Timeline for On holomorphic vector bundles over compact Kahler surfaces
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Nov 14, 2014 at 3:43 | vote | accept | Jiang | ||
| Nov 13, 2014 at 18:59 | comment | added | Matthias Wendt | @XinNie: unfortunately, there are situations where there is a big gap between moduli spaces and the classification of vector bundles. For instance, if $X=\mathbb{P}^2$, semistable holomorphic rank 2 bundles on $X$ satisfy $c_1^2-4c_2\leq 0$. However, every complex vector bundle has a holomorphic structure, independent of what the Chern classes are. In this case, I would say that the moduli space is empty although there is an enormous amount of holomorphic structures... | |
| Nov 13, 2014 at 15:46 | comment | added | Xin Nie | The moduli space $\mathcal{M}(E)$ of holomorphic structures on $E$ is a classic subject in gauge theory and algebraic geometry (and involves some stability issue). $\mathcal{M}(E)$ can be interpreted by differential-geometric means as the space of all integrable "partial connections" (a.k.a. pseudo connections, Dolbeault operators...) modulo gauge equivalence, see the book of Donaldson and Kronheimer Section 2.1.5. However, I can only ensure that $\mathcal{M}(E)$ is non-empty when $X$ is a curve. Higher dimensional cases should also be well known, I guess. | |
| Nov 13, 2014 at 15:33 | answer | added | Matthias Wendt | timeline score: 3 | |
| Nov 13, 2014 at 14:01 | history | asked | Jiang | CC BY-SA 3.0 |