imbeddings
Heisuke Hironaka · Illinois Journal of Mathematics · 1968
In this paper, the reader will find the following theorem" Let X be a smooth irreducible algebraic scheme over an algebraically closed field .Assume dim X >_ 2. Then an imbedding of X into a projective space P over is uniquely determined by the formal scheme which is obtained by completing P along X. (See Th.V, 2, for its precise meaning.)We actually prove that the field of "formal-rational" functions on/ coincides with the field of ra- tional functions on P. If k C (the complex number field), this result implies that for any connected open neighborhood U of X in P in the sense of the usual metric topology, every meromorphic function on U extends to a rational function on the entire P. (This implication is proven, for instance, by apply- ing the technique of GAGA, due to J. P. Serre, to the infinitesimal neighbor- hoods of X in P which are complex-analytic spaces.)A general problem I have in mind may be posed as follows" Let Z be a regular irreducible formal scheme over a field ]c, such that if I is a defining ideal sheaf of Z then the subschemes X of Z defined by y+l are proper over k.Let A H(Z, 0z) which is a k-algebra.We ask if there exists an A-morphism f" Z --T with an integral scheme of finite type (or finite presentation) over A such that if g Z ---> W is any A-morphism into an A-scheme of finite type (or finite presentation), then there exists a unique rational map h" T ---.W with g hr.In this paper, my interest is confined strictly to the case of ample normal bundle (e.g.X X0 is smooth and the dual of I/I as a sheaf of 0x-modules is an ample locally free sheaf on X).We have a satisfactory answer to the above question only in the case of codimension one, i.e., when I/I is an in- vertible sheaf on X. (See Theorems I, II, III, 1, and Theorems IV*, V*, 2.) The case of higher codimensions is still very little understood.Our result ia this case is done only for a very special kind of imbeddings, i.e., imbeddings into a projective space.This seems, however, to throw some encouraging light onto the general problem of higher codimensions.(See Theorems IV, V, 2.) For a certain technical reason we assume dim X >__ 2 throughout this paper.Some novel phenomena as well as gen- eralizations for the case of dim X 1 will be investigated in a future joint paper with Matsumura.