Classification of minimal symplectic Calabi–Yau surfaces with positive first Betti number

Determine whether every minimal symplectic Calabi–Yau surface with positive first Betti number is diffeomorphic to a torus bundle over a torus.

Background

The paper discusses the classification of minimal symplectic Calabi–Yau surfaces (SCYs). Such surfaces are known to have the rational homology type of a K3 surface, an Enriques surface, or a torus bundle over a torus. The author notes a folklore conjecture asserting that the positive-first-Betti-number case is exhausted, up to diffeomorphism, by torus bundles over tori.

The constructions in the paper produce genus-3 Lefschetz pencils whose total spaces are SCYs homeomorphic to torus bundles over tori, including examples without sections. These examples provide evidence relevant to the stated classification conjecture but do not establish it in full generality.