I have two questions which are somewhat related:
(a) It is a well known result (of Freedman?) in differential topology that R4 has exotic smooth structures. Apparently, it is known that this does not happen for any Rn, n≠4. In other words, every smooth structure on Rn, n≠4, is diffeomorphic to the standard structure. (Whose result is it?!)
MY QUESTION: What about analytic structures on Rn? or complex-analytic structures on Cn? Have these questions been answered?
(b) Let (M,OM) be a smooth (respectively, analytic) supermanifold (in the sense of Berezin, Kostant, Leites, Manin, etc.), that is, M is a manifold and OM is a sheaf of Z2-graded algebras which is locally isomorphic to C∞(Rm)⊗ΛRn (respectively, Cω(Rm)⊗ΛRn).
MY QUESTION: In his book "Gauge Field Theory and Complex Geometry", Manin defines the sheaf of ideals JM⊆OM by JM=O2M,1+OM,1 and calls it the the "ideal generated by odd elements" (see §4.1.3, page 182). Then he claims that for supermanifolds, JM 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 O2M,1+OM,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