Integral cohomology of quotients via toric geometry

TL;DR

The paper addresses the challenge of determining the integral cohomology of quotients $X/G$ where $G$ is a cyclic group of prime order acting with isolated fixed points. It develops a framework combining toric blow-ups and Boissière–Nieper-Wisskirchen–Sarti invariants to study the second-page degeneration of the equivariant cohomology spectral sequence, enabling explicit computations of $H^*(X/G)$ and its lattice structure. The main contributions include a criterion for degeneration at page 2, a description of the cohomology in terms of exceptional lattices, and explicit Beauville–Bogomolov forms for singular primitive symplectic quotients arising from Hilbert schemes of points on K3 surfaces under order-5 and order-7 symplectic automorphisms. The results extend to complete intersections and provide versatile tools for analyzing torsion, discriminants, and fixed-point data in a broad class of quotients, with significant implications for the study of singular irreducible symplectic spaces.

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.

Theorems & Definitions (148)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Lemma 2.2: cf. SmithTh
• Proposition 2.4
• Theorem 2.5
• proof
• Lemma 2.7
• proof