Complex line fields on almost-complex manifolds

TL;DR

This paper investigates the existence of complex line fields on almost-complex manifolds, establishing conditions under which such fields exist, and extends results to broader classes of complex bundles over CW complexes.

Contribution

It proves that vanishing virtual Chern classes are both necessary and sufficient for the existence of multiple complex line fields on certain manifolds and complex bundles.

Findings

Vanishing virtual Chern classes are necessary for complex line fields.

Vanishing virtual Chern classes are sufficient for 1-3 line fields on certain manifolds.

Results extend to complex bundles over CW complexes.

Abstract

We study linearly independent complex line fields on almost-complex manifolds, which is a topic of long-standing interest in differential topology and complex geometry. A necessary condition for the existence of such fields is the vanishing of appropriate virtual Chern classes. We prove that this condition is also sufficient for the existence of one, two, or three linearly independent complex line fields over certain manifolds. More generally, our results hold for a wider class of complex bundles over CW complexes.