Timeline for Artin vanishing for Stein manifolds and restriction maps
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 2, 2021 at 19:25 | vote | accept | Peter Scholze | ||
| Sep 2, 2021 at 18:59 | answer | added | Mohan Ramachandran | timeline score: 14 | |
| Sep 2, 2021 at 12:56 | comment | added | Peter Scholze | It doesn't have dense image: One argument is to observe that the residue at $0$ of a function on $\mathbb C^\times$ gives a nonzero continuous functional which vanishes on the image of $\mathcal{O}(\mathbb C)$. The same argument applies more generally to hyperplane complements. | |
| Sep 2, 2021 at 12:53 | comment | added | Donu Arapura | Interesting question. The inclusion of $\mathbb{C}^*\subset \mathbb{C}$ is an open immersion of Stein manifolds where the map on $H^d$ is not surjective, but I'm not sure about the density condition for function spaces. | |
| Sep 2, 2021 at 11:42 | comment | added | Peter Scholze | I'm using the Frechet topology on $\mathcal O(X)$ and $\mathcal O(U)$ (of uniform convergence on compact subsets), as I believe is the standard choice. (Sorry for not specifying.) | |
| Sep 2, 2021 at 11:40 | comment | added | Jason Starr | What topology are you using? For the open unit disk in the complex plane, the holomorphic restriction of $1/(1-z)$ is certainly not approximated by restrictions of entire holomorphic functions in the uniform topology (since these are bounded on the closure of the disk). | |
| Sep 2, 2021 at 11:31 | comment | added | Peter Scholze | @virkkunen If both $U$ and $X$ are affine algebraic, the hypothesis is basically never satisfied (as Jason Starr observed). But you are welcome to assume that $X$ is an affine algebraic variety. | |
| Sep 2, 2021 at 11:21 | comment | added | Peter Scholze | @JasonStarr An example would be a small open ball inside affine space | |
| Sep 2, 2021 at 11:16 | comment | added | Jason Starr | Shouldn't the closed complement of an open Stein submanifold of a (connected) Stein manifold be "generically" a hypersurface in the ambient Stein manifold? If so, it seems impossible to have the "dense image" condition, unless the open equals the entire ambient manifold or the empty set, i.e., the two cases where the conjecture is trivially true. | |
| Sep 2, 2021 at 10:47 | comment | added | user178279 | Is there an easy argument for smooth affine algebraic varieties? | |
| Sep 2, 2021 at 10:33 | history | asked | Peter Scholze | CC BY-SA 4.0 |