A generalization of the Castelnuovo-de Franchis inequality

TL;DR

This paper establishes a new lower bound on the Hodge number h^{2,0}(X) for irregular compact Kähler and projective varieties, generalizing classical inequalities and improving previous results for higher-dimensional varieties.

Contribution

It introduces a novel lower bound on h^{2,0}(X) based on the minimal rank in the kernel of a wedge product map, extending the Castelnuovo-de Franchis inequality to higher dimensions.

Findings

Provides a lower bound on h^{2,0}(X) in terms of kernel rank

Generalizes Castelnuovo-de Franchis inequality to higher dimensions

Improves previous bounds for threefolds and fourfolds

Abstract

We give a lower bound on the Hodge number h^{2,0}(X), where X is an irregular compact K\"ahler (or smooth complex projective) variety, in terms of the minimal rank of an element in the kernel of the wedge product map \psi_2: \Lambda^2 H^0(X,\Omega_X^1) -> H^0(X,\Omega_X^2). As a consequence, we obtain a generalization to higher dimensions of the Castelnuovo-de Franchis inequality for surfaces, improving some results of Lazarsfeld and Popa and Lombardi for threefolds and fourfolds.