A hyperbolic $4$-orbifold with underlying space $\mathbb{P}^2$

TL;DR

This paper constructs the first known example of a closed hyperbolic 4-orbifold with underlying space being the complex projective plane, linking hyperbolic geometry and symplectic structures.

Contribution

It demonstrates that $ ext{P}^2$ can serve as the underlying space for a closed hyperbolic 4-orbifold, addressing a key open question.

Findings

First example of a closed hyperbolic 4-orbifold with symplectic underlying space

Establishes a connection between hyperbolic geometry and symplectic structures in 4-dimensions

Provides new insights into the topology of hyperbolic 4-orbifolds

Abstract

This paper shows that the complex projective plane $\mathbb{P}^2$ can be realized as the underlying space for a closed hyperbolic $4$-orbifold. This is the first example of a closed hyperbolic $4$-orbifold whose underlying space is symplectic, which is related to the open question as to whether or not closed hyperbolic $4$-manifolds can admit symplectic structures.