An observation on the existence of stable generalized complex structures on ruled surfaces

TL;DR

This paper investigates the conditions under which stable generalized complex structures exist on ruled surfaces, specifically showing that on sphere bundles over surfaces with genus at least two, such structures must be of constant type.

Contribution

It establishes a necessary condition for stable generalized complex structures on certain ruled surfaces, advancing understanding of their geometric properties.

Findings

Stable generalized complex structures on sphere bundles over high-genus surfaces are of constant type.

The result constrains the possible structures on these ruled surfaces.

Provides insight into the geometric classification of generalized complex structures.

Abstract

We point out that any stable generalized complex structure on a sphere bundle over a closed surface of genus at least two must be of constant type.