[Paper Review] Un théoréme de Nakai-Moishezon pour certaines classes de type (1,1)

Philippe Eyssidieux|ArXiv.org|Nov 10, 1998

Algebraic Geometry and Number Theory3 references3 citations

TL;DR

This paper establishes a Nakai-Moishezon-type criterion for the ampleness of certain (1,1)-classes on compact Kähler manifolds. It proves that a cohomology class $[\omega] \in H^2(\pi_1(X),\mathbb{R})^{1,1} + NS(X) \otimes \mathbb{R}$ is Kähler if and only if its positive degree intersection with every reduced, closed, d-dimensional algebraic subvariety is positive, extending classical ampleness criteria to a broader class of cohomology classes on projective varieties.

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 $H^2(π_1(X),{\mathbb R}) o H^2(X)$ induced by the classifying map $X o Bπ_1(X)$. 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 $Z\subset X$, $[ω]^d.Z>0$.

Motivation & Objective

• To extend the classical Nakai-Moishezon criterion for ampleness to a broader class of (1,1)-cohomology classes on compact complex projective varieties.
• To characterize when a cohomology class in $H^2(\pi_1(X),\mathbb{R})^{1,1} + NS(X) \otimes \mathbb{R}$ is represented by a Kähler metric.
• To bridge the gap between topological invariants (via $\pi_1(X)$) and algebraic geometry by analyzing the interplay between fundamental group cohomology and Néron-Severi group in the context of Kähler classes.
• To provide a cohomological criterion for Kähler classes that generalizes the classical positivity condition on subvarieties.

Proposed method

• The paper studies the subspace $H^2(\pi_1(X),\mathbb{R})^{1,1} \subset H^{1,1}(X,\mathbb{R})$, defined as the intersection of the Hodge (1,1)-classes with the image of the classifying map $X \to B\pi_1(X)$.
• It considers cohomology classes $[\omega]$ in the sum $H^2(\pi_1(X),\mathbb{R})^{1,1} + NS(X) \otimes \mathbb{R}$, combining topological and algebraic parts.
• The main criterion is formulated in terms of the positivity of the top-degree intersection number $[\omega]^d \cdot Z > 0$ for all reduced, closed, d-dimensional subvarieties $Z \subset X$, where $d = \dim Z$.
• The proof relies on Hodge theory and the structure of the Néron-Severi group, using the fact that classes in $NS(X) \otimes \mathbb{R}$ are algebraic and thus amenable to intersection-theoretic analysis.
• The argument leverages the fact that the classifying map induces a map on cohomology, allowing the transfer of topological information into the Hodge-theoretic framework.
• The key technical step is showing that positivity on all subvarieties implies the existence of a Kähler metric representing $[\omega]$, using duality and the cone of effective cycles.

Experimental results

Research questions

• RQ1 Under what conditions is a cohomology class in $H^2(\pi_1(X),\mathbb{R})^{1,1} + NS(X) \otimes \mathbb{R}$ represented by a Kähler metric?
• RQ2 Can the classical Nakai-Moishezon criterion be extended to classes that are not purely algebraic but involve cohomology from the fundamental group?
• RQ3 Is the positivity of the intersection number $[\omega]^d \cdot Z$ for all reduced, closed subvarieties $Z$ of dimension $d$ sufficient to guarantee that $[\omega]$ is a Kähler class?
• RQ4 How do the cohomology classes arising from the fundamental group interact with the Néron-Severi group in the context of Kähler geometry?
• RQ5 What is the precise relationship between the topological structure of $X$ (via $\pi_1(X)$) and the existence of Kähler metrics on $X$?

Key findings

• A cohomology class $[\omega] \in H^2(\pi_1(X),\mathbb{R})^{1,1} + NS(X) \otimes \mathbb{R}$ is the cohomology class of a Kähler metric if and only if $[\omega]^d \cdot Z > 0$ for every reduced, closed, d-dimensional algebraic subvariety $Z \subset X$.
• The class $H^2(\pi_1(X),\mathbb{R})^{1,1}$ is a subspace of $H^{1,1}(X,\mathbb{R})$ that captures (1,1)-classes arising from the fundamental group, and its intersection with the Néron-Severi group is well-behaved in the context of positivity.
• The positivity condition on all subvarieties is both necessary and sufficient for the Kähler representability of $[\omega]$, generalizing the classical Nakai-Moishezon criterion.
• The result holds for smooth, compact, projective varieties over $\mathbb{C}$, and the class $[\omega]$ need not be algebraic, as it includes topological contributions from $H^2(\pi_1(X),\mathbb{R})^{1,1}$.
• The proof establishes that the cone of Kähler classes in this subspace is precisely characterized by the positivity of intersection numbers with all positive-dimensional subvarieties.
• The paper provides a cohomological criterion that unifies topological and algebraic aspects of Kähler geometry, showing that positivity on subvarieties determines Kähler representability even beyond the Néron-Severi group.