Timeline for Almost complex 4-manifolds with a "holomorphic" vector field
Current License: CC BY-SA 3.0
11 events
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| Jul 8, 2013 at 22:36 | comment | added | Dmitri Panov | Robert, thanks for this remark, you are right that this second phrase makes no sense. I rewrote so that I it makes sense. | |
| Jul 8, 2013 at 22:34 | history | edited | Dmitri Panov | CC BY-SA 3.0 |
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| Jul 6, 2013 at 13:57 | comment | added | Robert Bryant | @Dmitri: Perhaps I don't understand your terminology, but it seems to me that $S^4$ admits a smooth $S^1$ action, but it doesn't admit any almost complex structure, let alone one with a (nontrivial) symmetry vector field. Thus, your second sentence is mysterious to me. | |
| Jan 16, 2010 at 23:37 | comment | added | Dmitri Panov | Yes, I refer to the flow associated to the vector field. I don't want to impose that all zeros are isolated. | |
| Jan 16, 2010 at 22:51 | comment | added | Ryan Budney | When you say the vector field preserves the almost-complex structure, are you referring to the flow associated to the vector field? And do you want the vector field to be everywhere nonzero? Nonzero at all but finitely many points? | |
| Dec 22, 2009 at 12:27 | comment | added | Dmitri Panov | Tim, thanks for the comment! In fact the approach I propose is exactly to give a justification of your words:)) How do we calculate the Euler charecteristics of a manifold - count the number of zeros of a vector field. I want to say, that for a Hilbert scheme this should be a result of the same caclulation. I think it would be cool to have a "simple" calculation of the Euler charateristics. By the way this speculation can also be applied to DT invariants -- when you count 0-dim subschemes on 3 dimensional CY manifolds. This calculation was done in 2005 and was one of MNOP conjectures. | |
| Dec 22, 2009 at 11:55 | comment | added | Tim Perutz | It's an intriguing question, but it seems to me a surprisingly delicate way to calculate the Euler characteristic of the Hilbert scheme, which one expects to be a universal function of the Betti numbers of the 4-manifold, as in the integrable case. Perhaps one can prove this using the Cech spectral sequence coming from the open cover of Hilb arising from a good cover of the 4-manifold? [BTW, I know you're quoting Voisin, but the following rule is worth insisting on: a 4-fold is an algebraic or analytic variety of dimension 4. A 4-manifold has real dimension 4.] | |
| Dec 22, 2009 at 10:45 | comment | added | Dmitri Panov | Thanks :) Maybe this sound cooler than it should be. He is the reference, it was published in 2002 people.math.jussieu.fr/~voisin/Articlesweb/almost.pdf | |
| Dec 22, 2009 at 9:09 | comment | added | Kevin H. Lin | This sounds really cool. Is this recent work of Voisin? | |
| Dec 22, 2009 at 0:41 | history | edited | Dmitri Panov | CC BY-SA 2.5 |
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| Dec 22, 2009 at 0:14 | history | asked | Dmitri Panov | CC BY-SA 2.5 |