# Integral cohomology of quotients via toric geometry

## TL;DR

This paper computes the integral cohomology of quotients of complex manifolds by cyclic groups, using toric geometry techniques, and applies these results to specific cases like Hilbert schemes of K3 surfaces with symplectic automorphisms.

## Contribution

It introduces a method to determine the integral cohomology of quotients via toric blow-ups and spectral sequence analysis, with explicit computations for special geometric cases.

## Key findings

- Integral cohomology of quotients is characterized using toric geometry.
- Spectral sequence for equivariant cohomology degenerates under certain conditions.
- Explicit Beauville--Bogomolov form calculations for specific automorphisms.

## Abstract

We describe the integral cohomology of $X/G$ where $X$ is a compact complex manifold and $G$ a cyclic group of prime order with only isolated fixed points. As a preliminary step, we investigate the integral cohomology of toric blow-ups of quotients of $\mathbb{C}^n$. We also provide necessary and sufficient conditions for the spectral sequence of equivariant cohomology of $(X,G)$ to degenerate at the second page. As an application, we compute the Beauville--Bogomolov form of $X/G$ when $X$ is a Hilbert scheme of points on a K3 surface and $G$ a symplectic automorphism group of orders 5 or 7.