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

TL;DR

This paper extends the Nakai-Moishezon criterion to certain (1,1)-classes on smooth projective varieties, characterizing when these classes are represented by Kähler metrics via positivity on subvarieties.

Contribution

It proves a Nakai-Moishezon type theorem for classes in the intersection of the (1,1) cohomology and the image of the fundamental group's cohomology, linking algebraic and differential geometry.

Findings

Characterization of Kähler classes via positivity on algebraic subvarieties.

Extension of Nakai-Moishezon criterion to classes involving the fundamental group.

Conditions for classes to be represented by Kähler metrics.

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 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 $Z\subset X$, $[\omega]^d.Z>0$.