[Paper Review] Homology of Hilbert schemes of points on a locally planar curve

Jørgen Vold Rennemo|arXiv (Cornell University)|Aug 19, 2013

Algebraic Geometry and Number Theory15 references3 citations

TL;DR

This paper establishes a Weyl algebra action on the rational homology of Hilbert schemes of points on a locally planar curve, using creation/annihilation operators associated with a fixed point and the curve itself. It proves that the homology of the Hilbert scheme $C^{[n]}$ decomposes in terms of the compactified Jacobian $J$ and a new $D$-grading, recovering and refining recent cohomological formulas by Maulik–Yun and Migliorini–Shende.

ABSTRACT

Let C be a proper, integral, locally planar curve, and consider its Hilbert schemes of points C^[n]. We define 4 creation/annihilation operators acting on the rational homology groups of these Hilbert schemes and show that the operators satisfy the relations of a Weyl algebra. The action of this algebra is similar to that defined by Grojnowski and Nakajima for a smooth surface. As a corollary, we compute the cohomology of C^[n] in terms of the cohomology of the compactified Jacobian of C together with an auxiliary grading on the latter. This recovers and slightly strenghtens a formula recently obtained in a different way by Maulik and Yun and independently Migliorini and Shende.

Motivation & Objective

• To define and study creation/annihilation operators on the rational homology of Hilbert schemes $C^{[n]}$ for a proper, integral, locally planar curve $C$.
• To show that these operators satisfy the commutation relations of a Weyl algebra, generalizing Nakajima–Grojnowski constructions to singular curves.
• To express the homology of $C^{[n]}$ in terms of the compactified Jacobian $J$ and a new bigrading $D_m H_*(J)$, extending known results for $n \geq 2g-1$ to all $n$.
• To provide a new algebraic framework for computing $H_*(C^{[n]})$ via a Fock space construction from the kernel of annihilation operators.

Proposed method

• Define two pairs of operators: $\mu_{\pm}[\mathrm{pt}]$ for adding/removing a fixed nonsingular point $x$, and $\mu_{\pm}[C]$ for flag Hilbert schemes of pairs $(Z, Z')$ with $Z \subset Z'$.
• Use Gysin maps $p^!, q^!$ on the flag Hilbert scheme $C^{[n,n+1]}$ to define the operators, relying on the local planarity of $C$ to ensure well-definedness despite singularities.
• Construct a smooth family $\mathcal{C} \to B$ of curves over a smooth base $B$ such that the relative Hilbert schemes $\mathcal{C}^{[n]}$ are nonsingular, enabling standard correspondence calculus.
• Prove the Weyl algebra commutation relations by computing in the smooth family and restricting to the fiber $C$, using the fact that the operators lift to the family.
• Define the kernel $W = \ker \mu_-[\mathrm{pt}] \cap \ker \mu_-[C]$ and show that $V(C) \cong W \otimes \mathbb{Q}[\mu_+[\mathrm{pt}], \mu_+[C]]$, establishing a Fock space structure.
• Use the Abel–Jacobi map $AJ_*: V(C) \to H_*(J)$ to identify $W \cong H_*(J)$, inducing a bigrading $D_m H_i(J)$ on the compactified Jacobian.

Experimental results

Research questions

• RQ1 How can the homology of Hilbert schemes $C^{[n]}$ for a singular curve $C$ be described algebraically when $n < 2g-1$?
• RQ2 Can a Weyl algebra action be defined on the homology of $C^{[n]}$ analogous to Nakajima–Grojnowski's construction on smooth surfaces?
• RQ3 What is the precise relationship between $H_*(C^{[n]})$ and the cohomology of the compactified Jacobian $J$ beyond the known bundle structure for large $n$?
• RQ4 How does the new $D$-grading on $H_*(J)$ refine or recover existing formulas for $H_*(C^{[n]})$?
• RQ5 Can the cohomological structure of $C^{[n]}$ be reconstructed from the $Q$-filtration and $D$-grading on $H^*(J)$?

Key findings

• The four operators $\mu_{\pm}[\mathrm{pt}], \mu_{\pm}[C]$ on $V(C) = \bigoplus_n H_*(C^{[n]})$ satisfy the Weyl algebra relations $[\mu_-[\mathrm{pt}], \mu_+[C]] = [\mu_-[C], \mu_+[\mathrm{pt}]] = \mathrm{id}$, with all other pairs commuting.
• The homology $H_*(C^{[n]})$ is isomorphic to $\bigoplus_{m \leq n} D_m H_*(J) \otimes \mathrm{Sym}^{n-m}(\mathbb{Q} \oplus \mathbb{Q}[2])$, where $D_m H_*(J)$ is a new bigrading on the compactified Jacobian.
• The kernel $W = \ker \mu_-[\mathrm{pt}] \cap \ker \mu_-[C]$ is isomorphic to $H_*(J)$, and $V(C)$ is freely generated by $W$ and the creation operators.
• The $D$-grading on $H_*(J)$ is induced via the Abel–Jacobi map, and satisfies $D_m H_i(J) = 0$ unless $0 \leq m \leq 2g$, with $g$ the arithmetic genus of $C$.
• The cohomological version of the result shows that the $Q$-filtration on $H^*(C^{[n]})$ is preserved by the operators, and the generating functions for $H^*(C^{[n]})$ and $H^*(J)$ satisfy $F_J = F_W$, implying $D_{\leq i+j} H^i(J) = Q_{\leq j} H^i(J)$.
• The result recovers and strengthens the formula of Maulik–Yun and Migliorini–Shende by providing an algebraic, operator-theoretic framework and a new grading on $H_*(J)$.