Symplectic and projective small covers over products of polygons

TL;DR

This paper investigates symplectic and projective structures on small covers over products of polygons, introducing a new class and linking cohomology to geometric models.

Contribution

It introduces the factor-compatible class for small covers over polygons and establishes their smooth projective models and cohomological properties.

Findings

Every factor-compatible small cover admits a smooth projective model.

The graded mod 2 cohomology ring determines the Hodge diamond.

All such small covers admit an iterated equivariant bundle structure.

Abstract

We study symplectic and projective structures on small covers over products of polygons. We introduce the factor-compatible class for small covers over products of polygons and prove that every factor-compatible small cover admits a smooth projective model as a finite quotient of a product of curves. Furthermore, we show that the graded mod~$2$ cohomology ring determines the Hodge diamond of the associated projective model. We also prove that every factor-compatible small cover admits an iterated equivariant bundle structure.