# Formal completion of a complex normal bundle along the zero section

The total space of a complex normal bundle over a submanifold $Y$ in a complex manifold $X$ can be seen as an analytic scheme. If one blindly uses Hartshorne's definition about "formal schemes" from his book "Algebraic Geometry" to formally complete the normal bundle $N$ along the zero section $\epsilon$, the formal completion of $N$ would be of the form $(\epsilon, \lim_{\leftarrow}\mathcal{O}_N/\mathcal{I}^n)$ where $\mathcal{I}=\{f\in\mathcal{O}(N): f|_\epsilon=0\}$ is the ideal sheaf.

What worries me is that Hartshorne's definition in the category of algebraic schemes is specifically said to be valid for noetherian schemes, which the complex manifold N isn't. Is that a problem? Do we need the noetherian condition in the complex analytic case at all? How can we adapt the algebraic definition of "formal scheme" for analytic schemes?

Thank you for your help. I hope I was able to formulate clearly my problem.

• I suspect the theory goes through verbatim but one never knows with analytic spaces. If it helps, remember that the stalks of $\mathcal{O}$ are noetherian rings. – Denis Nardin Jan 31 '17 at 2:17
• Since analytic spaces are not locally built out of rings as schemes (or non-archimedean analytic spaces) are, it is definitely not the case that the arguments for setting up noetherian adic formal schemes can be imported "as is" to the context of complex-analytic spaces. Even to make useful definitions would require much thought and care. But there was a lot of activity in the late 1970's and 1980's to carry over to the complex-analytic setting Artin's deformation-theoretic approach to moduli problems in algebraic geometry, so that work might contain some useful discussion in this direction. – nfdc23 Jan 31 '17 at 2:46
• @nfdc23 I imagine you can still define formal analytic spaces as ind-analytic spaces that correspond (locally) to diagrams of nilpotent thickenings. I agree though that there are probably many more subtleties. – Denis Nardin Jan 31 '17 at 2:55
• @DenisNardin: The real theorems about formal schemes in EGA III make essential use of features that are very painful or impossible to express in the language of compatible systems of thickenings; e.g., flatness arguments, various devissage considerations that would interact poorly with reduction modulo ideals of definition, calculations with Ext-sheaves, etc. It is akin to trying to make the theory of completions of noetherian rings along ideals without forming the completion: flatness is lost and so many theorems cannot be expressed. So after making definitions, I think little could be done. – nfdc23 Jan 31 '17 at 3:05
• Excuse me but... Aren't analytic spaces Noetherian, if considered with the analytic Zariski topology? – Qfwfq Jan 31 '17 at 19:08

Instead of trying to adapt Hartshorne's definition for formal completion of noetherian schemes to the difficult case of analytic schemes, which are never noetherian, we could instead consider the holomorphic symmetric algebra bundle $\eta:Sym^{\bullet}(N^*)\rightarrow Y$ of the conormal bundle $N^*\rightarrow Y$ and formally complete this bundle.
According section 3 of the article "Formality theorems for Hochschild chains in the Lie algebroid setting" by D. Calaque, V. Dolgushev and G. Halbout one can formally complete the holomorphic symmetric algebra bundle $Sym^{\bullet}(N^*)$ of the complex conormal bundle $N^*$. The local sections of the completion $\widehat Sym^{\bullet}(N^*)$ are then given in local coordinates by $f=\sum_{l\geq0}f_{i_1\dots i_l}(y)w^{i_1}\dots w^{i_l}$, where $y_1, \dots, y_l$ are some local coordinates on $Y$ and $w_1, \dots, w_m$, m=codim(Y) are the local coordinates on the fibres of $N$. This bundle, I think, can be interpreted as the formal completion of the complex-analytic locally free quasi-coherent sheaf $\mathcal{Sym^{\bullet}(N^*)}$. Moreover, comparing Hartshorne with this, it seems that $\mathcal{\widehat Sym^{\bullet}(N^*)}$ is indeed the analytic version of the usual formal completion of an algebraic bundle $N$ along its zero section. It is an exercise to show that in the algebraic case $\mathcal{\widehat Sym^{\bullet}(N^*)}=\mathcal O_{\hat N}$ which is precisely Hartshorne's definition of formal completion of a noetherian scheme. I am not fooling myself, do I?
• Well, $\widehat Sym(N^*)$ in the analytic context seems the same as the formal completion of the algebraic normal bundle $N$ along the zero section in the algebraic set up. Which things does this construction fail to achieve? My sole goal is to formally complete the fiber of the normal bundle. Don't I achieve this by working with $\widehat Sym(N^*)$? – Flavius Aetius Feb 1 '17 at 22:36