[Paper Review] Cohomological Hasse principle and resolution of quotient singularities

Moritz Kerz, Shuji Saito|arXiv (Cornell University)|Nov 30, 2011

Algebraic Geometry and Number Theory22 references13 citations

TL;DR

This paper establishes a McKay principle for the homotopy type of the dual complex of exceptional divisors in resolutions of quotient singularities by reducing the geometric problem to Kato's cohomological Hasse principle. It proves that the dual complex of the exceptional divisor in a resolution of a quotient singularity is homotopy equivalent to the quotient of the dual complex from the equivariant resolution, leading to the key result that the dual complex is contractible for isolated quotient singularities over fields of characteristic zero or perfect fields with resolution of singularities.

ABSTRACT

In this paper we study weight homology of singular schemes. Weight homology is an invariant of a singular scheme defined in terms of hypercoverings of resolution of singularities. Our main result is McKay principle for weight homology of quotient singularities, i.e. we describe weight homology of a quotient scheme in terms of weight homology of an equivariant scheme. Our method is to reduce the geometric McKay principle for weight homology to Kato's cohomological Hasse principle for arithmetic schemes. The McKay principle for weight homology implies McKay principle for the homotopy type of the dual complex of the exceptional divisors of a resolution of a quotient singularity. As a consequence we show that the dual complex is contractible for isolated quotient singularities.

Motivation & Objective

• To establish a McKay principle for the homotopy type of the dual complex of exceptional divisors in resolutions of quotient singularities.
• To relate the geometry of the resolution of a quotient singularity $X/G$ to the $G$-equivariant geometry of $X$ via weight homology.
• To show that the dual complex $\Gamma(E_S)$ of the exceptional divisor in a resolution of $X/G$ is contractible when the singular locus is isolated.
• To reduce the geometric McKay principle to Kato's cohomological Hasse principle for arithmetic schemes.
• To prove that the canonical map $\phi: \Gamma(E_T)/G \to \Gamma(E_S)$ induces isomorphisms on homology and fundamental groups, implying homotopy equivalence under certain conditions.

Proposed method

• Uses weight homology as an invariant of singular schemes defined via hypercoverings of resolutions of singularities.
• Applies Kato's cohomological Hasse principle to reduce the geometric McKay principle to arithmetic descent properties.
• Employs equivariant weight homology and descent techniques for homology functors in the context of $G$-schemes.
• Utilizes the canonical resolution of singularities in characteristic zero or perfect fields to ensure existence of smooth $G$-equivariant resolutions.
• Constructs a commutative diagram relating $\widetilde{X}/G$, $\widetilde{Y}$, and $\widetilde{Y}'$ to relate dual complexes via $cs$-coverings and homotopy equivalences.
• Applies Whitehead and Hurewicz theorems to deduce homotopy equivalence from isomorphisms on homology and fundamental groups.

Experimental results

Research questions

• RQ1 Does the dual complex of the exceptional divisor in a resolution of a quotient singularity $X/G$ have a homotopy type determined by the $G$-equivariant geometry of $X$?
• RQ2 Can the McKay principle for the homotopy type of dual complexes be established using cohomological descent?
• RQ3 Is the dual complex $\Gamma(E_S)$ contractible when the singular locus of $X/G$ is isolated?
• RQ4 Does the canonical map $\phi: \Gamma(E_T)/G \to \Gamma(E_S)$ induce isomorphisms on homology and fundamental groups?
• RQ5 Can the weight homology of the equivariant resolution $\widetilde{X}$ be used to compute the weight homology of the resolution $\widetilde{Y}$ of $X/G$?

Key findings

• The canonical map $\phi: \Gamma(E_T)/G \to \Gamma(E_S)$ induces isomorphisms on all homology groups $H_a$ for $a \in \mathbb{Z}$.
• The map $\phi$ also induces an isomorphism on fundamental groups $\pi_1$.
• If $\Gamma(E_T)/G$ is simply connected, then $\phi$ is a homotopy equivalence.
• If $\Gamma(E_T)/G$ is contractible, then $\Gamma(E_S)$ is contractible.
• When $T$ is smooth (e.g., when the singular locus is isolated), $\Gamma(E_S)$ is contractible.
• The result holds over fields of characteristic zero or perfect fields with canonical resolution of singularities, generalizing Stepanov's theorem to the equivariant setting.