Un théoréme de Nakai-Moishezon pour certaines classes de type (1,1)

Abstract

Let $X$ be a smooth compact projective variety over $\mathbb C$. Let $H^2(π_1(X),\mathbb R)^{1,1}$ be the intersection of $H^{1,1}(X,{\mathbb R})$ with the image of the map $H^2(π_1(X),{\mathbb R})\to H^2(X)$ induced by the classifying map $X\to Bπ_1(X)$. Let $NS(X)$ be the Néron-Severi group of $X$. Let $[ω]\in H^2(π_1(X),\mathbb R)^{1,1}+ NS(X)\otimes {\mathbb R}$. In this note, we prove that $[ω]$ is the cohomology class of a Kähler metric if and only if for every $d$-dimensional reduced closed algebraic subvariety $Z\subset X$, $[ω]^d.Z>0$.