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

TL;DR

This paper provides a numerical description of the K"ahler cone of a compact K"ahler manifold, showing it depends only on intersection forms, Hodge structure, and homology classes, extending classical criteria.

Contribution

It generalizes the Nakai-Moishezon criterion to all compact K"ahler manifolds and introduces a new method to identify K"ahler classes using Monge-Ampère equations.

Findings

K"ahler cone characterized by positivity on analytic cycles

Extension of Nakai-Moishezon criterion to K"ahler manifolds

K"ahler cone remains invariant under deformations

Abstract

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.