Formal functions and formal embeddings

Abstract

This work contains two theorems, among others, which determine the field of formal-rational functions, $\hat{K}$ , along a closed algebraic set $X$ in a projective space $P$ and in an abelian variety $A$ , respectively. For obvious reasons, we assume that $X$ is connected and has a positive dimension. In the case of am- bient variety $p$ , the answer is that $\hat{K}$ is exactly the field of rational functions on $P$ . If $A$ is the ambient variety, $\hat{K}$ coincides with the field of rational func- tions on a certain abelian scheme $A^{*}$ over a certain complete local ring $R$ , which is derived from the given pair (X, $A$ ). For instance, if $X$ generates $A$ , then $R$ is nothing but the base field of $A$ and $A^{*}$ is the maximal one, say Al (X, $A$ ), among those \'etale and proper (hence abelian) extensions of $A$ which are dominated by the albanese variety of $X$ . In the general case, the origin of $A$ being chosen in $X$ with no loss of generality, let $A^{\prime}$ be the abelian sub- variety of $A$ which is generated by $X$ , and $A^{\prime\prime}=A/A^{\prime}$ . Then $R$ is the completion of the local ring of $A^{\prime\prime}$ at the origin, and $A^{*}$ is the unique etale extension of $A\times A^{g}Spec(R)$ that induces the covering Al (X, $A^{\prime}$ ) of the closed fibre $A^{\prime}$ . ( There exists a non-canonical isomorphism of $A^{*}$ with the product Al (X, $A^{\prime}$ ) $\times Spec(R).)$