Strong generic vanishing and a higher dimensional Castelnuovo-de Franchis inequality

TL;DR

This paper generalizes the Castelnuovo-de Franchis inequality to higher-dimensional manifolds using Generic Vanishing theory, higher regularity, and algebraic syzygy results, with implications for Kähler manifolds and Fourier-Mukai transforms.

Contribution

It extends classical inequalities to higher dimensions and provides new insights into vanishing theorems and Fourier-Mukai transforms in complex geometry.

Findings

Generalization of Castelnuovo-de Franchis inequality to arbitrary dimension

Positive answer to Green-Lazarsfeld question on higher direct images

Connections established between Generic Vanishing and Fourier-Mukai transforms

Abstract

We extend to manifolds of arbitrary dimension the Castelnuovo-de Franchis inequality for surfaces. The proof is based on the theory of Generic Vanishing and higher regularity, and on the Evans-Griffith Syzygy Theorem in commutative algebra. Along the way we give a positive answer, in the setting of K\"ahler manifolds, to a question of Green-Lazarsfeld on the vanishing of higher direct images of Poincar\'e bundles. We indicate generalizations to arbitrary Fourier-Mukai transforms.