Symplectic small covers in dimension four

TL;DR

This paper investigates symplectic structures on four-dimensional small covers, establishing asphericity, classifying certain cases, and constructing examples with non-product orbit polytopes.

Contribution

It proves all symplectic four-dimensional small covers are aspherical and classifies those over products of two polygons, linking symplecticity to factor-compatibility.

Findings

All symplectic four-dimensional small covers are aspherical.

Symplecticity over product polygons is equivalent to factor-compatibility.

Constructed a symplectic small cover with a non-product orbit polytope.

Abstract

We study symplectic structures on four-dimensional small covers. Our main result shows that every symplectic four-dimensional small cover is aspherical. We then classify symplectic small covers over products of two polygons, proving that symplecticity is equivalent to factor-compatibility. We also classify them up to diffeomorphism. Finally, we construct a symplectic four-dimensional small cover whose orbit polytope is not combinatorially equivalent to a product of two polygons.