Perverse Bundles and Calogero-Moser Spaces

TL;DR

This paper describes moduli spaces of D-bundles on smooth complex curves, linking them to twisted cotangent bundles and perverse vector bundles, extending known results about Calogero-Moser spaces.

Contribution

It generalizes the identification of ideal spaces in the Weyl algebra with Calogero-Moser varieties to moduli of D-bundles on arbitrary curves.

Findings

Moduli of D-bundles form twisted cotangent bundles to stacks of torsion sheaves.

Untwisted cotangent bundles correspond to moduli of perverse vector bundles on T^*X.

In the rank one case, these include Hilbert schemes of points on T^*X.

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.