Numerical characterization of the K\"ahler cone of a compact K\"ahler manifold

By J. Demailly, Mihai Paun. 2001

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The goal of this work is give a precise numerical description of the K\"ahler cone of a compact K\"ahler manifold. Our main result states that the K\"ahler 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 K\"ahler manifold, the K\"ahler 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. $\int_Y\alpha^p>0$ 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 $\int_X\alpha^n>0$ contains a K\"ahler current; this is done by using the Calabi-Yau theorem and a mass concentration technique for Monge-Amp\`ere 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 $\cX\to S$ of compact K\"ahler manifolds, the K\"ahler 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/19438513f3c40c13c503384d22a8cf2cb5ffb26b

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

An obstruction to the existence of constant scalar curvature K (2004, background): “n−1c 1(X). Take any c > 0 such that ω−ce has nonnegative volume on any analytic subvariety of Xb (if Xb is smooth then this is the condition that ω − ce be in the closure of the Ka¨hler cone of Xb [DP]). Then deﬁne the slope of IZ, with respect to ω and c, as before: µc(IZ) := Rc 0 a 1(x) + a′ 0”
Erratum to the paper: Compact hyperkaehler manifolds: basic results (2001, unknown): “The latter is the so called projectivity criterion for hyperk¨ahler manifolds. Here we will give a correct proof of these two results. The main input is a recent theorem of J.-P. Demailly and M. Paun [2]. All the other results of the paper remain unaﬀected. The projectivity criterion for hyperkahler”