I have two questions which are somewhat related:

(a) It is a well known result (of Freedman?) in differential topology that $\mathbb R^4$ has exotic smooth structures. Apparently, it is known that this does not happen for any $\mathbb R^n$, $n\neq 4$. In other words, every smooth structure on $\mathbb R^n$, $n\neq 4$, is diffeomorphic to the standard structure. (Whose result is it?!)

**MY QUESTION:** What about analytic structures on $\mathbb R^n$? or complex-analytic structures on $\mathbb C^n$? Have these questions been answered?

(b) Let $(M,O_M)$ be a smooth (respectively, analytic) supermanifold (in the sense of Berezin, Kostant, Leites, Manin, etc.), that is, $M$ is a manifold and $O_M$ is a sheaf of $\mathbb Z_2$-graded algebras which is locally isomorphic to $C^\infty(\mathbb R^m)\otimes\Lambda \mathbb R^n$ (respectively, $C^\omega(\mathbb R^m)\otimes\Lambda \mathbb R^n$).

**MY QUESTION**: In his book "Gauge Field Theory and Complex Geometry", Manin defines the sheaf of ideals $J_M\subseteq O_M$ by $J_M=O_{M,1}^2+O_{M,1}$ and calls it the the "ideal generated by odd elements" (see $\S4.1.3$, page 182). Then he claims that for supermanifolds, $J_M$ is equal to the sheaf of ideals of nilpotent elements. I can prove (using partition of unity) that this statement is correct for *smooth* supermanifolds, but not for *analytic* supermanifolds. I have seen this statement in various places, but without a proper explanation.

I am not even sure why $O_{M,1}^2+O_{M,1}$ is a sheaf! Of course one can consider its sheafification, but I am not convinced that this is what they are doing, and in any case I don't have a counterexample.

This post imported from StackExchange MathOverflow at 2015-03-21 18:37 (UTC), posted by SE-user Valerie