Homotopic properties of K\"ahler orbifolds

TL;DR

This paper extends classical K"ahler geometry results to orbifolds, proving formality and Betti number properties, and explores implications for symplectic and Sasakian orbifolds, including examples of non-K"ahler structures.

Contribution

It adapts proofs of formality and Betti number properties from K"ahler manifolds to orbifolds, and constructs new examples of non-formal Sasakian manifolds.

Findings

K"ahler orbifolds are formal with even Betti numbers in odd degrees

Existence of symplectic orbifolds without K"ahler structures

Construction of a non-formal quasi-regular Sasakian manifold

Abstract

We prove the formality and the evenness of odd-degree Betti numbers for compact K\"ahler orbifolds, by adapting the classical proofs for K\"ahler manifolds. As a consequence, we obtain examples of symplectic orbifolds not admitting any K\"ahler orbifold structure. We also review the known examples of non-formal simply connected Sasakian manifolds, and produce an example of a non-formal quasi-regular Sasakian manifold with Betti numbers $b_1=0$ and $b_2\,> 1$.