Timeline for Are spaces of holomorphic maps manifolds?
Current License: CC BY-SA 2.5
13 events
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| May 31, 2010 at 16:18 | comment | added | BCnrd | @Oblomov: I read "positive degree" in English rather than French sense ($> 0$, not $\ge 0$), so sounds like ignoring deg. 0 maps, which are in Hom-scheme. To show Hom-scheme has expected topology on $\mathbf{C}$-valued points, tedious to fit argument in comment box. Here's a hint: use valuative criterion for properness on each component to prove that a projective embedding on $Y$ induces a closed immersion on Hom-schemes, so reduce to case $Y = \mathbf{P}^n$. Then study line bundles on $X$, univ. property of $\mathbf{P}^n$, & topological meaning of pts of id. component of Pic scheme of $X$. | |
| May 31, 2010 at 16:00 | comment | added | Oblomov | @BCnrd: what do you mean "when corrected with constant maps"? Also: how do you indeed check that the topology on the complex points of the Hom-scheme is compact-open? | |
| May 31, 2010 at 15:38 | vote | accept | Oblomov | ||
| May 31, 2010 at 15:38 | vote | accept | Oblomov | ||
| May 31, 2010 at 15:38 | |||||
| May 31, 2010 at 14:33 | history | edited | Oblomov | CC BY-SA 2.5 |
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| May 31, 2010 at 13:57 | comment | added | BCnrd | Assume $X$, $Y$ projective. Justifying example with proj. line (when corrected with constant maps) uses ${\rm{Pic}}(\mathbf{P}^1_ A) = \mathbf{Z}$ for local $A$. Nice exercise to prove Grothendieck's Hom-scheme equips set of holomorphic maps with compact-open topology. Hom-scheme is (by construction) countable disjoint union of quasi-proj. schemes; identifying when two maps lie on same connected or even irreducible component looks hopeless. The tangent spaces can be described (using tangent bundle of $Y$), but there's no clean "formula" for their dimensions. Smoothness should be rare. | |
| May 31, 2010 at 13:55 | answer | added | Georges Elencwajg | timeline score: 11 | |
| May 31, 2010 at 13:54 | answer | added | Dmitri Panov | timeline score: 3 | |
| May 31, 2010 at 11:37 | comment | added | Oblomov | By the way, if the answer to my question were true for an other topology, then please let me know too! | |
| May 31, 2010 at 11:29 | comment | added | Oblomov | The topology of the subspace of the whole mapping space, equipped with the compact open topology. | |
| May 31, 2010 at 11:27 | history | edited | Oblomov | CC BY-SA 2.5 |
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| May 31, 2010 at 11:00 | comment | added | Harry Gindi | What topology are you equipping the space Hol(X(C),Y(C)) with? | |
| May 31, 2010 at 10:53 | history | asked | Oblomov | CC BY-SA 2.5 |