[Paper Review] Perverse Bundles and Calogero-Moser Spaces

Dani Ben‐Zvi, Thomas Nevins|ArXiv.org|Oct 3, 2006

Advanced Algebra and Geometry27 references3 citations

TL;DR

This paper establishes a geometric correspondence between moduli spaces of torsion-free ${\mathcal{D}}$-bundles on smooth complex curves and twisted cotangent bundles over moduli of torsion sheaves, generalizing the Calogero-Moser quiver variety description of ideals in the Weyl algebra. Using a noncommutative Beilinson transform and Koszul duality, it identifies the $n$-particle rational Calogero-Moser space on a curve $X$ as the moduli space of trivially framed ${\mathcal{D}}$-line bundles with second Chern class $n$, resolving a question of Ginzburg.

ABSTRACT

We present a simple description of moduli spaces of torsion-free D-modules (``D-bundles'') on general smooth complex curves X, generalizing the identification of the space of ideals in the Weyl algebra with Calogero-Moser quiver varieties. Namely, we show that the moduli of D-bundles form twisted cotangent bundles to stacks of torsion sheaves on X, answering a question of Ginzburg. The corresponding (untwisted) cotangent bundles are identified with moduli of ``perverse vector bundles'' on T^*X, which contain as open subsets the moduli of framed torsion-free sheaves (the Hilbert schemes (T^*X)^[n] in the rank one case). The proof is based on the description of the derived category of D-modules on X by a noncommutative version of the Beilinson transform on the projective line.

Motivation & Objective

• To generalize the identification of Hilbert schemes and Calogero-Moser spaces on $\mathbb{C}^2$ to higher-genus curves.
• To resolve Ginzburg's question on the higher-genus analog of the Berest-Wilson theorem for ideals in the Weyl algebra.
• To describe moduli spaces of torsion-free ${\mathcal{D}}$-modules on smooth curves using derived categories and noncommutative geometry.
• To establish a correspondence between ${\mathcal{D}}$-bundles and quiver data satisfying the Calogero-Moser relation via homotopy classes of Koszul complexes.

Proposed method

• Uses a noncommutative version of the Beilinson transform on $\mathbb{P}^1$ to describe the derived category of ${\mathcal{D}}$-modules on a curve $X$.
• Constructs Koszul data from ${\mathcal{D}}$-bundle structures via complexes of $\mathcal{D}^1$-modules and $\mathbb{C}[x]$-modules.
• Applies homotopy theory to show that each ${\mathcal{D}}$-bundle corresponds to a unique quadruple $(X,Y,i,j)$ satisfying $[X,Y] + ij = I$, the Calogero-Moser relation.
• Identifies the moduli of ${\mathcal{D}}$-bundles as twisted cotangent bundles over moduli of torsion sheaves via Hamiltonian reduction.
• Uses perverse vector bundles on the compactified cotangent bundle $S = \mathbb{P}(T_X \oplus \mathcal{O}_X)$ to realize the moduli of framed torsion-free sheaves.
• Demonstrates that the $n$th Calogero-Moser space $\mathsf{CM}_n(X)$ is the moduli space of trivially framed ${\mathcal{D}}$-line bundles with second Chern class $n$.

Experimental results

Research questions

• RQ1 How can the moduli space of ideals in the Weyl algebra be generalized to higher-genus curves?
• RQ2 What is the geometric structure of the moduli space of torsion-free ${\mathcal{D}}$-modules on a smooth curve $X$?
• RQ3 How does the Calogero-Moser space $\mathsf{CM}_n(X)$ on a curve $X$ relate to the geometry of ${\mathcal{D}}$-bundles?
• RQ4 Can the noncommutative Beilinson transform be used to describe ${\mathcal{D}}$-modules on curves via quiver-like data?
• RQ5 What is the precise correspondence between $\mathsf{CM}_n(X)$ and framed ${\mathcal{D}}$-line bundles with second Chern class $n$?

Key findings

• The moduli space $\mathsf{CM}_n(X)$ of the $n$-particle rational Calogero-Moser system on a smooth curve $X$ is identified as the moduli space of trivially framed ${\mathcal{D}}$-line bundles with second Chern class $n$, resolving Ginzburg's question.
• The moduli of torsion-free ${\mathcal{D}}$-bundles on $X$ are shown to form a twisted cotangent bundle over the moduli space of torsion sheaves of length $n$, generalizing the classical $\mathbb{C}^2$ case.
• The space $\mathsf{CM}_n(X)$ is realized as the Hamiltonian reduction of $T^*Q_n(X)$ at the identity, where $Q_n(X)$ parametrizes pairs $(\mathcal{O}_X^n \to Q, i \in H^0(Q))$ with $Q$ a torsion sheaf of length $n$.
• A Koszul duality construction provides a one-to-one correspondence between ${\mathcal{D}}$-bundles and quadruples $(X,Y,i,j)$ satisfying the Calogero-Moser relation $[X,Y] + ij = I$, up to homotopy.
• Each homotopy class of Koszul data corresponds to a unique CM quadruple, and the correspondence is bijective after quotienting by homotopy, proving the classification.
• The untwisted cotangent bundles over the moduli of torsion sheaves are identified with moduli of perverse vector bundles on $\mathbb{P}(T_X \oplus \mathcal{O}_X)$, with the Hilbert scheme $T^*X^{[n]}$ as an open subset in the rank-one case.