| | |
| | |
Stat |
Members: 3643 Articles: 2'487'895 Articles rated: 2609
28 March 2024 |
|
| | | |
|
Article overview
| |
|
Numerical characterization of the Kähler cone of a compact Kähler manifold | Jean-Pierre Demailly
; Mihai Paun
; | Date: |
22 May 2001 | Subject: | Algebraic Geometry MSC-class: 14C30, 14J45, 32C17, 32C30, 32F07, 32J27 | math.AG | Affiliation: | Université Joseph Fourier, Grenoble, France), Mihai Paun (Université Louis Pasteur, Strasbourg, France | Abstract: | The goal of this work is give a precise numerical description of the Kähler cone of a compact Kähler manifold. Our main result states that the Kähler cone depends only on the intersection form of the cohomology ring, the Hodge structure and the homology classes of analytic cycles: if $X$ is a compact Kähler manifold, the Kähler cone $cK$ of $X$ is one of the connected components of the set $cP$ of real $(1,1)$ cohomology classes ${alpha}$ which are numerically positive on analytic cycles, i.e. $int_Yalpha^p>0$ for every irreducible analytic set $Y$ in $X$, hbox{$p=dim Y$}. This result is new even in the case of projective manifolds, where it can be seen as a generalization of the well-known Nakai-Moishezon criterion, and it also extends previous results by Campana-Peternell and Eyssidieux. The principal technical step is to show that every nef class ${alpha}$ which has positive highest self-intersection number $int_Xalpha^n>0$ contains a Kähler current; this is done by using the Calabi-Yau theorem and a mass concentration technique for Monge-Ampère equations. The main result admits a number of variants and corollaries, including a description of the cone of numerically effective $(1,1)$ classes and their dual cone. Another important consequence is the fact that for an arbitrary deformation $cX o S$ of compact Kähler manifolds, the Kähler cone of a very general fibre $X_t$ is ``independent’’ of $t$, i.e. invariant by parallel transport under the $(1,1)$-component of the Gauss-Manin connection. | Source: | arXiv, math.AG/0105176 | Services: | Forum | Review | PDF | Favorites |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser claudebot
|
| |
|
|
|
| News, job offers and information for researchers and scientists:
| |