Timeline for References on Moishezon spaces in English/French
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 5, 2017 at 13:11 | comment | added | user21574 | The main question for research is that under which conditions, the Moieshezon manifold is scheme. ? For some cases it has been solved, but in general it is largely open problem | |
| Dec 5, 2017 at 5:36 | comment | added | user21574 | Let $\pi :X\to Δ$ be a complex analytic family of compact complex manifolds such that the fibre$ X_t=π^{-1}(t)$ is projective for every $t\in Δ^∗=Δ\setminus \{0\}$. Suppose that the Hodge number of central fibre satisfies in $h^{0,1}(X_0)=h^{0,1}(X_t)$ for $t$ close to $0$. Then $ X_0=π^{-1}(0)$ is Moishezon. This result is due to Dan Popovici link.springer.com/article/10.1007/s00222-013-0449-0 . I think if fibers are Kahler and we have such assumption on Hodge numbers, then central fiber is of Fujiki class $\mathcal C$. | |
| Nov 27, 2017 at 22:36 | comment | added | user21574 | Let $X$ be a compact complex manifold. Assume that on a dense Zariski-open subset of $X$ there exists a complex polarized variation of Hodge structure whose period map is immersive at one point. Then $X$ is Moishezon. This is due to Griffiths and Schmid result projecteuclid.org/euclid.acta/1485889630 . See arxiv.org/pdf/1707.01327.pdf | |
| Jul 23, 2017 at 4:17 | comment | added | user21574 | Moishezon manifolds are balanced | |
| Jul 23, 2017 at 3:37 | comment | added | user21574 | Any complex Moishezon manifold homeomorphic to $P^n_{\mathbb C}$ is isomorphic to $P^n_{\mathbb C}$. Any complex analytic global deformation of $P^n_{\mathbb C}$ is isomorphic to $P^n_{\mathbb C}$ | |
| Jul 22, 2017 at 16:43 | comment | added | user21574 | There is a new projectivity criterion for Moishezon 3-folds $X$ due to Kollár which says that $X$ is projective if and only if there is no irreducible curve $C⊂X$ homologous to zero and $NE(X)\cap−\overline{NE(X)}=0$, where $NE(X)$ is the cone of effective curves in the vector space of 1-cycles modulo numerical equivalence. | |
| Jul 22, 2017 at 16:40 | comment | added | user21574 | A Moishezon manifold is projective if and only if it is a Kähler manifold or if and only if it has a line bundle whose curvature is semi-positive and positive in at least one point due to Siu and Demailly | |
| Jul 20, 2017 at 23:24 | comment | added | user21574 | Note that Moishezon surfaces are not important since every smooth Moishezon surface is projective: So the higher dimension can be useful . In fact for some singular cases, the projectivity of Moishezon surface is known, for example if $S$ be a normal Moishezon surface with at worst rational singularites then it is projective | |
| Jan 11, 2011 at 19:35 | vote | accept | shenghao | ||
| Jan 11, 2011 at 12:01 | comment | added | Gjergji Zaimi | There is a chapter on modifications in the book "Several complex variables VII: sheaf-theoretical methods in complex analysis" by Grauert and Peternell. they don't give the proofs but refer to the english translation of Moishezon's paper that Georges mentions in his answer. | |
| Jan 11, 2011 at 11:40 | answer | added | Georges Elencwajg | timeline score: 9 | |
| Jan 11, 2011 at 0:38 | history | asked | shenghao | CC BY-SA 2.5 |