Asymptotically holomorphic theory for symplectic orbifolds

TL;DR

This paper extends asymptotically holomorphic techniques to symplectic orbifolds, constructing symplectic suborbifolds and deriving topological properties similar to smooth cases.

Contribution

It generalizes Donaldson's methods to symplectic orbifolds, establishing existence of symplectic suborbifolds and Lefschetz-type theorems in this setting.

Findings

Existence of symplectic suborbifolds via asymptotically holomorphic sections

Lefschetz hyperplane theorem for orbifolds

Hard Lefschetz and formality properties for suborbifolds

Abstract

We extend Donaldson's asymptotically holomorphic techniques to symplectic orbifolds. More precisely, given a symplectic orbifold such that the symplectic form defines an integer cohomology class, we prove that there exist sections of large tensor powers of the prequantizable line bundle such that their zero sets are symplectic suborbifolds. We then derive a Lefschetz hyperplane theorem for these suborbifolds, that computes their real cohomology up to middle dimension. We also get the hard Lefschetz and formality properties for them, when the ambient manifold satisfies those properties.