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 $H^2({\pi }_1(X),\mathbb{R}{)}^{1,1}$ 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 Néron-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 Kähler metric if and only if for every $d$-dimensional reduced closed algebraic subvariety $Z\subset X$, $[\omega {]}^d.Z>0$.

13 pages, latex2e