Un th\'eor\'eme de Nakai-Moishezon pour certaines classes de type (1,1)

Let $X$ be a smooth compact projective variety over $\mathbb C$. Let $H^2(\pi_1(X),\mathbb R)^{1,1}$ be the intersection of $H^{1,1}(X,{\mathbb R})$ with the image of the map $H^2(\pi_1(X),{\mathbb R})\to H^2(X)$ induced by the classifying map $X\to B\pi_1(X)$. Let $NS(X)$ be the N\'eron-Severi group of $X$. Let $[\omega]\in H^2(\pi_1(X),\mathbb R)^{1,1}+ NS(X)\otimes {\mathbb R}$. In this note, we prove that $[\omega]$ is the cohomology class of a K\"ahler metric if and only if for every $d$-dimensional reduced closed algebraic subvariety $Z\subset X$, $[\omega]^d.Z>0$.