[Paper Review] Symplectic Resolutions for Quotient Singularities

Baohua Fu|ArXiv.org|Jun 27, 2002

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper establishes that if the cotangent bundle of a smooth complex variety $X$, denoted $T^*X$, is quotiented by a finite group $G \subset \mathrm{Aut}(X)$ preserving the symplectic form, and the resulting quotient $T^*X/G$ admits a symplectic resolution, then $X/G$ must be smooth and the resolution contains an open subset isomorphic to $T^*(X/G)$. The key contribution is a McKay-type correspondence for Hodge numbers of such resolutions when $X = \mathbb{P}^n$ and $G \subset SL(n+1,\mathbb{C})$, expressed via eigenvalue multiplicities of group elements.

ABSTRACT

We give some necessary conditions for the existence of a symplectic resolution for quotient singularities. The McKay correspondence is also worked out for these resolutions.

Motivation & Objective

• To determine necessary conditions for the existence of symplectic resolutions of quotient symplectic singularities $T^*X/G$.
• To establish a geometric structure of the resolution $Z$ when $T^*X/G$ admits a symplectic resolution.
• To develop a McKay correspondence-type formula for the Hodge numbers of the resolution $Z$ in the case $X = \mathbb{P}^n$ and $G \subset SL(n+1,\mathbb{C})$.
• To derive topological invariants of $Z$, such as the Euler characteristic, from group-theoretic data of $G$.

Proposed method

• Use the ${\mathbb{C}}^*$-action on $T^*X/G$ and its lift to the resolution $Z$ to analyze the symplectic form's weight and apply crepant resolution techniques.
• Apply results from Kaledin and Fu on symplectic resolutions and ${\mathbb{Q}}$-factoriality to constrain the fixed locus $\bigcup_{g \neq 1} \mathrm{Fix}(g)$.
• Employ orbit E-functions and E-polynomials to compute Hodge-theoretic invariants of the resolution $Z$.
• Relate the Hodge polynomial of $Z$ to the conjugacy classes of $G$ and the eigenvalue multiplicities $k_i(g)$ of elements $g \in G \subset SL(n+1,\mathbb{C})$.
• Use the fibration structure of fixed point sets $X^g$ and their quotients by centralizers to compute E-polynomials.
• Apply Poincaré duality and change of variable to derive the generating function for Betti numbers of $Hilb^n(T^*\Sigma)$.

Experimental results

Research questions

• RQ1 Under what conditions does $T^*X/G$ admit a symplectic resolution when $G$ acts symplectically on $T^*X$?
• RQ2 What geometric structure must the resolution $Z$ of $T^*X/G$ possess if such a resolution exists?
• RQ3 Can a McKay correspondence-type formula be established for the Hodge numbers of $Z$ in the case $X = \mathbb{P}^n$ and $G \subset SL(n+1,\mathbb{C})$?
• RQ4 How do the Betti numbers and Euler characteristic of $Z$ relate to the conjugacy classes and eigenvalue data of $G$?
• RQ5 What is the cohomology of the Hilbert scheme $Hilb^n(T^*\Sigma)$ for a smooth projective curve $\Sigma$?

Key findings

• If $T^*X/G$ admits a symplectic resolution, then $X/G$ is smooth and the resolution $Z$ contains an open subset isomorphic to $T^*(X/G)$.
• For $X = \mathbb{P}^n$ and $G \subset SL(n+1,\mathbb{C})$, the Hodge polynomial of the resolution $Z$ is given by $\sum_{p,q}(-1)^{p+q}h^{p,q}(Z)u^p v^q = (uv)^n \sum_{\{g\}} \sum_i \frac{(uv)^{k_i(g)} - 1}{uv - 1}$, where $k_i(g)$ are eigenvalue multiplicities of $g \in G$.
• The Euler characteristic of $Z$ is $e(Z) = (n+1) c(G)$, where $c(G)$ is the number of conjugacy classes of $G$ in $\mathrm{Aut}(\mathbb{P}^n)$.
• The cohomology $H^{2j+1}(Z,\mathbb{C}) = 0$ for all $j$, and $H^{2i}(Z,\mathbb{C})$ has only $(i,i)$-type Hodge components for all $i$, with $H^{2i}(Z,\mathbb{C}) = 0$ for $i < n$.
• The generating function for the Betti numbers of $Hilb^n(T^*\Sigma)$ is $\sum_{n=0}^\infty P_t(Hilb^n(T^*\Sigma)) q^n = \prod_{d=1}^\infty \frac{(1 + t^{2d-1} q^d)^{b_1(\Sigma)}}{(1 - t^{2d-2} q^d)^{b_0(\Sigma)} (1 - t^{2d} q^d)^{b_2(\Sigma)}}$, recovering Göttsche's formula.
• The resolution $Z$ is simply connected and rational when $X = \mathbb{C}^n$ and $G$ acts symplectically.