# Orbifold cohomology of abelian symplectic reductions and the case of weighted projective spaces

## TL;DR

This paper explores the topology and orbifold cohomology of symplectic quotients, especially weighted projective spaces, demonstrating simplified computational methods and providing new explicit cohomology ring calculations.

## Contribution

It introduces direct computational techniques for orbifold cohomology of abelian symplectic reductions, specifically for weighted projective spaces, bypassing complex combinatorial tools.

## Key findings

- Computed Chen-Ruan orbifold cohomology rings for weighted projective spaces
- Demonstrated simplified methods for topological computations in symplectic quotients
- Compared different cohomology rings associated with weighted projective spaces

## Abstract

These notes accompany a lecture about the topology of symplectic (and other) quotients. The aim is two-fold: first to advertise the ease of computation in the symplectic category; and second to give an account of some new computations for weighted projective spaces. We start with a brief exposition of how orbifolds arise in the symplectic category, and discuss the techniques used to understand their topology. We then show how these results can be used to compute the Chen-Ruan orbifold cohomology ring of abelian symplectic reductions. We conclude by comparing the several rings associated to a weighted projective space. We make these computations directly, avoiding any mention of a stacky fan or of a labeled moment polytope.