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.

• 14F43 - Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
• 14L30 - Group actions on varieties or schemes (quotients)
• 14M25 - Toric varieties, Newton polyhedra, Okounkov bodies
• 53C26 - Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry
• 55N10 - Singular homology and cohomology theory